UniformSampleCone 2

Percentage Accurate: 98.9% → 99.0%

Time: 11.4s

Alternatives: 21

Speedup: 1.0×

Specification

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(\frac{2328306437}{10000000000000000000} \leq ux \land ux \leq 1\right)\right) \land \left(\frac{2328306437}{10000000000000000000} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

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 ux) maxCos) ux))
       (t_1 (sqrt (- 1 (* t_0 t_0))))
       (t_2 (* (* uy 2) 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: 98.9% 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 ux) maxCos) ux))
       (t_1 (sqrt (- 1 (* t_0 t_0))))
       (t_2 (* (* uy 2) 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: 99.0% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (let* ((t_0 (* (* (- 1 ux) maxCos) ux))
       (t_1 (sqrt (- 1 (* t_0 t_0)))))
  (+
   (+
    (* (* (sin (* uy (+ (* -2 PI) (* 1/2 (/ PI uy))))) t_1) xi)
    (* (* (sin (* (* uy 2) 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((uy * ((-2.0f * ((float) M_PI)) + (0.5f * (((float) M_PI) / uy))))) * 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(uy * Float32(Float32(Float32(-2.0) * Float32(pi)) + Float32(Float32(0.5) * Float32(Float32(pi) / uy))))) * t_1) * xi) + Float32(Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * 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)));
	tmp = (((sin((uy * ((single(-2.0) * single(pi)) + (single(0.5) * (single(pi) / uy))))) * t_1) * xi) + ((sin(((uy * single(2.0)) * single(pi))) * t_1) * yi)) + (t_0 * zi);
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
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. lower-+.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

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

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

   10. lower-*.f32 N/A

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

   11. lower-neg.f32 N/A

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

   12. count-2-rev N/A

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

   13. lower-+.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. mult-flip N/A

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

   16. metadata-eval N/A

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

   17. lower-*.f32 99.0%

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

3. Applied rewrites 99.0%

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

4. Taylor expanded in uy around inf

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-PI.f32 98.9%

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

6. Applied rewrites 98.9%

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

Alternative 2: 98.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (let* ((t_0 (* (* (- 1 ux) maxCos) ux))
       (t_1 (sqrt (- 1 (* t_0 t_0)))))
  (+
   (+
    (* (* (sin (+ (* (- uy) (+ PI PI)) (* PI 1/2))) t_1) xi)
    (* (* (sin (* (* uy 2) 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(((-uy * (((float) M_PI) + ((float) M_PI))) + (((float) M_PI) * 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(Float32(-uy) * Float32(Float32(pi) + Float32(pi))) + Float32(Float32(pi) * 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

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)));
	tmp = (((sin(((-uy * (single(pi) + single(pi))) + (single(pi) * single(0.5)))) * t_1) * xi) + ((sin(((uy * single(2.0)) * single(pi))) * t_1) * yi)) + (t_0 * zi);
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
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. lower-+.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

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

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

   10. lower-*.f32 N/A

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

   11. lower-neg.f32 N/A

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

   12. count-2-rev N/A

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

   13. lower-+.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. mult-flip N/A

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

   16. metadata-eval N/A

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

   17. lower-*.f32 99.0%

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

3. Applied rewrites 99.0%

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

Alternative 3: 98.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (let* ((t_0 (* (+ uy uy) PI))
       (t_1
        (sqrt
         (-
          1
          (*
           (* (* (* (* maxCos (- 1 ux)) maxCos) (- 1 ux)) ux)
           ux)))))
  (+
   (+ (* (* zi (- 1 ux)) (* maxCos ux)) (* (* (cos t_0) xi) t_1))
   (* (* (sin t_0) yi) t_1))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
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. lower-+.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

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

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

   10. lower-*.f32 N/A

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

   11. lower-neg.f32 N/A

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

   12. count-2-rev N/A

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

   13. lower-+.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. mult-flip N/A

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

   16. metadata-eval N/A

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

   17. lower-*.f32 99.0%

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

3. Applied rewrites 99.0%

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

5. Applied rewrites 98.9%

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

Alternative 4: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \left(uy + uy\right)\\ t_1 := maxCos \cdot \left(1 - ux\right)\\ t_2 := t\_1 \cdot ux\\ t_3 := \sqrt{1 - \left(t\_2 \cdot ux\right) \cdot t\_1}\\ \left(zi \cdot t\_2 + \left(xi \cdot t\_3\right) \cdot \cos t\_0\right) + \left(yi \cdot t\_3\right) \cdot \sin t\_0 \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (let* ((t_0 (* PI (+ uy uy)))
       (t_1 (* maxCos (- 1 ux)))
       (t_2 (* t_1 ux))
       (t_3 (sqrt (- 1 (* (* t_2 ux) t_1)))))
  (+
   (+ (* zi t_2) (* (* xi t_3) (cos t_0)))
   (* (* yi t_3) (sin 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);
	float t_1 = maxCos * (1.0f - ux);
	float t_2 = t_1 * ux;
	float t_3 = sqrtf((1.0f - ((t_2 * ux) * t_1)));
	return ((zi * t_2) + ((xi * t_3) * cosf(t_0))) + ((yi * t_3) * sinf(t_0));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(uy + uy))
	t_1 = Float32(maxCos * Float32(Float32(1.0) - ux))
	t_2 = Float32(t_1 * ux)
	t_3 = sqrt(Float32(Float32(1.0) - Float32(Float32(t_2 * ux) * t_1)))
	return Float32(Float32(Float32(zi * t_2) + Float32(Float32(xi * t_3) * cos(t_0))) + Float32(Float32(yi * t_3) * sin(t_0)))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = single(pi) * (uy + uy);
	t_1 = maxCos * (single(1.0) - ux);
	t_2 = t_1 * ux;
	t_3 = sqrt((single(1.0) - ((t_2 * ux) * t_1)));
	tmp = ((zi * t_2) + ((xi * t_3) * cos(t_0))) + ((yi * t_3) * sin(t_0));
end

Derivation

1. Initial program 98.9%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-+.f32 N/A

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

3. Applied rewrites 98.9%

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

Alternative 5: 98.8% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (let* ((t_0 (* (* (- 1 ux) maxCos) ux)))
  (+
   (+
    (* (cos (* 2 (* uy PI))) xi)
    (* (* (sin (* (* uy 2) PI)) (sqrt (- 1 (* t_0 t_0)))) 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;
	return ((cosf((2.0f * (uy * ((float) M_PI)))) * xi) + ((sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((1.0f - (t_0 * t_0)))) * yi)) + (t_0 * zi);
}

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in ux around 0

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

   1. lower-cos.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 98.8%

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

4. Applied rewrites 98.8%

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

Alternative 6: 98.8% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (let* ((t_0 (* (+ uy uy) PI)))
  (-
   (+ (* (sin t_0) yi) (* (cos t_0) xi))
   (* (* (* zi ux) (- ux 1)) maxCos))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

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-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-+.f32 N/A

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

4. Applied rewrites 98.8%

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

6. Applied rewrites 98.8%

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

Alternative 7: 95.7% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (let* ((t_0 (* (+ uy uy) PI)))
  (+
   (+ (* (sin (- (* PI 1/2) t_0)) xi) (* (* zi ux) maxCos))
   (* (sin t_0) yi))))

C

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

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = (uy + uy) * single(pi);
	tmp = ((sin(((single(pi) * single(0.5)) - t_0)) * xi) + ((zi * ux) * maxCos)) + (sin(t_0) * yi);
end

Derivation

1. Initial program 98.9%

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

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-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-+.f32 N/A

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

4. Applied rewrites 95.7%

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

   1. lift-+.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. associate-+r+ N/A

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

   4. lower-+.f32 N/A

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

6. Applied rewrites 95.7%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

   3. lift-*.f32 N/A

      \[\leadsto \left(\cos \left(\mathsf{neg}\left(\left(uy + uy\right) \cdot \pi\right)\right) \cdot xi + \left(zi \cdot ux\right) \cdot maxCos\right) + \sin \left(\left(uy + uy\right) \cdot \pi\right) \cdot yi \]

   4. lift-+.f32 N/A

      \[\leadsto \left(\cos \left(\mathsf{neg}\left(\left(uy + uy\right) \cdot \pi\right)\right) \cdot xi + \left(zi \cdot ux\right) \cdot maxCos\right) + \sin \left(\left(uy + uy\right) \cdot \pi\right) \cdot yi \]

   5. count-2 N/A

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

   6. associate-*r* N/A

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

   7. lift-*.f32 N/A

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

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

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. distribute-rgt-neg-out N/A

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

   12. count-2 N/A

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

   13. distribute-rgt-in N/A

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

   14. lift-PI.f32 N/A

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

   15. lift-PI.f32 N/A

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

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

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

   17. lower-sin.f32 N/A

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

   18. +-commutative N/A

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

   19. fp-cancel-sub-sign N/A

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

8. Applied rewrites 95.7%

   \[\leadsto \left(\sin \left(\pi \cdot \frac{1}{2} - \left(uy + uy\right) \cdot \pi\right) \cdot xi + \left(zi \cdot ux\right) \cdot maxCos\right) + \sin \left(\color{blue}{\left(uy + uy\right)} \cdot \pi\right) \cdot yi \]
9. Add Preprocessing

Alternative 8: 95.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = (uy + uy) * single(pi);
	tmp = ((cos(t_0) * xi) + ((zi * ux) * maxCos)) + (sin(t_0) * yi);
end

Derivation

1. Initial program 98.9%

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

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-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-+.f32 N/A

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

4. Applied rewrites 95.7%

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

   1. lift-+.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. associate-+r+ N/A

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

   4. lower-+.f32 N/A

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

6. Applied rewrites 95.7%

   \[\leadsto \left(\cos \left(\left(uy + uy\right) \cdot \pi\right) \cdot xi + \left(zi \cdot ux\right) \cdot maxCos\right) + \color{blue}{\sin \left(\left(uy + uy\right) \cdot \pi\right) \cdot yi} \]
7. Add Preprocessing

Alternative 9: 89.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := maxCos \cdot \left(1 - ux\right)\\ \mathbf{if}\;uy \leq \frac{15182709}{1073741824}:\\ \;\;\;\;xi + \left(maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\pi}^{2}\right)\right) + 2 \cdot \left(yi \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{2} \cdot \frac{xi}{uy \cdot \left(yi \cdot \pi\right)}\right) \cdot \left(\left(yi \cdot \sqrt{1 - \left(\left(t\_0 \cdot ux\right) \cdot ux\right) \cdot t\_0}\right) \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\\ \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (let* ((t_0 (* maxCos (- 1 ux))))
  (if (<= uy 15182709/1073741824)
    (+
     xi
     (+
      (* maxCos (* ux (* zi (- 1 ux))))
      (* uy (+ (* -2 (* uy (* xi (pow PI 2)))) (* 2 (* yi PI))))))
    (+
     (*
      (+ 1 (* 1/2 (/ xi (* uy (* yi PI)))))
      (*
       (* yi (sqrt (- 1 (* (* (* t_0 ux) ux) t_0))))
       (sin (* PI (+ uy uy)))))
     (* (* (* (- 1 ux) maxCos) ux) zi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = maxCos * (1.0f - ux);
	float tmp;
	if (uy <= 0.014139999635517597f) {
		tmp = xi + ((maxCos * (ux * (zi * (1.0f - ux)))) + (uy * ((-2.0f * (uy * (xi * powf(((float) M_PI), 2.0f)))) + (2.0f * (yi * ((float) M_PI))))));
	} else {
		tmp = ((1.0f + (0.5f * (xi / (uy * (yi * ((float) M_PI)))))) * ((yi * sqrtf((1.0f - (((t_0 * ux) * ux) * t_0)))) * sinf((((float) M_PI) * (uy + uy))))) + ((((1.0f - ux) * maxCos) * ux) * zi);
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(maxCos * Float32(Float32(1.0) - ux))
	tmp = Float32(0.0)
	if (uy <= Float32(0.014139999635517597))
		tmp = Float32(xi + Float32(Float32(maxCos * Float32(ux * Float32(zi * Float32(Float32(1.0) - ux)))) + Float32(uy * Float32(Float32(Float32(-2.0) * Float32(uy * Float32(xi * (Float32(pi) ^ Float32(2.0))))) + Float32(Float32(2.0) * Float32(yi * Float32(pi)))))));
	else
		tmp = Float32(Float32(Float32(Float32(1.0) + Float32(Float32(0.5) * Float32(xi / Float32(uy * Float32(yi * Float32(pi)))))) * Float32(Float32(yi * sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(t_0 * ux) * ux) * t_0)))) * sin(Float32(Float32(pi) * Float32(uy + uy))))) + Float32(Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux) * zi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = maxCos * (single(1.0) - ux);
	tmp = single(0.0);
	if (uy <= single(0.014139999635517597))
		tmp = xi + ((maxCos * (ux * (zi * (single(1.0) - ux)))) + (uy * ((single(-2.0) * (uy * (xi * (single(pi) ^ single(2.0))))) + (single(2.0) * (yi * single(pi))))));
	else
		tmp = ((single(1.0) + (single(0.5) * (xi / (uy * (yi * single(pi)))))) * ((yi * sqrt((single(1.0) - (((t_0 * ux) * ux) * t_0)))) * sin((single(pi) * (uy + uy))))) + ((((single(1.0) - ux) * maxCos) * ux) * zi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0141399996

   1. Initial program 98.9%

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

   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-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-+.f32 N/A

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in uy around 0

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

      1. lower-+.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-+.f32 N/A

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

   7. Applied rewrites 86.1%

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

   if 0.0141399996 < uy

   1. Initial program 98.9%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult N/A

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

      4. lower-unsound-*.f32 N/A

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

   3. Applied rewrites 96.8%

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

   4. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-/.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-PI.f32 85.3%

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

   6. Applied rewrites 85.3%

      \[\leadsto \left(1 + \color{blue}{\frac{1}{2} \cdot \frac{xi}{uy \cdot \left(yi \cdot \pi\right)}}\right) \cdot \left(\left(yi \cdot \sqrt{1 - \left(\left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux\right) \cdot \left(maxCos \cdot \left(1 - ux\right)\right)}\right) \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 87.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := maxCos \cdot \left(1 - ux\right)\\ \mathbf{if}\;uy \leq \frac{15182709}{1073741824}:\\ \;\;\;\;xi + \left(maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\pi}^{2}\right)\right) + 2 \cdot \left(yi \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(yi \cdot \sqrt{1 - \left(\left(t\_0 \cdot ux\right) \cdot ux\right) \cdot t\_0}\right) \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\\ \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (let* ((t_0 (* maxCos (- 1 ux))))
  (if (<= uy 15182709/1073741824)
    (+
     xi
     (+
      (* maxCos (* ux (* zi (- 1 ux))))
      (* uy (+ (* -2 (* uy (* xi (pow PI 2)))) (* 2 (* yi PI))))))
    (+
     (*
      1
      (*
       (* yi (sqrt (- 1 (* (* (* t_0 ux) ux) t_0))))
       (sin (* PI (+ uy uy)))))
     (* (* (* (- 1 ux) maxCos) ux) zi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = maxCos * (1.0f - ux);
	float tmp;
	if (uy <= 0.014139999635517597f) {
		tmp = xi + ((maxCos * (ux * (zi * (1.0f - ux)))) + (uy * ((-2.0f * (uy * (xi * powf(((float) M_PI), 2.0f)))) + (2.0f * (yi * ((float) M_PI))))));
	} else {
		tmp = (1.0f * ((yi * sqrtf((1.0f - (((t_0 * ux) * ux) * t_0)))) * sinf((((float) M_PI) * (uy + uy))))) + ((((1.0f - ux) * maxCos) * ux) * zi);
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(maxCos * Float32(Float32(1.0) - ux))
	tmp = Float32(0.0)
	if (uy <= Float32(0.014139999635517597))
		tmp = Float32(xi + Float32(Float32(maxCos * Float32(ux * Float32(zi * Float32(Float32(1.0) - ux)))) + Float32(uy * Float32(Float32(Float32(-2.0) * Float32(uy * Float32(xi * (Float32(pi) ^ Float32(2.0))))) + Float32(Float32(2.0) * Float32(yi * Float32(pi)))))));
	else
		tmp = Float32(Float32(Float32(1.0) * Float32(Float32(yi * sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(t_0 * ux) * ux) * t_0)))) * sin(Float32(Float32(pi) * Float32(uy + uy))))) + Float32(Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux) * zi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = maxCos * (single(1.0) - ux);
	tmp = single(0.0);
	if (uy <= single(0.014139999635517597))
		tmp = xi + ((maxCos * (ux * (zi * (single(1.0) - ux)))) + (uy * ((single(-2.0) * (uy * (xi * (single(pi) ^ single(2.0))))) + (single(2.0) * (yi * single(pi))))));
	else
		tmp = (single(1.0) * ((yi * sqrt((single(1.0) - (((t_0 * ux) * ux) * t_0)))) * sin((single(pi) * (uy + uy))))) + ((((single(1.0) - ux) * maxCos) * ux) * zi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0141399996

   1. Initial program 98.9%

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

   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-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-+.f32 N/A

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in uy around 0

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

      1. lower-+.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-+.f32 N/A

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

   7. Applied rewrites 86.1%

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

   if 0.0141399996 < uy

   1. Initial program 98.9%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult N/A

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

      4. lower-unsound-*.f32 N/A

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

   3. Applied rewrites 96.8%

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

   4. Taylor expanded in xi around 0

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

   6. Applied rewrites 43.4%

      \[\leadsto \color{blue}{1} \cdot \left(\left(yi \cdot \sqrt{1 - \left(\left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux\right) \cdot \left(maxCos \cdot \left(1 - ux\right)\right)}\right) \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 87.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} t_0 := maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\\ \mathbf{if}\;uy \leq \frac{15182709}{1073741824}:\\ \;\;\;\;xi + \left(t\_0 + uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\pi}^{2}\right)\right) + 2 \cdot \left(yi \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (let* ((t_0 (* maxCos (* ux (* zi (- 1 ux))))))
  (if (<= uy 15182709/1073741824)
    (+
     xi
     (+
      t_0
      (* uy (+ (* -2 (* uy (* xi (pow PI 2)))) (* 2 (* yi PI))))))
    (+ t_0 (* yi (sin (* 2 (* uy PI))))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = maxCos * (ux * (zi * (1.0f - ux)));
	float tmp;
	if (uy <= 0.014139999635517597f) {
		tmp = xi + (t_0 + (uy * ((-2.0f * (uy * (xi * powf(((float) M_PI), 2.0f)))) + (2.0f * (yi * ((float) M_PI))))));
	} else {
		tmp = t_0 + (yi * sinf((2.0f * (uy * ((float) M_PI)))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(maxCos * Float32(ux * Float32(zi * Float32(Float32(1.0) - ux))))
	tmp = Float32(0.0)
	if (uy <= Float32(0.014139999635517597))
		tmp = Float32(xi + Float32(t_0 + Float32(uy * Float32(Float32(Float32(-2.0) * Float32(uy * Float32(xi * (Float32(pi) ^ Float32(2.0))))) + Float32(Float32(2.0) * Float32(yi * Float32(pi)))))));
	else
		tmp = Float32(t_0 + Float32(yi * sin(Float32(Float32(2.0) * Float32(uy * Float32(pi))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = maxCos * (ux * (zi * (single(1.0) - ux)));
	tmp = single(0.0);
	if (uy <= single(0.014139999635517597))
		tmp = xi + (t_0 + (uy * ((single(-2.0) * (uy * (xi * (single(pi) ^ single(2.0))))) + (single(2.0) * (yi * single(pi))))));
	else
		tmp = t_0 + (yi * sin((single(2.0) * (uy * single(pi)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0141399996

   1. Initial program 98.9%

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

   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-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-+.f32 N/A

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in uy around 0

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

      1. lower-+.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-+.f32 N/A

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

   7. Applied rewrites 86.1%

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

   if 0.0141399996 < uy

   1. Initial program 98.9%

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

   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-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-+.f32 N/A

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in xi around 0

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-sin.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 43.3%

          \[\leadsto maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \]

   7. Applied rewrites 43.3%

      \[\leadsto maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \color{blue}{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 85.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;uy \leq \frac{8589935}{268435456}:\\ \;\;\;\;\left(\cos \left(\left(uy + uy\right) \cdot \pi\right) \cdot xi + \left(zi \cdot ux\right) \cdot maxCos\right) + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (if (<= uy 8589935/268435456)
  (+
   (+ (* (cos (* (+ uy uy) PI)) xi) (* (* zi ux) maxCos))
   (* 2 (* uy (* yi PI))))
  (+ (* maxCos (* ux (* zi (- 1 ux)))) (* yi (sin (* 2 (* uy PI)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.03200000151991844f) {
		tmp = ((cosf(((uy + uy) * ((float) M_PI))) * xi) + ((zi * ux) * maxCos)) + (2.0f * (uy * (yi * ((float) M_PI))));
	} else {
		tmp = (maxCos * (ux * (zi * (1.0f - ux)))) + (yi * sinf((2.0f * (uy * ((float) M_PI)))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.03200000151991844))
		tmp = Float32(Float32(Float32(cos(Float32(Float32(uy + uy) * Float32(pi))) * xi) + Float32(Float32(zi * ux) * maxCos)) + Float32(Float32(2.0) * Float32(uy * Float32(yi * Float32(pi)))));
	else
		tmp = Float32(Float32(maxCos * Float32(ux * Float32(zi * Float32(Float32(1.0) - ux)))) + Float32(yi * sin(Float32(Float32(2.0) * Float32(uy * Float32(pi))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.03200000151991844))
		tmp = ((cos(((uy + uy) * single(pi))) * xi) + ((zi * ux) * maxCos)) + (single(2.0) * (uy * (yi * single(pi))));
	else
		tmp = (maxCos * (ux * (zi * (single(1.0) - ux)))) + (yi * sin((single(2.0) * (uy * single(pi)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0320000015

   1. Initial program 98.9%

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

   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-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-+.f32 N/A

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

   4. Applied rewrites 95.7%

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

      1. lift-+.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. associate-+r+ N/A

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

      4. lower-+.f32 N/A

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

   6. Applied rewrites 95.7%

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

   7. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 87.6%

         \[\leadsto \left(\cos \left(\left(uy + uy\right) \cdot \pi\right) \cdot xi + \left(zi \cdot ux\right) \cdot maxCos\right) + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) \]

   9. Applied rewrites 87.6%

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

   if 0.0320000015 < uy

   1. Initial program 98.9%

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

   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-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-+.f32 N/A

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in xi around 0

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-sin.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 43.3%

          \[\leadsto maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \]

   7. Applied rewrites 43.3%

      \[\leadsto maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \color{blue}{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 84.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} t_0 := maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\\ \mathbf{if}\;uy \leq \frac{15182709}{1073741824}:\\ \;\;\;\;xi + \left(2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) + t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (let* ((t_0 (* maxCos (* ux (* zi (- 1 ux))))))
  (if (<= uy 15182709/1073741824)
    (+ xi (+ (* 2 (* uy (* yi PI))) t_0))
    (+ t_0 (* yi (sin (* 2 (* uy PI))))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = maxCos * (ux * (zi * (1.0f - ux)));
	float tmp;
	if (uy <= 0.014139999635517597f) {
		tmp = xi + ((2.0f * (uy * (yi * ((float) M_PI)))) + t_0);
	} else {
		tmp = t_0 + (yi * sinf((2.0f * (uy * ((float) M_PI)))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(maxCos * Float32(ux * Float32(zi * Float32(Float32(1.0) - ux))))
	tmp = Float32(0.0)
	if (uy <= Float32(0.014139999635517597))
		tmp = Float32(xi + Float32(Float32(Float32(2.0) * Float32(uy * Float32(yi * Float32(pi)))) + t_0));
	else
		tmp = Float32(t_0 + Float32(yi * sin(Float32(Float32(2.0) * Float32(uy * Float32(pi))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = maxCos * (ux * (zi * (single(1.0) - ux)));
	tmp = single(0.0);
	if (uy <= single(0.014139999635517597))
		tmp = xi + ((single(2.0) * (uy * (yi * single(pi)))) + t_0);
	else
		tmp = t_0 + (yi * sin((single(2.0) * (uy * single(pi)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0141399996

   1. Initial program 98.9%

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

   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-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-+.f32 N/A

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in uy around 0

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

      1. lower-+.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 N/A

         \[\leadsto xi + \left(2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

      7. lower-*.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower--.f32 82.0%

          \[\leadsto xi + \left(2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   7. Applied rewrites 82.0%

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

   if 0.0141399996 < uy

   1. Initial program 98.9%

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

   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-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-+.f32 N/A

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in xi around 0

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-sin.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 43.3%

          \[\leadsto maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \]

   7. Applied rewrites 43.3%

      \[\leadsto maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \color{blue}{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 84.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;uy \leq \frac{15182709}{1073741824}:\\ \;\;\;\;xi + \left(2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;maxCos \cdot \left(ux \cdot zi\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (if (<= uy 15182709/1073741824)
  (+ xi (+ (* 2 (* uy (* yi PI))) (* maxCos (* ux (* zi (- 1 ux))))))
  (+ (* maxCos (* ux zi)) (* yi (sin (* 2 (* uy PI)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.014139999635517597f) {
		tmp = xi + ((2.0f * (uy * (yi * ((float) M_PI)))) + (maxCos * (ux * (zi * (1.0f - ux)))));
	} else {
		tmp = (maxCos * (ux * zi)) + (yi * sinf((2.0f * (uy * ((float) M_PI)))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.014139999635517597))
		tmp = Float32(xi + Float32(Float32(Float32(2.0) * Float32(uy * Float32(yi * Float32(pi)))) + Float32(maxCos * Float32(ux * Float32(zi * Float32(Float32(1.0) - ux))))));
	else
		tmp = Float32(Float32(maxCos * Float32(ux * zi)) + Float32(yi * sin(Float32(Float32(2.0) * Float32(uy * Float32(pi))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.014139999635517597))
		tmp = xi + ((single(2.0) * (uy * (yi * single(pi)))) + (maxCos * (ux * (zi * (single(1.0) - ux)))));
	else
		tmp = (maxCos * (ux * zi)) + (yi * sin((single(2.0) * (uy * single(pi)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0141399996

   1. Initial program 98.9%

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

   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-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-+.f32 N/A

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in uy around 0

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

      1. lower-+.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 N/A

         \[\leadsto xi + \left(2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

      7. lower-*.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower--.f32 82.0%

          \[\leadsto xi + \left(2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   7. Applied rewrites 82.0%

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

   if 0.0141399996 < uy

   1. Initial program 98.9%

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

   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-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-+.f32 N/A

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

   4. Applied rewrites 95.7%

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

   5. Taylor expanded in xi around 0

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

      1. lower-*.f32 N/A

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

      2. lower-sin.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-PI.f32 41.3%

         \[\leadsto maxCos \cdot \left(ux \cdot zi\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \]

   7. Applied rewrites 41.3%

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) + yi \cdot \color{blue}{\sin \left(2 \cdot \left(uy \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 82.0% accurate, 8.8× speedup

Math

\[xi + \left(2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (+ xi (+ (* 2 (* uy (* yi PI))) (* maxCos (* ux (* zi (- 1 ux)))))))

C

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

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = xi + ((single(2.0) * (uy * (yi * single(pi)))) + (maxCos * (ux * (zi * (single(1.0) - ux)))));
end

Derivation

1. Initial program 98.9%

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

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-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-+.f32 N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in uy around 0

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

   1. lower-+.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 N/A

      \[\leadsto xi + \left(2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower--.f32 82.0%

       \[\leadsto xi + \left(2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

7. Applied rewrites 82.0%

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

Alternative 16: 79.2% accurate, 11.0× speedup

Math

\[maxCos \cdot \left(ux \cdot zi\right) + \left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (+ (* maxCos (* ux zi)) (+ xi (* 2 (* uy (* yi PI))))))

C

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

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = (maxCos * (ux * zi)) + (xi + (single(2.0) * (uy * (yi * single(pi)))));
end

Derivation

1. Initial program 98.9%

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

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-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-+.f32 N/A

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

4. Applied rewrites 95.7%

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

5. Taylor expanded in uy around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 79.2%

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) + \left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right) \]

7. Applied rewrites 79.2%

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

Alternative 17: 79.2% accurate, 11.0× speedup

Math

\[xi + \left(2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) + maxCos \cdot \left(ux \cdot zi\right)\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (+ xi (+ (* 2 (* uy (* yi PI))) (* maxCos (* ux zi)))))

C

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

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = xi + ((single(2.0) * (uy * (yi * single(pi)))) + (maxCos * (ux * zi)));
end

Derivation

1. Initial program 98.9%

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

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-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-+.f32 N/A

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

4. Applied rewrites 95.7%

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

5. Taylor expanded in uy around 0

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
6. 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 50.4%

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

7. Applied rewrites 50.4%

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

8. Taylor expanded in xi around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 12.0%

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

10. Applied rewrites 12.0%

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

11. Taylor expanded in uy around 0

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

    1. lower-+.f32 N/A

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

    2. lower-+.f32 N/A

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

    3. lower-*.f32 N/A

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

    4. lower-*.f32 N/A

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

    5. lower-*.f32 N/A

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

    6. lower-PI.f32 N/A

       \[\leadsto xi + \left(2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) + maxCos \cdot \left(ux \cdot zi\right)\right) \]

    7. lower-*.f32 N/A

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

    8. lower-*.f32 79.2%

       \[\leadsto xi + \left(2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) + maxCos \cdot \left(ux \cdot zi\right)\right) \]

13. Applied rewrites 79.2%

    \[\leadsto xi + \color{blue}{\left(2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) + maxCos \cdot \left(ux \cdot zi\right)\right)} \]
14. Add Preprocessing

Alternative 18: 52.5% accurate, 16.0× speedup

Math

\[xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (+ xi (* maxCos (* ux (* zi (- 1 ux))))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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 * (1.0e0 - ux))))
end function

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = xi + (maxCos * (ux * (zi * (single(1.0) - ux))));
end

Derivation

1. Initial program 98.9%

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

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-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-+.f32 N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in uy around 0

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto xi + maxCos \cdot \color{blue}{\left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right)\right) \]

   4. lower-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right)\right) \]

   5. lower--.f32 52.5%

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) \]

7. Applied rewrites 52.5%

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]
8. Add Preprocessing

Alternative 19: 50.4% accurate, 25.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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 * zi) * ux)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

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-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-+.f32 N/A

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

4. Applied rewrites 95.7%

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

5. Taylor expanded in uy around 0

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
6. 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 50.4%

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

7. Applied rewrites 50.4%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 50.4%

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

9. Applied rewrites 50.4%

   \[\leadsto xi + \left(maxCos \cdot zi\right) \cdot ux \]
10. Add Preprocessing

Alternative 20: 50.4% accurate, 25.2× 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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 98.9%

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

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-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-+.f32 N/A

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

4. Applied rewrites 95.7%

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

5. Taylor expanded in uy around 0

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
6. 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 50.4%

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

7. Applied rewrites 50.4%

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

Alternative 21: 12.0% accurate, 32.1× 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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 98.9%

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

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-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-+.f32 N/A

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

4. Applied rewrites 95.7%

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

5. Taylor expanded in uy around 0

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
6. 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 50.4%

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

7. Applied rewrites 50.4%

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

8. Taylor expanded in xi around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 12.0%

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

10. Applied rewrites 12.0%

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

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore (xi yi zi ux uy maxCos)
  :name "UniformSampleCone 2"
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000 xi) (<= xi 10000)) (and (<= -10000 yi) (<= yi 10000))) (and (<= -10000 zi) (<= zi 10000))) (and (<= 2328306437/10000000000000000000 ux) (<= ux 1))) (and (<= 2328306437/10000000000000000000 uy) (<= uy 1))) (and (<= 0 maxCos) (<= maxCos 1)))
  (+ (+ (* (* (cos (* (* uy 2) PI)) (sqrt (- 1 (* (* (* (- 1 ux) maxCos) ux) (* (* (- 1 ux) maxCos) ux))))) xi) (* (* (sin (* (* uy 2) PI)) (sqrt (- 1 (* (* (* (- 1 ux) maxCos) ux) (* (* (- 1 ux) maxCos) ux))))) yi)) (* (* (* (- 1 ux) maxCos) ux) zi)))