UniformSampleCone 2

Percentage Accurate: 99.0% → 99.1%

Time: 27.8s

Alternatives: 16

Speedup: 1.2×

• unsoundness detected in the egraph

Specification

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\begin{array}{l} t_0 := \left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\\ t_1 := \sqrt{1 - t\_0 \cdot t\_0}\\ t_2 := \left(uy \cdot 2\right) \cdot \pi\\ \left(\left(\cos t\_2 \cdot t\_1\right) \cdot xi + \left(\sin t\_2 \cdot t\_1\right) \cdot yi\right) + t\_0 \cdot zi \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux))
       (t_1 (sqrt (- 1.0 (* t_0 t_0))))
       (t_2 (* (* uy 2.0) PI)))
  (+
   (+ (* (* (cos t_2) t_1) xi) (* (* (sin t_2) t_1) yi))
   (* t_0 zi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((1.0f - ux) * maxCos) * ux;
	float t_1 = sqrtf((1.0f - (t_0 * t_0)));
	float t_2 = (uy * 2.0f) * ((float) M_PI);
	return (((cosf(t_2) * t_1) * xi) + ((sinf(t_2) * t_1) * yi)) + (t_0 * zi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux)
	t_1 = sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))
	t_2 = Float32(Float32(uy * Float32(2.0)) * Float32(pi))
	return Float32(Float32(Float32(Float32(cos(t_2) * t_1) * xi) + Float32(Float32(sin(t_2) * t_1) * yi)) + Float32(t_0 * zi))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = ((single(1.0) - ux) * maxCos) * ux;
	t_1 = sqrt((single(1.0) - (t_0 * t_0)));
	t_2 = (uy * single(2.0)) * single(pi);
	tmp = (((cos(t_2) * t_1) * xi) + ((sin(t_2) * t_1) * yi)) + (t_0 * zi);
end

Initial Program: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\\ t_1 := \sqrt{1 - t\_0 \cdot t\_0}\\ t_2 := \left(uy \cdot 2\right) \cdot \pi\\ \left(\left(\cos t\_2 \cdot t\_1\right) \cdot xi + \left(\sin t\_2 \cdot t\_1\right) \cdot yi\right) + t\_0 \cdot zi \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux))
       (t_1 (sqrt (- 1.0 (* t_0 t_0))))
       (t_2 (* (* uy 2.0) PI)))
  (+
   (+ (* (* (cos t_2) t_1) xi) (* (* (sin t_2) t_1) yi))
   (* t_0 zi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((1.0f - ux) * maxCos) * ux;
	float t_1 = sqrtf((1.0f - (t_0 * t_0)));
	float t_2 = (uy * 2.0f) * ((float) M_PI);
	return (((cosf(t_2) * t_1) * xi) + ((sinf(t_2) * t_1) * yi)) + (t_0 * zi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux)
	t_1 = sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))
	t_2 = Float32(Float32(uy * Float32(2.0)) * Float32(pi))
	return Float32(Float32(Float32(Float32(cos(t_2) * t_1) * xi) + Float32(Float32(sin(t_2) * t_1) * yi)) + Float32(t_0 * zi))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = ((single(1.0) - ux) * maxCos) * ux;
	t_1 = sqrt((single(1.0) - (t_0 * t_0)));
	t_2 = (uy * single(2.0)) * single(pi);
	tmp = (((cos(t_2) * t_1) * xi) + ((sin(t_2) * t_1) * yi)) + (t_0 * zi);
end

Alternative 1: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\\ t_1 := \sqrt{1 - t\_0 \cdot t\_0}\\ \left(\left(\sin \left(\pi \cdot \mathsf{fma}\left(-2, uy, 0.5\right)\right) \cdot t\_1\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot t\_1\right) \cdot yi\right) + t\_0 \cdot zi \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux))
       (t_1 (sqrt (- 1.0 (* t_0 t_0)))))
  (+
   (+
    (* (* (sin (* PI (fma -2.0 uy 0.5))) t_1) xi)
    (* (* (sin (* (* uy 2.0) PI)) t_1) yi))
   (* t_0 zi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((1.0f - ux) * maxCos) * ux;
	float t_1 = sqrtf((1.0f - (t_0 * t_0)));
	return (((sinf((((float) M_PI) * fmaf(-2.0f, uy, 0.5f))) * t_1) * xi) + ((sinf(((uy * 2.0f) * ((float) M_PI))) * t_1) * yi)) + (t_0 * zi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux)
	t_1 = sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))
	return Float32(Float32(Float32(Float32(sin(Float32(Float32(pi) * fma(Float32(-2.0), uy, Float32(0.5)))) * t_1) * xi) + Float32(Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * t_1) * yi)) + Float32(t_0 * zi))
end

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Step-by-step derivation

   1. lift-cos.f32 N/A

      \[\leadsto \left(\left(\color{blue}{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   2. cos-neg-rev N/A

      \[\leadsto \left(\left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(uy \cdot 2\right) \cdot \pi\right)\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   3. sin-+PI/2-rev N/A

      \[\leadsto \left(\left(\color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(uy \cdot 2\right) \cdot \pi\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   4. lower-sin.f32 N/A

      \[\leadsto \left(\left(\color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(uy \cdot 2\right) \cdot \pi\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   5. lift-*.f32 N/A

      \[\leadsto \left(\left(\sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(uy \cdot 2\right) \cdot \pi}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   6. *-commutative N/A

      \[\leadsto \left(\left(\sin \left(\left(\mathsf{neg}\left(\color{blue}{\pi \cdot \left(uy \cdot 2\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   7. distribute-rgt-neg-in N/A

      \[\leadsto \left(\left(\sin \left(\color{blue}{\pi \cdot \left(\mathsf{neg}\left(uy \cdot 2\right)\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   8. lift-PI.f32 N/A

      \[\leadsto \left(\left(\sin \left(\pi \cdot \left(\mathsf{neg}\left(uy \cdot 2\right)\right) + \frac{\color{blue}{\pi}}{2}\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   9. mult-flip N/A

      \[\leadsto \left(\left(\sin \left(\pi \cdot \left(\mathsf{neg}\left(uy \cdot 2\right)\right) + \color{blue}{\pi \cdot \frac{1}{2}}\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   10. metadata-eval N/A

       \[\leadsto \left(\left(\sin \left(\pi \cdot \left(\mathsf{neg}\left(uy \cdot 2\right)\right) + \pi \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   11. distribute-lft-out N/A

       \[\leadsto \left(\left(\sin \color{blue}{\left(\pi \cdot \left(\left(\mathsf{neg}\left(uy \cdot 2\right)\right) + \frac{1}{2}\right)\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   12. lower-*.f32 N/A

       \[\leadsto \left(\left(\sin \color{blue}{\left(\pi \cdot \left(\left(\mathsf{neg}\left(uy \cdot 2\right)\right) + \frac{1}{2}\right)\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   13. lift-*.f32 N/A

       \[\leadsto \left(\left(\sin \left(\pi \cdot \left(\left(\mathsf{neg}\left(\color{blue}{uy \cdot 2}\right)\right) + \frac{1}{2}\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   14. *-commutative N/A

       \[\leadsto \left(\left(\sin \left(\pi \cdot \left(\left(\mathsf{neg}\left(\color{blue}{2 \cdot uy}\right)\right) + \frac{1}{2}\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   15. distribute-lft-neg-in N/A

       \[\leadsto \left(\left(\sin \left(\pi \cdot \left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot uy} + \frac{1}{2}\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   16. lower-fma.f32 N/A

       \[\leadsto \left(\left(\sin \left(\pi \cdot \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(2\right), uy, \frac{1}{2}\right)}\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   17. metadata-eval 99.1%

       \[\leadsto \left(\left(\sin \left(\pi \cdot \mathsf{fma}\left(\color{blue}{-2}, uy, 0.5\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Applied rewrites 99.1%

   \[\leadsto \left(\left(\color{blue}{\sin \left(\pi \cdot \mathsf{fma}\left(-2, uy, 0.5\right)\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
4. Add Preprocessing

Alternative 2: 99.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \left(uy + uy\right)\\ \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin t\_0, xi \cdot \cos t\_0\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (let* ((t_0 (* PI (+ uy uy))))
  (fma
   (- 1.0 ux)
   (* (* maxCos ux) zi)
   (*
    (sqrt
     (fma
      (* (- ux 1.0) maxCos)
      (* (* (* maxCos (- 1.0 ux)) ux) ux)
      1.0))
    (fma yi (sin t_0) (* xi (cos t_0)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((float) M_PI) * (uy + uy);
	return fmaf((1.0f - ux), ((maxCos * ux) * zi), (sqrtf(fmaf(((ux - 1.0f) * maxCos), (((maxCos * (1.0f - ux)) * ux) * ux), 1.0f)) * fmaf(yi, sinf(t_0), (xi * cosf(t_0)))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(uy + uy))
	return fma(Float32(Float32(1.0) - ux), Float32(Float32(maxCos * ux) * zi), Float32(sqrt(fma(Float32(Float32(ux - Float32(1.0)) * maxCos), Float32(Float32(Float32(maxCos * Float32(Float32(1.0) - ux)) * ux) * ux), Float32(1.0))) * fma(yi, sin(t_0), Float32(xi * cos(t_0)))))
end

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\right)\right)} \]
4. Add Preprocessing

Alternative 3: 99.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \left(uy + uy\right)\\ \mathsf{fma}\left(zi \cdot \left(1 - ux\right), maxCos \cdot ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin t\_0, xi \cdot \cos t\_0\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (let* ((t_0 (* PI (+ uy uy))))
  (fma
   (* zi (- 1.0 ux))
   (* maxCos ux)
   (*
    (sqrt
     (fma
      (* (- ux 1.0) maxCos)
      (* (* (* maxCos (- 1.0 ux)) ux) ux)
      1.0))
    (fma yi (sin t_0) (* xi (cos t_0)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((float) M_PI) * (uy + uy);
	return fmaf((zi * (1.0f - ux)), (maxCos * ux), (sqrtf(fmaf(((ux - 1.0f) * maxCos), (((maxCos * (1.0f - ux)) * ux) * ux), 1.0f)) * fmaf(yi, sinf(t_0), (xi * cosf(t_0)))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(uy + uy))
	return fma(Float32(zi * Float32(Float32(1.0) - ux)), Float32(maxCos * ux), Float32(sqrt(fma(Float32(Float32(ux - Float32(1.0)) * maxCos), Float32(Float32(Float32(maxCos * Float32(Float32(1.0) - ux)) * ux) * ux), Float32(1.0))) * fma(yi, sin(t_0), Float32(xi * cos(t_0)))))
end

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(zi \cdot \left(1 - ux\right), maxCos \cdot ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\right)\right)} \]
4. Add Preprocessing

Alternative 4: 98.8% accurate, 1.3× speedup

Math

\[\mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(-1 \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (fma
 (*
  yi
  (sqrt
   (fma (* -1.0 maxCos) (* (* (* maxCos (- 1.0 ux)) ux) ux) 1.0)))
 (sin (* PI (+ uy uy)))
 (fma maxCos (* ux (* zi (- 1.0 ux))) (* xi (cos (* 2.0 (* uy PI)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf((yi * sqrtf(fmaf((-1.0f * maxCos), (((maxCos * (1.0f - ux)) * ux) * ux), 1.0f))), sinf((((float) M_PI) * (uy + uy))), fmaf(maxCos, (ux * (zi * (1.0f - ux))), (xi * cosf((2.0f * (uy * ((float) M_PI)))))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(yi * sqrt(fma(Float32(Float32(-1.0) * maxCos), Float32(Float32(Float32(maxCos * Float32(Float32(1.0) - ux)) * ux) * ux), Float32(1.0)))), sin(Float32(Float32(pi) * Float32(uy + uy))), fma(maxCos, Float32(ux * Float32(zi * Float32(Float32(1.0) - ux))), Float32(xi * cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))))))
end

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), \mathsf{fma}\left(xi \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right), \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right)} \]

4. Taylor expanded in maxCos around 0

   \[\leadsto \mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}\right) \]
5. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   4. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   6. lower-cos.f32 N/A

      \[\leadsto \mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   7. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   8. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   9. lower-PI.f32 98.9%

      \[\leadsto \mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right) \]

6. Applied rewrites 98.9%

   \[\leadsto \mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)}\right) \]

7. Taylor expanded in ux around 0

   \[\leadsto \mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\color{blue}{-1} \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right) \]
8. Step-by-step derivation

9. Applied rewrites 98.8%

   \[\leadsto \mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\color{blue}{-1} \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 5: 98.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos t\_0, yi \cdot \sin t\_0\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (let* ((t_0 (* 2.0 (* uy PI))))
  (fma
   maxCos
   (* ux (* zi (- 1.0 ux)))
   (fma xi (cos t_0) (* yi (sin t_0))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 2.0f * (uy * ((float) M_PI));
	return fmaf(maxCos, (ux * (zi * (1.0f - ux))), fmaf(xi, cosf(t_0), (yi * sinf(t_0))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	return fma(maxCos, Float32(ux * Float32(zi * Float32(Float32(1.0) - ux))), fma(xi, cos(t_0), Float32(yi * sin(t_0))))
end

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   4. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

4. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]
5. Add Preprocessing

Alternative 6: 96.2% accurate, 1.6× speedup

Math

\[\mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \sin \left(\pi \cdot \left(0.5 + -2 \cdot uy\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (fma
 maxCos
 (* ux zi)
 (fma
  xi
  (sin (* PI (+ 0.5 (* -2.0 uy))))
  (* yi (sin (* 2.0 (* uy PI)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf(maxCos, (ux * zi), fmaf(xi, sinf((((float) M_PI) * (0.5f + (-2.0f * uy)))), (yi * sinf((2.0f * (uy * ((float) M_PI)))))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(maxCos, Float32(ux * zi), fma(xi, sin(Float32(Float32(pi) * Float32(Float32(0.5) + Float32(Float32(-2.0) * uy)))), Float32(yi * sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))))))
end

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Step-by-step derivation

   1. lift-cos.f32 N/A

      \[\leadsto \left(\left(\color{blue}{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   2. cos-neg-rev N/A

      \[\leadsto \left(\left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(uy \cdot 2\right) \cdot \pi\right)\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   3. sin-+PI/2-rev N/A

      \[\leadsto \left(\left(\color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(uy \cdot 2\right) \cdot \pi\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   4. lower-sin.f32 N/A

      \[\leadsto \left(\left(\color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(uy \cdot 2\right) \cdot \pi\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   5. lift-*.f32 N/A

      \[\leadsto \left(\left(\sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(uy \cdot 2\right) \cdot \pi}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   6. *-commutative N/A

      \[\leadsto \left(\left(\sin \left(\left(\mathsf{neg}\left(\color{blue}{\pi \cdot \left(uy \cdot 2\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   7. distribute-rgt-neg-in N/A

      \[\leadsto \left(\left(\sin \left(\color{blue}{\pi \cdot \left(\mathsf{neg}\left(uy \cdot 2\right)\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   8. lift-PI.f32 N/A

      \[\leadsto \left(\left(\sin \left(\pi \cdot \left(\mathsf{neg}\left(uy \cdot 2\right)\right) + \frac{\color{blue}{\pi}}{2}\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   9. mult-flip N/A

      \[\leadsto \left(\left(\sin \left(\pi \cdot \left(\mathsf{neg}\left(uy \cdot 2\right)\right) + \color{blue}{\pi \cdot \frac{1}{2}}\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   10. metadata-eval N/A

       \[\leadsto \left(\left(\sin \left(\pi \cdot \left(\mathsf{neg}\left(uy \cdot 2\right)\right) + \pi \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   11. distribute-lft-out N/A

       \[\leadsto \left(\left(\sin \color{blue}{\left(\pi \cdot \left(\left(\mathsf{neg}\left(uy \cdot 2\right)\right) + \frac{1}{2}\right)\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   12. lower-*.f32 N/A

       \[\leadsto \left(\left(\sin \color{blue}{\left(\pi \cdot \left(\left(\mathsf{neg}\left(uy \cdot 2\right)\right) + \frac{1}{2}\right)\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   13. lift-*.f32 N/A

       \[\leadsto \left(\left(\sin \left(\pi \cdot \left(\left(\mathsf{neg}\left(\color{blue}{uy \cdot 2}\right)\right) + \frac{1}{2}\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   14. *-commutative N/A

       \[\leadsto \left(\left(\sin \left(\pi \cdot \left(\left(\mathsf{neg}\left(\color{blue}{2 \cdot uy}\right)\right) + \frac{1}{2}\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   15. distribute-lft-neg-in N/A

       \[\leadsto \left(\left(\sin \left(\pi \cdot \left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot uy} + \frac{1}{2}\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   16. lower-fma.f32 N/A

       \[\leadsto \left(\left(\sin \left(\pi \cdot \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(2\right), uy, \frac{1}{2}\right)}\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   17. metadata-eval 99.1%

       \[\leadsto \left(\left(\sin \left(\pi \cdot \mathsf{fma}\left(\color{blue}{-2}, uy, 0.5\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Applied rewrites 99.1%

   \[\leadsto \left(\left(\color{blue}{\sin \left(\pi \cdot \mathsf{fma}\left(-2, uy, 0.5\right)\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

4. Taylor expanded in ux around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + \left(xi \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + -2 \cdot uy\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]
5. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot zi}, xi \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + -2 \cdot uy\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{zi}, xi \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + -2 \cdot uy\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \sin \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + -2 \cdot uy\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

6. Applied rewrites 96.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \sin \left(\pi \cdot \left(0.5 + -2 \cdot uy\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]
7. Add Preprocessing

Alternative 7: 96.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos t\_0, yi \cdot \sin t\_0\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (let* ((t_0 (* 2.0 (* uy PI))))
  (fma maxCos (* ux zi) (fma xi (cos t_0) (* yi (sin t_0))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 2.0f * (uy * ((float) M_PI));
	return fmaf(maxCos, (ux * zi), fmaf(xi, cosf(t_0), (yi * sinf(t_0))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	return fma(maxCos, Float32(ux * zi), fma(xi, cos(t_0), Float32(yi * sin(t_0))))
end

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in ux around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot zi}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{zi}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   4. lower-cos.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   7. lower-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   8. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

4. Applied rewrites 96.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]
5. Add Preprocessing

Alternative 8: 90.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;xi \leq 1.0000000168623835 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (if (<= xi 1.0000000168623835e-16)
  (fma
   (*
    yi
    (sqrt
     (fma
      (* (- ux 1.0) maxCos)
      (* (* (* maxCos (- 1.0 ux)) ux) ux)
      1.0)))
   (sin (* PI (+ uy uy)))
   (+ xi (* maxCos (* ux (* zi (- 1.0 ux))))))
  (fma
   maxCos
   (* ux zi)
   (fma xi (cos (* 2.0 (* uy PI))) (* 2.0 (* uy (* yi PI)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (xi <= 1.0000000168623835e-16f) {
		tmp = fmaf((yi * sqrtf(fmaf(((ux - 1.0f) * maxCos), (((maxCos * (1.0f - ux)) * ux) * ux), 1.0f))), sinf((((float) M_PI) * (uy + uy))), (xi + (maxCos * (ux * (zi * (1.0f - ux))))));
	} else {
		tmp = fmaf(maxCos, (ux * zi), fmaf(xi, cosf((2.0f * (uy * ((float) M_PI)))), (2.0f * (uy * (yi * ((float) M_PI))))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (xi <= Float32(1.0000000168623835e-16))
		tmp = fma(Float32(yi * sqrt(fma(Float32(Float32(ux - Float32(1.0)) * maxCos), Float32(Float32(Float32(maxCos * Float32(Float32(1.0) - ux)) * ux) * ux), Float32(1.0)))), sin(Float32(Float32(pi) * Float32(uy + uy))), Float32(xi + Float32(maxCos * Float32(ux * Float32(zi * Float32(Float32(1.0) - ux))))));
	else
		tmp = fma(maxCos, Float32(ux * zi), fma(xi, cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))), Float32(Float32(2.0) * Float32(uy * Float32(yi * Float32(pi))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if xi < 1.00000002e-16

   1. Initial program 99.0%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   3. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), \mathsf{fma}\left(xi \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right), \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right)} \]

   4. Taylor expanded in maxCos around 0

      \[\leadsto \mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}\right) \]
   5. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      4. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      6. lower-cos.f32 N/A

         \[\leadsto \mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      7. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      9. lower-PI.f32 98.9%

         \[\leadsto \mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right) \]

   6. Applied rewrites 98.9%

      \[\leadsto \mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)}\right) \]

   7. Taylor expanded in uy around 0

      \[\leadsto \mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi + \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)}\right) \]
   8. Step-by-step derivation

      1. lower-+.f32 N/A

         \[\leadsto \mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi + maxCos \cdot \color{blue}{\left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi + maxCos \cdot \left(ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right)\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right)\right)\right) \]

      5. lower--.f32 88.8%

         \[\leadsto \mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   9. Applied rewrites 88.8%

      \[\leadsto \mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi + \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)}\right) \]

   if 1.00000002e-16 < xi

   1. Initial program 99.0%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   2. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot zi}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{zi}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      4. lower-cos.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      7. lower-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   4. Applied rewrites 96.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]

   5. Taylor expanded in uy around 0

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      4. lower-PI.f32 87.5%

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)\right) \]

   7. Applied rewrites 87.5%

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 87.5% accurate, 2.5× speedup

Math

\[\mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (fma
 maxCos
 (* ux zi)
 (fma xi (cos (* 2.0 (* uy PI))) (* 2.0 (* uy (* yi PI))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf(maxCos, (ux * zi), fmaf(xi, cosf((2.0f * (uy * ((float) M_PI)))), (2.0f * (uy * (yi * ((float) M_PI))))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(maxCos, Float32(ux * zi), fma(xi, cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))), Float32(Float32(2.0) * Float32(uy * Float32(yi * Float32(pi))))))
end

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in ux around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot zi}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{zi}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   4. lower-cos.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   7. lower-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   8. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

4. Applied rewrites 96.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]

5. Taylor expanded in uy around 0

   \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   4. lower-PI.f32 87.5%

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)\right) \]

7. Applied rewrites 87.5%

   \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)\right) \]
8. Add Preprocessing

Alternative 10: 84.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;uy \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(maxCos, ux \cdot zi, xi \cdot \sin \left(\pi \cdot \left(0.5 + -2 \cdot uy\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (if (<= uy 0.20000000298023224)
  (fma
   (- 1.0 ux)
   (* (* maxCos ux) zi)
   (*
    (sqrt
     (fma
      (* (- ux 1.0) maxCos)
      (* (* (* maxCos (- 1.0 ux)) ux) ux)
      1.0))
    (+ xi (* 2.0 (* uy (* yi PI))))))
  (fma maxCos (* ux zi) (* xi (sin (* PI (+ 0.5 (* -2.0 uy))))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.20000000298023224f) {
		tmp = fmaf((1.0f - ux), ((maxCos * ux) * zi), (sqrtf(fmaf(((ux - 1.0f) * maxCos), (((maxCos * (1.0f - ux)) * ux) * ux), 1.0f)) * (xi + (2.0f * (uy * (yi * ((float) M_PI)))))));
	} else {
		tmp = fmaf(maxCos, (ux * zi), (xi * sinf((((float) M_PI) * (0.5f + (-2.0f * uy))))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.20000000298023224))
		tmp = fma(Float32(Float32(1.0) - ux), Float32(Float32(maxCos * ux) * zi), Float32(sqrt(fma(Float32(Float32(ux - Float32(1.0)) * maxCos), Float32(Float32(Float32(maxCos * Float32(Float32(1.0) - ux)) * ux) * ux), Float32(1.0))) * Float32(xi + Float32(Float32(2.0) * Float32(uy * Float32(yi * Float32(pi)))))));
	else
		tmp = fma(maxCos, Float32(ux * zi), Float32(xi * sin(Float32(Float32(pi) * Float32(Float32(0.5) + Float32(Float32(-2.0) * uy))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.200000003

   1. Initial program 99.0%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   3. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\right)\right)} \]

   4. Taylor expanded in uy around 0

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)}\right) \]
   5. Step-by-step derivation

      1. lower-+.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + \color{blue}{2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \color{blue}{\left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \color{blue}{\left(yi \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

      5. lower-PI.f32 81.9%

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)\right) \]

   6. Applied rewrites 81.9%

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)}\right) \]

   if 0.200000003 < uy

   1. Initial program 99.0%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   2. Step-by-step derivation

      1. lift-cos.f32 N/A

         \[\leadsto \left(\left(\color{blue}{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      2. cos-neg-rev N/A

         \[\leadsto \left(\left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(uy \cdot 2\right) \cdot \pi\right)\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \left(\left(\color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(uy \cdot 2\right) \cdot \pi\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      4. lower-sin.f32 N/A

         \[\leadsto \left(\left(\color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(uy \cdot 2\right) \cdot \pi\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      5. lift-*.f32 N/A

         \[\leadsto \left(\left(\sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(uy \cdot 2\right) \cdot \pi}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      6. *-commutative N/A

         \[\leadsto \left(\left(\sin \left(\left(\mathsf{neg}\left(\color{blue}{\pi \cdot \left(uy \cdot 2\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto \left(\left(\sin \left(\color{blue}{\pi \cdot \left(\mathsf{neg}\left(uy \cdot 2\right)\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      8. lift-PI.f32 N/A

         \[\leadsto \left(\left(\sin \left(\pi \cdot \left(\mathsf{neg}\left(uy \cdot 2\right)\right) + \frac{\color{blue}{\pi}}{2}\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      9. mult-flip N/A

         \[\leadsto \left(\left(\sin \left(\pi \cdot \left(\mathsf{neg}\left(uy \cdot 2\right)\right) + \color{blue}{\pi \cdot \frac{1}{2}}\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      10. metadata-eval N/A

          \[\leadsto \left(\left(\sin \left(\pi \cdot \left(\mathsf{neg}\left(uy \cdot 2\right)\right) + \pi \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      11. distribute-lft-out N/A

          \[\leadsto \left(\left(\sin \color{blue}{\left(\pi \cdot \left(\left(\mathsf{neg}\left(uy \cdot 2\right)\right) + \frac{1}{2}\right)\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      12. lower-*.f32 N/A

          \[\leadsto \left(\left(\sin \color{blue}{\left(\pi \cdot \left(\left(\mathsf{neg}\left(uy \cdot 2\right)\right) + \frac{1}{2}\right)\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      13. lift-*.f32 N/A

          \[\leadsto \left(\left(\sin \left(\pi \cdot \left(\left(\mathsf{neg}\left(\color{blue}{uy \cdot 2}\right)\right) + \frac{1}{2}\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      14. *-commutative N/A

          \[\leadsto \left(\left(\sin \left(\pi \cdot \left(\left(\mathsf{neg}\left(\color{blue}{2 \cdot uy}\right)\right) + \frac{1}{2}\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      15. distribute-lft-neg-in N/A

          \[\leadsto \left(\left(\sin \left(\pi \cdot \left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot uy} + \frac{1}{2}\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      16. lower-fma.f32 N/A

          \[\leadsto \left(\left(\sin \left(\pi \cdot \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(2\right), uy, \frac{1}{2}\right)}\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      17. metadata-eval 99.1%

          \[\leadsto \left(\left(\sin \left(\pi \cdot \mathsf{fma}\left(\color{blue}{-2}, uy, 0.5\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   3. Applied rewrites 99.1%

      \[\leadsto \left(\left(\color{blue}{\sin \left(\pi \cdot \mathsf{fma}\left(-2, uy, 0.5\right)\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   4. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + \left(xi \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + -2 \cdot uy\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot zi}, xi \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + -2 \cdot uy\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{zi}, xi \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + -2 \cdot uy\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \sin \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + -2 \cdot uy\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   6. Applied rewrites 96.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \sin \left(\pi \cdot \left(0.5 + -2 \cdot uy\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]

   7. Taylor expanded in xi around inf

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, xi \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + -2 \cdot uy\right)\right)\right) \]
   8. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, xi \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + -2 \cdot uy\right)\right)\right) \]

      2. lower-sin.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, xi \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + -2 \cdot uy\right)\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, xi \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + -2 \cdot uy\right)\right)\right) \]

      4. lower-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, xi \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + -2 \cdot uy\right)\right)\right) \]

      5. lower-+.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, xi \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + -2 \cdot uy\right)\right)\right) \]

      6. lower-*.f32 58.1%

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, xi \cdot \sin \left(\pi \cdot \left(0.5 + -2 \cdot uy\right)\right)\right) \]

   9. Applied rewrites 58.1%

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, xi \cdot \sin \left(\pi \cdot \left(0.5 + -2 \cdot uy\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 84.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;uy \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(maxCos, ux \cdot zi, xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (if (<= uy 0.20000000298023224)
  (fma
   (- 1.0 ux)
   (* (* maxCos ux) zi)
   (*
    (sqrt
     (fma
      (* (- ux 1.0) maxCos)
      (* (* (* maxCos (- 1.0 ux)) ux) ux)
      1.0))
    (+ xi (* 2.0 (* uy (* yi PI))))))
  (fma maxCos (* ux zi) (* xi (cos (* 2.0 (* uy PI)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.20000000298023224f) {
		tmp = fmaf((1.0f - ux), ((maxCos * ux) * zi), (sqrtf(fmaf(((ux - 1.0f) * maxCos), (((maxCos * (1.0f - ux)) * ux) * ux), 1.0f)) * (xi + (2.0f * (uy * (yi * ((float) M_PI)))))));
	} else {
		tmp = fmaf(maxCos, (ux * zi), (xi * cosf((2.0f * (uy * ((float) M_PI))))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.20000000298023224))
		tmp = fma(Float32(Float32(1.0) - ux), Float32(Float32(maxCos * ux) * zi), Float32(sqrt(fma(Float32(Float32(ux - Float32(1.0)) * maxCos), Float32(Float32(Float32(maxCos * Float32(Float32(1.0) - ux)) * ux) * ux), Float32(1.0))) * Float32(xi + Float32(Float32(2.0) * Float32(uy * Float32(yi * Float32(pi)))))));
	else
		tmp = fma(maxCos, Float32(ux * zi), Float32(xi * cos(Float32(Float32(2.0) * Float32(uy * Float32(pi))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.200000003

   1. Initial program 99.0%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   3. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\right)\right)} \]

   4. Taylor expanded in uy around 0

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)}\right) \]
   5. Step-by-step derivation

      1. lower-+.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + \color{blue}{2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \color{blue}{\left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \color{blue}{\left(yi \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

      5. lower-PI.f32 81.9%

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)\right) \]

   6. Applied rewrites 81.9%

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)}\right) \]

   if 0.200000003 < uy

   1. Initial program 99.0%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   2. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot zi}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{zi}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      4. lower-cos.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      7. lower-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   4. Applied rewrites 96.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]

   5. Taylor expanded in xi around inf

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      2. lower-cos.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. lower-PI.f32 58.0%

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   7. Applied rewrites 58.0%

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 82.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;uy \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{fma}\left(1, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot 1\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(maxCos, ux \cdot zi, xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (if (<= uy 0.05000000074505806)
  (fma
   1.0
   (* (* maxCos ux) zi)
   (*
    (sqrt
     (fma (* (- ux 1.0) maxCos) (* (* (* maxCos 1.0) ux) ux) 1.0))
    (+ xi (* 2.0 (* uy (* yi PI))))))
  (fma maxCos (* ux zi) (* xi (cos (* 2.0 (* uy PI)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.05000000074505806f) {
		tmp = fmaf(1.0f, ((maxCos * ux) * zi), (sqrtf(fmaf(((ux - 1.0f) * maxCos), (((maxCos * 1.0f) * ux) * ux), 1.0f)) * (xi + (2.0f * (uy * (yi * ((float) M_PI)))))));
	} else {
		tmp = fmaf(maxCos, (ux * zi), (xi * cosf((2.0f * (uy * ((float) M_PI))))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.05000000074505806))
		tmp = fma(Float32(1.0), Float32(Float32(maxCos * ux) * zi), Float32(sqrt(fma(Float32(Float32(ux - Float32(1.0)) * maxCos), Float32(Float32(Float32(maxCos * Float32(1.0)) * ux) * ux), Float32(1.0))) * Float32(xi + Float32(Float32(2.0) * Float32(uy * Float32(yi * Float32(pi)))))));
	else
		tmp = fma(maxCos, Float32(ux * zi), Float32(xi * cos(Float32(Float32(2.0) * Float32(uy * Float32(pi))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0500000007

   1. Initial program 99.0%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   3. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\right)\right)} \]

   4. Taylor expanded in ux around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 96.2%

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\right)\right) \]

   7. Taylor expanded in ux around 0

      \[\leadsto \mathsf{fma}\left(1, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \color{blue}{1}\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\right)\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 96.1%

      \[\leadsto \mathsf{fma}\left(1, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \color{blue}{1}\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\right)\right) \]

   10. Taylor expanded in uy around 0

       \[\leadsto \mathsf{fma}\left(1, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot 1\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)}\right) \]
   11. Step-by-step derivation

       1. lower-+.f32 N/A

          \[\leadsto \mathsf{fma}\left(1, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot 1\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + \color{blue}{2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

       2. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(1, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot 1\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \color{blue}{\left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

       3. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(1, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot 1\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \color{blue}{\left(yi \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \]

       4. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(1, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot 1\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

       5. lower-PI.f32 79.4%

          \[\leadsto \mathsf{fma}\left(1, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot 1\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)\right) \]

   12. Applied rewrites 79.4%

       \[\leadsto \mathsf{fma}\left(1, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot 1\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)}\right) \]

   if 0.0500000007 < uy

   1. Initial program 99.0%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   2. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot zi}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{zi}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      4. lower-cos.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      7. lower-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   4. Applied rewrites 96.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]

   5. Taylor expanded in xi around inf

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      2. lower-cos.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. lower-PI.f32 58.0%

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   7. Applied rewrites 58.0%

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 79.4% accurate, 3.3× speedup

Math

\[\mathsf{fma}\left(1, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot 1\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (fma
 1.0
 (* (* maxCos ux) zi)
 (*
  (sqrt (fma (* (- ux 1.0) maxCos) (* (* (* maxCos 1.0) ux) ux) 1.0))
  (+ xi (* 2.0 (* uy (* yi PI)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf(1.0f, ((maxCos * ux) * zi), (sqrtf(fmaf(((ux - 1.0f) * maxCos), (((maxCos * 1.0f) * ux) * ux), 1.0f)) * (xi + (2.0f * (uy * (yi * ((float) M_PI)))))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(1.0), Float32(Float32(maxCos * ux) * zi), Float32(sqrt(fma(Float32(Float32(ux - Float32(1.0)) * maxCos), Float32(Float32(Float32(maxCos * Float32(1.0)) * ux) * ux), Float32(1.0))) * Float32(xi + Float32(Float32(2.0) * Float32(uy * Float32(yi * Float32(pi)))))))
end

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\right)\right)} \]

4. Taylor expanded in ux around 0

   \[\leadsto \mathsf{fma}\left(\color{blue}{1}, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\right)\right) \]
5. Step-by-step derivation

6. Applied rewrites 96.2%

   \[\leadsto \mathsf{fma}\left(\color{blue}{1}, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\right)\right) \]

7. Taylor expanded in ux around 0

   \[\leadsto \mathsf{fma}\left(1, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \color{blue}{1}\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\right)\right) \]
8. Step-by-step derivation

9. Applied rewrites 96.1%

   \[\leadsto \mathsf{fma}\left(1, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \color{blue}{1}\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(yi, \sin \left(\pi \cdot \left(uy + uy\right)\right), xi \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\right)\right) \]

10. Taylor expanded in uy around 0

    \[\leadsto \mathsf{fma}\left(1, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot 1\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)}\right) \]
11. Step-by-step derivation

    1. lower-+.f32 N/A

       \[\leadsto \mathsf{fma}\left(1, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot 1\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + \color{blue}{2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

    2. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(1, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot 1\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \color{blue}{\left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

    3. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(1, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot 1\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \color{blue}{\left(yi \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \]

    4. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(1, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot 1\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

    5. lower-PI.f32 79.4%

       \[\leadsto \mathsf{fma}\left(1, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot 1\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)\right) \]

12. Applied rewrites 79.4%

    \[\leadsto \mathsf{fma}\left(1, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot 1\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)}\right) \]
13. Add Preprocessing

Alternative 14: 79.4% accurate, 7.7× speedup

Math

\[\mathsf{fma}\left(maxCos, ux \cdot zi, xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (fma maxCos (* ux zi) (+ xi (* 2.0 (* uy (* yi PI))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf(maxCos, (ux * zi), (xi + (2.0f * (uy * (yi * ((float) M_PI))))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(maxCos, Float32(ux * zi), Float32(xi + Float32(Float32(2.0) * Float32(uy * Float32(yi * Float32(pi))))))
end

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in ux around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot zi}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{zi}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   4. lower-cos.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   7. lower-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   8. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

4. Applied rewrites 96.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]

5. Taylor expanded in uy around 0

   \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right) \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, xi + 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, xi + 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, xi + 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   4. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, xi + 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   5. lower-PI.f32 79.4%

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right) \]

7. Applied rewrites 79.4%

   \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right) \]
8. Add Preprocessing

Alternative 15: 50.7% accurate, 16.6× speedup

Math

\[xi + maxCos \cdot \left(ux \cdot zi\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (+ xi (* maxCos (* ux zi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return xi + (maxCos * (ux * zi));
}

Fortran

real(4) function code(xi, yi, zi, ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: xi
    real(4), intent (in) :: yi
    real(4), intent (in) :: zi
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = xi + (maxcos * (ux * zi))
end function

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(xi + Float32(maxCos * Float32(ux * zi)))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = xi + (maxCos * (ux * zi));
end

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in ux around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot zi}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{zi}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   4. lower-cos.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   7. lower-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   8. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

4. Applied rewrites 96.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]

5. Taylor expanded in zi around inf

   \[\leadsto maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]
6. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]

   2. lower-*.f32 11.8%

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) \]

7. Applied rewrites 11.8%

   \[\leadsto maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]

8. Taylor expanded in uy around 0

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
9. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto xi + maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]

   3. lower-*.f32 50.7%

      \[\leadsto xi + maxCos \cdot \left(ux \cdot zi\right) \]

10. Applied rewrites 50.7%

    \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
11. Add Preprocessing

Alternative 16: 11.8% accurate, 22.8× speedup

Math

\[maxCos \cdot \left(ux \cdot zi\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (* maxCos (* ux zi)))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return maxCos * (ux * zi);
}

Fortran

real(4) function code(xi, yi, zi, ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: xi
    real(4), intent (in) :: yi
    real(4), intent (in) :: zi
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = maxcos * (ux * zi)
end function

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(maxCos * Float32(ux * zi))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = maxCos * (ux * zi);
end

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in ux around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot zi}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{zi}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   4. lower-cos.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   7. lower-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   8. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

4. Applied rewrites 96.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]

5. Taylor expanded in zi around inf

   \[\leadsto maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]
6. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]

   2. lower-*.f32 11.8%

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) \]

7. Applied rewrites 11.8%

   \[\leadsto maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025322 
(FPCore (xi yi zi ux uy maxCos)
  :name "UniformSampleCone 2"
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0)) (and (<= -10000.0 yi) (<= yi 10000.0))) (and (<= -10000.0 zi) (<= zi 10000.0))) (and (<= 2.328306437e-10 ux) (<= ux 1.0))) (and (<= 2.328306437e-10 uy) (<= uy 1.0))) (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (+ (+ (* (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (* (* (- 1.0 ux) maxCos) ux) (* (* (- 1.0 ux) maxCos) ux))))) xi) (* (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (* (* (- 1.0 ux) maxCos) ux) (* (* (- 1.0 ux) maxCos) ux))))) yi)) (* (* (* (- 1.0 ux) maxCos) ux) zi)))