improve speed of z->z²+c iteration and also parameterize z
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768028d45f
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07964396e0
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1BBA026E13FC0495
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@ -13,9 +13,8 @@ find_package(LibProf 2.4 REQUIRED)
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set(SOURCES
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src/main.cpp
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src/mandelbrot.s
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src/iteration.S
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src/mandelbrotshader.cpp
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src/julia.s
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src/juliashader.cpp
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src/utilities.cpp
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# ...
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@ -0,0 +1,101 @@
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/* Iteration of a quadratic function z -> z²+c on complex inputs. This version
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is by Lephe (2023) and fairly optimized.
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Notes for optimizers:
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- The entire loop is an obvious EX chain, the only stuff that parallelizes
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are the sts and the branches.
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- I gained 15% speed from the original by getting as many sts to parallelize
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as possible, which required tightening up dmuls.l -> delay -> sts chains.
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- The code is strongly affected by alignment, is somehow very sensitive to
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the initialization code, and does not behave as I expect it to in most
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ways. Probably none of it executes as I envision. */
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.global _quadratic_iteration_32
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.text
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.balign 4
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#define _zx r0 /* Re(z) */
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#define _zy r1 /* Im(z) */
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#define _zx2 r2 /* _zx² */
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#define _zy2 r3 /* _zy² */
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#define _cx r4 /* Re(c) */
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#define _cy r5 /* Im(c) */
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_quadratic_iteration_32:
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/* Initialize zx and zy from arguments, r6/r7 from stack. Compute the
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initial values of zx² and zy². Start the loop pipeline by preparing
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the multiplication zx⋅zy. */
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mov.l r8, @-r15
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dmuls.l r6, r6
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mov.l r9, @-r15
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mov r6, _zx
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sts mach, r8
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mov r7, _zy
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sts macl, _zx2
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dmuls.l r7, r7
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mov.l @(8,r15), r6
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xtrct r8, _zx2
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sts mach, r9
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nop
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sts macl, _zy2
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dmuls.l _zx, _zy
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mov.l @(12,r15), r7
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xtrct r9, _zy2
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.loop:
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/* We want to compute z²+c:
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(zx + i·zy)(zx + i·zy) + (cx + i⋅cy)
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= (zx² - zy² + cx) + i(2⋅zx⋅zy + cy)
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Due to pipelining, zx * zy is currently being computed in MAC. */
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/* Update zx while retrieving zx⋅zy and preparing the next zx² */
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sts mach, r8
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sub _zy2, _zx2
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sts macl, _zy
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add _cx, _zx2
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mov _zx2, _zx
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dmuls.l _zx, _zx
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/* Update zy and zx² while preparing the next zy² */
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xtrct r8, _zy
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sts mach, r8
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shll _zy
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sts macl, _zx2
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add _cy, _zy
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dmuls.l r1, r1
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xtrct r8, r2
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/* Update zy² and compute |z|² = zx² + zy² in 32:0 format. Compare |z|²
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to t² and return if the threshold is reached. */
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sts mach, r9
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sts macl, _zy2
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add r9, r8
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cmp/ge r6, r8
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bt.s .end
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dt r7
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/* Start zx⋅zy early for next frame. */
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dmuls.l _zx, _zy
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/* Continue looping */
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bf.s .loop
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xtrct r9, _zy2
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.end:
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mov.l @r15+, r9
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mov.l @r15+, r8
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rts
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mov r7, r0
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@ -0,0 +1,28 @@
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#ifndef ITERATION_H
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#define ITERATION_H
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#ifdef __cplusplus
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extern "C" {
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#endif
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/* Quadratic iteration. This function iterates z -> z²+c for z, c complex
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inputs, until a fixed number of steps is reached or |z| becomes larger than
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a predefined threshold t. Computation is carried in 16:16 fixed-point.
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@Re_c @Im_c Constant c [num32]
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@Re_z @Im_z Initial value of z [num32]
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@t_squared Squared threshold t² [num32]
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@steps Maximum number of steps [int]
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Returns the number of steps remaining at the time the threshold was crossed,
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or 0 if it was never reached. */
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int quadratic_iteration_32(
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int32_t Re_c, int32_t Im_c,
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int32_t Re_z, int32_t Im_z,
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uint32_t t_squared, int steps);
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#ifdef __cplusplus
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}
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#endif
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#endif /* ITERATION_H */
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31
src/julia.s
31
src/julia.s
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@ -1,31 +0,0 @@
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/*
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int julia(int32_t cr, int32_t ci, uint32_t t_squared, int steps, int32_t zr, int32_t zi);
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(Pour moi : t_squared = (2 * 2) << 16, steps = 50)
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Si renvoie 0 -> dans la fractale
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Sinon renvoie un nombre entre 1...steps indiquant la "distance" (pour le dégradé)
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*/
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.global _julia
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.text
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/* Iteration of z = z²+c with threshold checks
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@r4 Real part of c, cr
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@r5 Imaginary part of c, ci
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@r6 Squared threshold t²
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@r7 Number of steps
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If c is in the set, returns 0. Otherwise, returns the remaining number of
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iterations to be checked before the process finised. */
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_julia:
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mov.l r8, @-r15
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mov.l r9, @-r15
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mov.l r10, @-r15
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mov.l r11, @-r15
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.loop:
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.end:
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rts
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nop
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@ -1,74 +0,0 @@
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/*
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int mandelbrot(int32_t cr, int32_t ci, uint32_t t_squared, int steps);
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(Pour moi : t_squared = (2 * 2) << 16, steps = 50)
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Si renvoie 0 -> dans la fractale
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Sinon renvoie un nombre entre 1...steps indiquant la "distance" (pour le dégradé)
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*/
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.global _mandelbrot
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.text
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/* Iteration of z = z²+c with threshold checks
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@r4 Real part of c, cr
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@r5 Imaginary part of c, ci
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@r6 Squared threshold t²
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@r7 Number of steps
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If c is in the set, returns 0. Otherwise, returns the remaining number of
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iterations to be checked before the process finised. */
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_mandelbrot:
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mov.l r8, @-r15
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mov.l r9, @-r15
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/* Start with z=z²=0 (z=x+iy) */
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mov #0, r0
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mov #0, r1
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mov #0, r2
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mov #0, r3
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.loop:
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/* Start xy */
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dmuls.l r0, r1
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/* Update x = x²-y²+cr */
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mov r2, r0
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sub r3, r0
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add r4, r0
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/* Get y = 2xy+ci while starting x²
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TODO: Hope there's no overflow in that [shll r1]. It might be
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TODO: provable because 2xy overflowing probably implies a lower
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TODO: bound on x²+y² at the previous iteration. */
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sts mach, r8
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sts macl, r1
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dmuls.l r0, r0
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xtrct r8, r1
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shll r1
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add r5, r1
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/* Get x² and keep the integer part at full precision for ∥z∥²; at the
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same time, start y² */
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sts mach, r8
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sts macl, r2
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dmuls.l r1, r1
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xtrct r8, r2
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/* Get y² and compute ∥z∥² = x²+y² in 32:0 format */
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sts mach, r9
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sts macl, r3
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add r9, r8
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xtrct r9, r3
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/* Compare ∥z∥² to t², return if the threshold is reached */
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cmp/ge r6, r8
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bt .end
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/* Continue looping */
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dt r7
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bf .loop
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.end:
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mov.l @r15+, r9
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mov.l @r15+, r8
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rts
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mov r7, r0
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@ -1,5 +1,6 @@
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#include <azur/gint/render.h>
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#include "fractalshaders.h"
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#include "iteration.h"
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#include <cstdlib>
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#include <cstdio>
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@ -110,7 +111,7 @@ void azrp_shader_mandelbrot( void *uniforms, void *comnd, void *fragment )
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zr = zrsave;
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zi = zisave;
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u = mandelbrot( zr.v, zi.v, (2*2)<<16, maxiter );
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u = quadratic_iteration_32( zr.v, zi.v, 0, 0, (2*2)<<16, maxiter );
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if (u==0) frag[offset + i] = C_RGB(0,0,0);
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else
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