UniformSampleCone 2

Percentage Accurate: 99.0% → 99.0%

Time: 8.9s

Alternatives: 20

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux))
        (t_1 (sqrt (- 1.0 (* t_0 t_0))))
        (t_2 (* (* uy 2.0) PI)))
   (+
    (+ (* (* (cos t_2) t_1) xi) (* (* (sin t_2) t_1) yi))
    (* (* (fma (- maxCos) ux maxCos) ux) 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)) + ((fmaf(-maxCos, ux, maxCos) * ux) * 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(Float32(fma(Float32(-maxCos), ux, maxCos) * ux) * zi))
end

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in ux around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f32 98.7

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

5. Applied rewrites 98.7%

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

Alternative 2: 98.9% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux)))
   (+
    (+
     (* (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* t_0 t_0)))) xi)
     (* (sin (* PI (* 2.0 uy))) 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(((uy * 2.0f) * ((float) M_PI))) * sqrtf((1.0f - (t_0 * t_0)))) * xi) + (sinf((((float) M_PI) * (2.0f * uy))) * 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(Float32(cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))) * xi) + Float32(sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) * 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(((uy * single(2.0)) * single(pi))) * sqrt((single(1.0) - (t_0 * t_0)))) * xi) + (sin((single(pi) * (single(2.0) * uy))) * yi)) + (t_0 * zi);
end

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in ux around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-sin.f32 98.7

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

   7. lift-PI.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lift-PI.f32 98.7

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 98.7

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

5. Applied rewrites 98.7%

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

Alternative 3: 98.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in maxCos around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift--.f32 N/A

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

   7. *-commutative N/A

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

5. Applied rewrites 98.6%

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

Alternative 4: 95.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. cos-2 N/A

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

   8. lower--.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-cos.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lift-PI.f32 N/A

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

   14. lower-cos.f32 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 N/A

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

   17. lift-PI.f32 N/A

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

   18. lower-*.f32 N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 95.6%

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

Alternative 5: 95.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in ux around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

5. Applied rewrites 95.6%

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

Alternative 6: 96.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(2 \cdot uy\right)\\ \mathbf{if}\;uy \leq 0.014999999664723873:\\ \;\;\;\;\left(1 \cdot \left(1 + uy \cdot \mathsf{fma}\left(2, \frac{yi \cdot \pi}{xi}, uy \cdot \mathsf{fma}\left(-2, \pi \cdot \pi, -1.3333333333333333 \cdot \frac{uy \cdot \left(yi \cdot {\pi}^{3}\right)}{xi}\right)\right)\right)\right) \cdot xi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos t\_0, xi, \sin t\_0 \cdot yi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* PI (* 2.0 uy))))
   (if (<= uy 0.014999999664723873)
     (+
      (*
       (*
        1.0
        (+
         1.0
         (*
          uy
          (fma
           2.0
           (/ (* yi PI) xi)
           (*
            uy
            (fma
             -2.0
             (* PI PI)
             (* -1.3333333333333333 (/ (* uy (* yi (pow PI 3.0))) xi))))))))
       xi)
      (* (* (* (- 1.0 ux) maxCos) ux) zi))
     (fma (cos t_0) xi (* (sin t_0) yi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((float) M_PI) * (2.0f * uy);
	float tmp;
	if (uy <= 0.014999999664723873f) {
		tmp = ((1.0f * (1.0f + (uy * fmaf(2.0f, ((yi * ((float) M_PI)) / xi), (uy * fmaf(-2.0f, (((float) M_PI) * ((float) M_PI)), (-1.3333333333333333f * ((uy * (yi * powf(((float) M_PI), 3.0f))) / xi)))))))) * xi) + ((((1.0f - ux) * maxCos) * ux) * zi);
	} else {
		tmp = fmaf(cosf(t_0), xi, (sinf(t_0) * yi));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(Float32(2.0) * uy))
	tmp = Float32(0.0)
	if (uy <= Float32(0.014999999664723873))
		tmp = Float32(Float32(Float32(Float32(1.0) * Float32(Float32(1.0) + Float32(uy * fma(Float32(2.0), Float32(Float32(yi * Float32(pi)) / xi), Float32(uy * fma(Float32(-2.0), Float32(Float32(pi) * Float32(pi)), Float32(Float32(-1.3333333333333333) * Float32(Float32(uy * Float32(yi * (Float32(pi) ^ Float32(3.0)))) / xi)))))))) * xi) + Float32(Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux) * zi));
	else
		tmp = fma(cos(t_0), xi, Float32(sin(t_0) * yi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0149999997

   1. Initial program 99.0%

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

   3. Taylor expanded in xi around inf

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in ux around 0

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

   8. Applied rewrites 98.7%

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

   9. Taylor expanded in uy around 0

      \[\leadsto \left(1 \cdot \left(1 + uy \cdot \left(2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi} + uy \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-4}{3} \cdot \frac{uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)}{xi}\right)\right)\right)\right) \cdot xi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   10. Step-by-step derivation

       1. lower-+.f32 N/A

          \[\leadsto \left(1 \cdot \left(1 + uy \cdot \left(2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi} + uy \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-4}{3} \cdot \frac{uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)}{xi}\right)\right)\right)\right) \cdot xi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

       2. lower-*.f32 N/A

          \[\leadsto \left(1 \cdot \left(1 + uy \cdot \left(2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi} + uy \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-4}{3} \cdot \frac{uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)}{xi}\right)\right)\right)\right) \cdot xi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

       3. lower-fma.f32 N/A

          \[\leadsto \left(1 \cdot \left(1 + uy \cdot \mathsf{fma}\left(2, \frac{yi \cdot \mathsf{PI}\left(\right)}{xi}, uy \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-4}{3} \cdot \frac{uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)}{xi}\right)\right)\right)\right) \cdot xi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

       4. lower-/.f32 N/A

          \[\leadsto \left(1 \cdot \left(1 + uy \cdot \mathsf{fma}\left(2, \frac{yi \cdot \mathsf{PI}\left(\right)}{xi}, uy \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-4}{3} \cdot \frac{uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)}{xi}\right)\right)\right)\right) \cdot xi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

       5. lower-*.f32 N/A

          \[\leadsto \left(1 \cdot \left(1 + uy \cdot \mathsf{fma}\left(2, \frac{yi \cdot \mathsf{PI}\left(\right)}{xi}, uy \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-4}{3} \cdot \frac{uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)}{xi}\right)\right)\right)\right) \cdot xi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

       6. lift-PI.f32 N/A

          \[\leadsto \left(1 \cdot \left(1 + uy \cdot \mathsf{fma}\left(2, \frac{yi \cdot \pi}{xi}, uy \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-4}{3} \cdot \frac{uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)}{xi}\right)\right)\right)\right) \cdot xi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

       7. lower-*.f32 N/A

          \[\leadsto \left(1 \cdot \left(1 + uy \cdot \mathsf{fma}\left(2, \frac{yi \cdot \pi}{xi}, uy \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-4}{3} \cdot \frac{uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)}{xi}\right)\right)\right)\right) \cdot xi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   11. Applied rewrites 97.8%

       \[\leadsto \left(1 \cdot \left(1 + uy \cdot \mathsf{fma}\left(2, \frac{yi \cdot \pi}{xi}, uy \cdot \mathsf{fma}\left(-2, \pi \cdot \pi, -1.3333333333333333 \cdot \frac{uy \cdot \left(yi \cdot {\pi}^{3}\right)}{xi}\right)\right)\right)\right) \cdot xi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   if 0.0149999997 < uy

   1. Initial program 97.4%

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

   3. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

   5. Applied rewrites 90.0%

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

Alternative 7: 92.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\\ t_1 := t\_0 \cdot zi\\ t_2 := \sqrt{1 - t\_0 \cdot t\_0}\\ \mathbf{if}\;yi \leq 7.399999700872703 \cdot 10^{-26}:\\ \;\;\;\;\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot t\_2\right) \cdot xi + \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 \cdot t\_2\right) \cdot xi + \sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot yi\right) + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux))
        (t_1 (* t_0 zi))
        (t_2 (sqrt (- 1.0 (* t_0 t_0)))))
   (if (<= yi 7.399999700872703e-26)
     (+
      (+ (* (* (cos (* (* uy 2.0) PI)) t_2) xi) (* (* 2.0 (* uy PI)) yi))
      t_1)
     (+ (+ (* (* 1.0 t_2) xi) (* (sin (* PI (* 2.0 uy))) yi)) t_1))))

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 = t_0 * zi;
	float t_2 = sqrtf((1.0f - (t_0 * t_0)));
	float tmp;
	if (yi <= 7.399999700872703e-26f) {
		tmp = (((cosf(((uy * 2.0f) * ((float) M_PI))) * t_2) * xi) + ((2.0f * (uy * ((float) M_PI))) * yi)) + t_1;
	} else {
		tmp = (((1.0f * t_2) * xi) + (sinf((((float) M_PI) * (2.0f * uy))) * yi)) + t_1;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux)
	t_1 = Float32(t_0 * zi)
	t_2 = sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))
	tmp = Float32(0.0)
	if (yi <= Float32(7.399999700872703e-26))
		tmp = Float32(Float32(Float32(Float32(cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * t_2) * xi) + Float32(Float32(Float32(2.0) * Float32(uy * Float32(pi))) * yi)) + t_1);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(1.0) * t_2) * xi) + Float32(sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) * yi)) + t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = ((single(1.0) - ux) * maxCos) * ux;
	t_1 = t_0 * zi;
	t_2 = sqrt((single(1.0) - (t_0 * t_0)));
	tmp = single(0.0);
	if (yi <= single(7.399999700872703e-26))
		tmp = (((cos(((uy * single(2.0)) * single(pi))) * t_2) * xi) + ((single(2.0) * (uy * single(pi))) * yi)) + t_1;
	else
		tmp = (((single(1.0) * t_2) * xi) + (sin((single(pi) * (single(2.0) * uy))) * yi)) + t_1;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if yi < 7.3999997e-26

   1. Initial program 98.8%

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

   3. Taylor expanded in ux around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-sin.f32 98.8

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

      7. lift-PI.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lift-PI.f32 98.8

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in uy around 0

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

      1. lift-*.f32 N/A

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

      2. lift-PI.f32 N/A

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

      3. lift-*.f32 92.8

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

   8. Applied rewrites 92.8%

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

   if 7.3999997e-26 < yi

   1. Initial program 98.6%

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

   3. Taylor expanded in ux around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-sin.f32 98.6

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

      7. lift-PI.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lift-PI.f32 98.6

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 94.2%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;yi \leq 7.399999700872703 \cdot 10^{-26}:\\ \;\;\;\;\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(2 \cdot uy\right)\\ t_1 := \left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\\ t_2 := t\_1 \cdot zi\\ \mathbf{if}\;yi \leq 7.399999700872703 \cdot 10^{-26}:\\ \;\;\;\;\left(1 \cdot \left(\cos t\_0 + yi \cdot \frac{2 \cdot \left(uy \cdot \pi\right)}{xi}\right)\right) \cdot xi + t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 \cdot \sqrt{1 - t\_1 \cdot t\_1}\right) \cdot xi + \sin t\_0 \cdot yi\right) + t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* PI (* 2.0 uy)))
        (t_1 (* (* (- 1.0 ux) maxCos) ux))
        (t_2 (* t_1 zi)))
   (if (<= yi 7.399999700872703e-26)
     (+ (* (* 1.0 (+ (cos t_0) (* yi (/ (* 2.0 (* uy PI)) xi)))) xi) t_2)
     (+ (+ (* (* 1.0 (sqrt (- 1.0 (* t_1 t_1)))) xi) (* (sin t_0) yi)) t_2))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((float) M_PI) * (2.0f * uy);
	float t_1 = ((1.0f - ux) * maxCos) * ux;
	float t_2 = t_1 * zi;
	float tmp;
	if (yi <= 7.399999700872703e-26f) {
		tmp = ((1.0f * (cosf(t_0) + (yi * ((2.0f * (uy * ((float) M_PI))) / xi)))) * xi) + t_2;
	} else {
		tmp = (((1.0f * sqrtf((1.0f - (t_1 * t_1)))) * xi) + (sinf(t_0) * yi)) + t_2;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(Float32(2.0) * uy))
	t_1 = Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux)
	t_2 = Float32(t_1 * zi)
	tmp = Float32(0.0)
	if (yi <= Float32(7.399999700872703e-26))
		tmp = Float32(Float32(Float32(Float32(1.0) * Float32(cos(t_0) + Float32(yi * Float32(Float32(Float32(2.0) * Float32(uy * Float32(pi))) / xi)))) * xi) + t_2);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(1.0) * sqrt(Float32(Float32(1.0) - Float32(t_1 * t_1)))) * xi) + Float32(sin(t_0) * yi)) + t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = single(pi) * (single(2.0) * uy);
	t_1 = ((single(1.0) - ux) * maxCos) * ux;
	t_2 = t_1 * zi;
	tmp = single(0.0);
	if (yi <= single(7.399999700872703e-26))
		tmp = ((single(1.0) * (cos(t_0) + (yi * ((single(2.0) * (uy * single(pi))) / xi)))) * xi) + t_2;
	else
		tmp = (((single(1.0) * sqrt((single(1.0) - (t_1 * t_1)))) * xi) + (sin(t_0) * yi)) + t_2;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if yi < 7.3999997e-26

   1. Initial program 98.8%

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

   3. Taylor expanded in xi around inf

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in ux around 0

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

   8. Applied rewrites 98.5%

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

   9. Taylor expanded in uy around 0

      \[\leadsto \left(1 \cdot \left(\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) + yi \cdot \frac{2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)}{xi}\right)\right) \cdot xi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   10. Step-by-step derivation

       1. lift-*.f32 N/A

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

       2. lift-PI.f32 N/A

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

       3. lift-*.f32 92.6

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

   11. Applied rewrites 92.6%

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

   if 7.3999997e-26 < yi

   1. Initial program 98.6%

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

   3. Taylor expanded in ux around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-sin.f32 98.6

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

      7. lift-PI.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lift-PI.f32 98.6

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 94.2%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;yi \leq 7.399999700872703 \cdot 10^{-26}:\\ \;\;\;\;\left(1 \cdot \left(\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) + yi \cdot \frac{2 \cdot \left(uy \cdot \pi\right)}{xi}\right)\right) \cdot xi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(2 \cdot uy\right)\\ t_1 := \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\\ \mathbf{if}\;yi \leq 3.99999987306209 \cdot 10^{-20}:\\ \;\;\;\;\left(1 \cdot \left(\cos t\_0 + yi \cdot \frac{2 \cdot \left(uy \cdot \pi\right)}{xi}\right)\right) \cdot xi + t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot \left(1 + yi \cdot \frac{\sin t\_0}{xi}\right)\right) \cdot xi + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* PI (* 2.0 uy))) (t_1 (* (* (* (- 1.0 ux) maxCos) ux) zi)))
   (if (<= yi 3.99999987306209e-20)
     (+ (* (* 1.0 (+ (cos t_0) (* yi (/ (* 2.0 (* uy PI)) xi)))) xi) t_1)
     (+ (* (* 1.0 (+ 1.0 (* yi (/ (sin t_0) xi)))) xi) t_1))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((float) M_PI) * (2.0f * uy);
	float t_1 = (((1.0f - ux) * maxCos) * ux) * zi;
	float tmp;
	if (yi <= 3.99999987306209e-20f) {
		tmp = ((1.0f * (cosf(t_0) + (yi * ((2.0f * (uy * ((float) M_PI))) / xi)))) * xi) + t_1;
	} else {
		tmp = ((1.0f * (1.0f + (yi * (sinf(t_0) / xi)))) * xi) + t_1;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(Float32(2.0) * uy))
	t_1 = Float32(Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux) * zi)
	tmp = Float32(0.0)
	if (yi <= Float32(3.99999987306209e-20))
		tmp = Float32(Float32(Float32(Float32(1.0) * Float32(cos(t_0) + Float32(yi * Float32(Float32(Float32(2.0) * Float32(uy * Float32(pi))) / xi)))) * xi) + t_1);
	else
		tmp = Float32(Float32(Float32(Float32(1.0) * Float32(Float32(1.0) + Float32(yi * Float32(sin(t_0) / xi)))) * xi) + t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = single(pi) * (single(2.0) * uy);
	t_1 = (((single(1.0) - ux) * maxCos) * ux) * zi;
	tmp = single(0.0);
	if (yi <= single(3.99999987306209e-20))
		tmp = ((single(1.0) * (cos(t_0) + (yi * ((single(2.0) * (uy * single(pi))) / xi)))) * xi) + t_1;
	else
		tmp = ((single(1.0) * (single(1.0) + (yi * (sin(t_0) / xi)))) * xi) + t_1;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if yi < 3.99999987e-20

   1. Initial program 98.7%

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

   3. Taylor expanded in xi around inf

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in ux around 0

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

   8. Applied rewrites 98.4%

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

   9. Taylor expanded in uy around 0

      \[\leadsto \left(1 \cdot \left(\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) + yi \cdot \frac{2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)}{xi}\right)\right) \cdot xi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   10. Step-by-step derivation

       1. lift-*.f32 N/A

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

       2. lift-PI.f32 N/A

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

       3. lift-*.f32 92.2

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

   11. Applied rewrites 92.2%

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

   if 3.99999987e-20 < yi

   1. Initial program 98.6%

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

   3. Taylor expanded in xi around inf

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in ux around 0

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

   8. Applied rewrites 98.7%

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

   9. Taylor expanded in uy around 0

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

   11. Applied rewrites 95.6%

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

Alternative 10: 88.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in xi around inf

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

5. Applied rewrites 98.6%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 98.5%

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

9. Taylor expanded in uy around 0

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

11. Applied rewrites 89.0%

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

Alternative 11: 85.8% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in xi around inf

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

5. Applied rewrites 98.6%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 98.5%

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

9. Taylor expanded in uy around 0

   \[\leadsto \left(1 \cdot \left(1 + uy \cdot \left(-2 \cdot \left(uy \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi}\right)\right)\right) \cdot xi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
10. Step-by-step derivation

    1. lower-+.f32 N/A

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

    2. lower-*.f32 N/A

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

    3. lower-fma.f32 N/A

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

    4. lower-*.f32 N/A

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

    5. unpow2 N/A

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

    6. lower-*.f32 N/A

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

    7. lift-PI.f32 N/A

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

    8. lift-PI.f32 N/A

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

    9. lower-*.f32 N/A

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

    10. lower-/.f32 N/A

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

    11. lower-*.f32 N/A

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

    12. lift-PI.f32 84.7

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

11. Applied rewrites 84.7%

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

Alternative 12: 82.3% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in xi around inf

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

5. Applied rewrites 98.6%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 98.5%

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

9. Taylor expanded in uy around 0

   \[\leadsto \left(1 \cdot \left(1 + 2 \cdot \frac{uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)}{xi}\right)\right) \cdot xi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
10. Step-by-step derivation

    1. lower-+.f32 N/A

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

    2. lower-*.f32 N/A

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

    3. lower-/.f32 N/A

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

    4. lower-*.f32 N/A

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

    5. lower-*.f32 N/A

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

    6. lift-PI.f32 82.0

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

11. Applied rewrites 82.0%

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

12. Taylor expanded in maxCos around 0

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

    1. lower-*.f32 N/A

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

    2. lower-*.f32 N/A

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

    3. lift--.f32 82.0

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

14. Applied rewrites 82.0%

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

Alternative 13: 82.4% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in xi around inf

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

5. Applied rewrites 98.6%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 98.5%

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

9. Taylor expanded in uy around 0

   \[\leadsto \left(1 \cdot \left(1 + 2 \cdot \frac{uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)}{xi}\right)\right) \cdot xi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
10. Step-by-step derivation

    1. lower-+.f32 N/A

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

    2. lower-*.f32 N/A

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

    3. lower-/.f32 N/A

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

    4. lower-*.f32 N/A

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

    5. lower-*.f32 N/A

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

    6. lift-PI.f32 82.0

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

11. Applied rewrites 82.0%

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

12. Taylor expanded in xi around 0

    \[\leadsto \left(1 \cdot \frac{xi + 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
13. Step-by-step derivation

    1. lower-/.f32 N/A

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

    2. lower-+.f32 N/A

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

    3. lower-*.f32 N/A

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

    4. lift-*.f32 N/A

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

    5. lift-PI.f32 N/A

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

    6. lift-*.f32 82.0

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

14. Applied rewrites 82.0%

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

Alternative 14: 79.7% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \left(1 \cdot \left(1 + 2 \cdot \frac{uy \cdot \left(yi \cdot \pi\right)}{xi}\right)\right) \cdot xi + \left(maxCos \cdot ux\right) \cdot zi \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in xi around inf

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

5. Applied rewrites 98.6%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 98.5%

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

9. Taylor expanded in uy around 0

   \[\leadsto \left(1 \cdot \left(1 + 2 \cdot \frac{uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)}{xi}\right)\right) \cdot xi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
10. Step-by-step derivation

    1. lower-+.f32 N/A

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

    2. lower-*.f32 N/A

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

    3. lower-/.f32 N/A

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

    4. lower-*.f32 N/A

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

    5. lower-*.f32 N/A

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

    6. lift-PI.f32 82.0

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

11. Applied rewrites 82.0%

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

12. Taylor expanded in ux around 0

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

14. Applied rewrites 79.2%

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

Alternative 15: 79.7% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \left(1 \cdot \left(1 + 2 \cdot \frac{uy \cdot \left(yi \cdot \pi\right)}{xi}\right)\right) \cdot xi + maxCos \cdot \left(ux \cdot zi\right) \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in xi around inf

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

5. Applied rewrites 98.6%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 98.5%

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

9. Taylor expanded in uy around 0

   \[\leadsto \left(1 \cdot \left(1 + 2 \cdot \frac{uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)}{xi}\right)\right) \cdot xi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
10. Step-by-step derivation

    1. lower-+.f32 N/A

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

    2. lower-*.f32 N/A

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

    3. lower-/.f32 N/A

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

    4. lower-*.f32 N/A

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

    5. lower-*.f32 N/A

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

    6. lift-PI.f32 82.0

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

11. Applied rewrites 82.0%

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

12. Taylor expanded in ux around 0

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

    1. lower-*.f32 N/A

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

    2. lower-*.f32 79.2

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

14. Applied rewrites 79.2%

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

Alternative 16: 51.8% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ xi + ux \cdot \mathsf{fma}\left(maxCos, zi, ux \cdot \mathsf{fma}\left(-1, maxCos \cdot zi, -0.5 \cdot \left(\left(maxCos \cdot maxCos\right) \cdot xi\right)\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (+
  xi
  (*
   ux
   (fma
    maxCos
    zi
    (* ux (fma -1.0 (* maxCos zi) (* -0.5 (* (* maxCos maxCos) xi))))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return xi + (ux * fmaf(maxCos, zi, (ux * fmaf(-1.0f, (maxCos * zi), (-0.5f * ((maxCos * maxCos) * xi))))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(xi + Float32(ux * fma(maxCos, zi, Float32(ux * fma(Float32(-1.0), Float32(maxCos * zi), Float32(Float32(-0.5) * Float32(Float32(maxCos * maxCos) * xi)))))))
end

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in ux around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f32 98.7

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

5. Applied rewrites 98.7%

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

6. Taylor expanded in uy around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lift--.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-sqrt.f32 N/A

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

   7. lower--.f32 N/A

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

8. Applied rewrites 50.1%

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

9. Taylor expanded in ux around 0

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

    1. lower-+.f32 N/A

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

    2. lower-*.f32 N/A

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

    3. lower-fma.f32 N/A

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

    4. lower-*.f32 N/A

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

    5. lower-fma.f32 N/A

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

    6. lower-*.f32 N/A

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

    7. lower-*.f32 N/A

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

    8. lower-*.f32 N/A

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

    9. pow2 N/A

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

    10. lift-*.f32 50.0

        \[\leadsto xi + ux \cdot \mathsf{fma}\left(maxCos, zi, ux \cdot \mathsf{fma}\left(-1, maxCos \cdot zi, -0.5 \cdot \left(\left(maxCos \cdot maxCos\right) \cdot xi\right)\right)\right) \]

11. Applied rewrites 50.0%

    \[\leadsto xi + \color{blue}{ux \cdot \mathsf{fma}\left(maxCos, zi, ux \cdot \mathsf{fma}\left(-1, maxCos \cdot zi, -0.5 \cdot \left(\left(maxCos \cdot maxCos\right) \cdot xi\right)\right)\right)} \]
12. Add Preprocessing

Alternative 17: 51.7% accurate, 16.0× speedup

Math

\[\begin{array}{l} \\ xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (+ xi (* maxCos (* ux (* zi (- 1.0 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 98.7%

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

3. Taylor expanded in ux around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f32 98.7

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

5. Applied rewrites 98.7%

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

6. Taylor expanded in uy around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lift--.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-sqrt.f32 N/A

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

   7. lower--.f32 N/A

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

8. Applied rewrites 50.1%

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

9. Taylor expanded in maxCos around 0

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]
10. 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. lift-*.f32 N/A

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

    4. lift--.f32 N/A

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

    5. lift-*.f32 50.0

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

11. Applied rewrites 50.0%

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

Alternative 18: 51.7% accurate, 17.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi\right) \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in ux around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f32 98.7

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

5. Applied rewrites 98.7%

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

6. Taylor expanded in uy around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lift--.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-sqrt.f32 N/A

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

   7. lower--.f32 N/A

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

8. Applied rewrites 50.1%

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

9. Taylor expanded in ux around 0

   \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi\right) \]
10. Step-by-step derivation

11. Applied rewrites 49.9%

    \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi\right) \]
12. Add Preprocessing

Alternative 19: 49.7% accurate, 25.2× speedup

Math

\[\begin{array}{l} \\ xi + maxCos \cdot \left(ux \cdot zi\right) \end{array} \]

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 98.7%

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

3. Taylor expanded in ux around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f32 98.7

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

5. Applied rewrites 98.7%

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

6. Taylor expanded in uy around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lift--.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-sqrt.f32 N/A

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

   7. lower--.f32 N/A

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

8. Applied rewrites 50.1%

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

9. Taylor expanded in ux around 0

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
10. 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 47.8

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

11. Applied rewrites 47.8%

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

Alternative 20: 45.6% accurate, 353.0× speedup

Math

\[\begin{array}{l} \\ xi \end{array} \]

FPCore

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

C

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

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
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in ux around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f32 98.7

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

5. Applied rewrites 98.7%

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

6. Taylor expanded in uy around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lift--.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-sqrt.f32 N/A

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

   7. lower--.f32 N/A

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

8. Applied rewrites 50.1%

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

9. Taylor expanded in ux around 0

   \[\leadsto xi \]
10. Step-by-step derivation

11. Applied rewrites 42.9%

    \[\leadsto xi \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025085 
(FPCore (xi yi zi ux uy maxCos)
  :name "UniformSampleCone 2"
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0)) (and (<= -10000.0 yi) (<= yi 10000.0))) (and (<= -10000.0 zi) (<= zi 10000.0))) (and (<= 2.328306437e-10 ux) (<= ux 1.0))) (and (<= 2.328306437e-10 uy) (<= uy 1.0))) (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (+ (+ (* (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (* (* (- 1.0 ux) maxCos) ux) (* (* (- 1.0 ux) maxCos) ux))))) xi) (* (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (* (* (- 1.0 ux) maxCos) ux) (* (* (- 1.0 ux) maxCos) ux))))) yi)) (* (* (* (- 1.0 ux) maxCos) ux) zi)))