UniformSampleCone 2

Percentage Accurate: 98.9% → 99.0%

Time: 19.8s

Alternatives: 21

Speedup: 1.0×

Specification

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

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: 98.9% 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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. associate-*r* N/A

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

4. Applied rewrites 99.0%

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

5. Final simplification 99.0%

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

Alternative 2: 98.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in xi around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 98.0%

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

7. Applied rewrites 98.7%

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

8. Final simplification 98.7%

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

Alternative 3: 98.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in maxCos around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-cos.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-sin.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 N/A

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

   16. lower-PI.f32 98.7

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

5. Applied rewrites 98.7%

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

6. Final simplification 98.7%

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

Alternative 4: 98.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

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

   2. associate-+l+ N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-cos.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f32 N/A

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

5. Applied rewrites 98.7%

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

6. Final simplification 98.7%

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

Alternative 5: 97.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(uy \cdot \pi\right) \cdot 2\\ t_1 := \left(1 - ux\right) \cdot ux\\ t_2 := \sqrt{1 - \left(maxCos \cdot maxCos\right) \cdot \left(t\_1 \cdot t\_1\right)}\\ \mathbf{if}\;uy \cdot 2 \leq 0.019500000402331352:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.3333333333333333, \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot yi\right) \cdot uy, \left(\left(\pi \cdot \pi\right) \cdot xi\right) \cdot -2\right) \cdot t\_2, uy, \left(\left(\pi \cdot yi\right) \cdot 2\right) \cdot t\_2\right), uy, t\_2 \cdot xi\right) - \left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos t\_0, xi, \sin t\_0 \cdot yi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* uy PI) 2.0))
        (t_1 (* (- 1.0 ux) ux))
        (t_2 (sqrt (- 1.0 (* (* maxCos maxCos) (* t_1 t_1))))))
   (if (<= (* uy 2.0) 0.019500000402331352)
     (-
      (fma
       (fma
        (*
         (fma
          -1.3333333333333333
          (* (* (* (* PI PI) PI) yi) uy)
          (* (* (* PI PI) xi) -2.0))
         t_2)
        uy
        (* (* (* PI yi) 2.0) t_2))
       uy
       (* t_2 xi))
      (* (* (* (- ux 1.0) 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 = (uy * ((float) M_PI)) * 2.0f;
	float t_1 = (1.0f - ux) * ux;
	float t_2 = sqrtf((1.0f - ((maxCos * maxCos) * (t_1 * t_1))));
	float tmp;
	if ((uy * 2.0f) <= 0.019500000402331352f) {
		tmp = fmaf(fmaf((fmaf(-1.3333333333333333f, ((((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)) * yi) * uy), (((((float) M_PI) * ((float) M_PI)) * xi) * -2.0f)) * t_2), uy, (((((float) M_PI) * yi) * 2.0f) * t_2)), uy, (t_2 * xi)) - ((((ux - 1.0f) * 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(uy * Float32(pi)) * Float32(2.0))
	t_1 = Float32(Float32(Float32(1.0) - ux) * ux)
	t_2 = sqrt(Float32(Float32(1.0) - Float32(Float32(maxCos * maxCos) * Float32(t_1 * t_1))))
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.019500000402331352))
		tmp = Float32(fma(fma(Float32(fma(Float32(-1.3333333333333333), Float32(Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)) * yi) * uy), Float32(Float32(Float32(Float32(pi) * Float32(pi)) * xi) * Float32(-2.0))) * t_2), uy, Float32(Float32(Float32(Float32(pi) * yi) * Float32(2.0)) * t_2)), uy, Float32(t_2 * xi)) - Float32(Float32(Float32(Float32(ux - Float32(1.0)) * 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 (*.f32 uy #s(literal 2 binary32)) < 0.0195000004

   1. Initial program 99.2%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 99.4%

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

   if 0.0195000004 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 97.8%

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

   3. Taylor expanded in maxCos 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

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

      2. lower-fma.f32 N/A

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

      3. lower-cos.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-sin.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lower-PI.f32 93.9

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

   5. Applied rewrites 93.9%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.019500000402331352:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.3333333333333333, \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot yi\right) \cdot uy, \left(\left(\pi \cdot \pi\right) \cdot xi\right) \cdot -2\right) \cdot \sqrt{1 - \left(maxCos \cdot maxCos\right) \cdot \left(\left(\left(1 - ux\right) \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot ux\right)\right)}, uy, \left(\left(\pi \cdot yi\right) \cdot 2\right) \cdot \sqrt{1 - \left(maxCos \cdot maxCos\right) \cdot \left(\left(\left(1 - ux\right) \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot ux\right)\right)}\right), uy, \sqrt{1 - \left(maxCos \cdot maxCos\right) \cdot \left(\left(\left(1 - ux\right) \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot ux\right)\right)} \cdot xi\right) - \left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(\left(uy \cdot \pi\right) \cdot 2\right), xi, \sin \left(\left(uy \cdot \pi\right) \cdot 2\right) \cdot yi\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) \cdot ux\\ t_1 := \sqrt{1 - \left(maxCos \cdot maxCos\right) \cdot \left(t\_0 \cdot t\_0\right)}\\ t_2 := \left(uy \cdot \pi\right) \cdot 2\\ \mathbf{if}\;uy \cdot 2 \leq 0.019500000402331352:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.3333333333333333, \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot yi\right) \cdot uy, \left(\left(\pi \cdot \pi\right) \cdot xi\right) \cdot -2\right) \cdot t\_1, uy, \left(\left(\pi \cdot yi\right) \cdot 2\right) \cdot t\_1\right), uy, \mathsf{fma}\left(xi, t\_1, \left(\left(\left(1 - ux\right) \cdot zi\right) \cdot ux\right) \cdot maxCos\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos t\_2, xi, \sin t\_2 \cdot yi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (- 1.0 ux) ux))
        (t_1 (sqrt (- 1.0 (* (* maxCos maxCos) (* t_0 t_0)))))
        (t_2 (* (* uy PI) 2.0)))
   (if (<= (* uy 2.0) 0.019500000402331352)
     (fma
      (fma
       (*
        (fma
         -1.3333333333333333
         (* (* (* (* PI PI) PI) yi) uy)
         (* (* (* PI PI) xi) -2.0))
        t_1)
       uy
       (* (* (* PI yi) 2.0) t_1))
      uy
      (fma xi t_1 (* (* (* (- 1.0 ux) zi) ux) maxCos)))
     (fma (cos t_2) xi (* (sin t_2) yi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) * ux;
	float t_1 = sqrtf((1.0f - ((maxCos * maxCos) * (t_0 * t_0))));
	float t_2 = (uy * ((float) M_PI)) * 2.0f;
	float tmp;
	if ((uy * 2.0f) <= 0.019500000402331352f) {
		tmp = fmaf(fmaf((fmaf(-1.3333333333333333f, ((((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)) * yi) * uy), (((((float) M_PI) * ((float) M_PI)) * xi) * -2.0f)) * t_1), uy, (((((float) M_PI) * yi) * 2.0f) * t_1)), uy, fmaf(xi, t_1, ((((1.0f - ux) * zi) * ux) * maxCos)));
	} else {
		tmp = fmaf(cosf(t_2), xi, (sinf(t_2) * yi));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) * ux)
	t_1 = sqrt(Float32(Float32(1.0) - Float32(Float32(maxCos * maxCos) * Float32(t_0 * t_0))))
	t_2 = Float32(Float32(uy * Float32(pi)) * Float32(2.0))
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.019500000402331352))
		tmp = fma(fma(Float32(fma(Float32(-1.3333333333333333), Float32(Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)) * yi) * uy), Float32(Float32(Float32(Float32(pi) * Float32(pi)) * xi) * Float32(-2.0))) * t_1), uy, Float32(Float32(Float32(Float32(pi) * yi) * Float32(2.0)) * t_1)), uy, fma(xi, t_1, Float32(Float32(Float32(Float32(Float32(1.0) - ux) * zi) * ux) * maxCos)));
	else
		tmp = fma(cos(t_2), xi, Float32(sin(t_2) * yi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.0195000004

   1. Initial program 99.2%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{1 - \left(\left(\left(1 - ux\right) \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot ux\right)\right) \cdot \left(maxCos \cdot maxCos\right)} \cdot \mathsf{fma}\left(-1.3333333333333333, \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot yi\right) \cdot uy, -2 \cdot \left(\left(\pi \cdot \pi\right) \cdot xi\right)\right), uy, \left(2 \cdot \left(\pi \cdot yi\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot ux\right)\right) \cdot \left(maxCos \cdot maxCos\right)}\right), uy, \mathsf{fma}\left(xi, \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot ux\right)\right) \cdot \left(maxCos \cdot maxCos\right)}, \left(\left(zi \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot maxCos\right)\right)} \]

   if 0.0195000004 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 97.8%

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

   3. Taylor expanded in maxCos 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

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

      2. lower-fma.f32 N/A

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

      3. lower-cos.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-sin.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lower-PI.f32 93.9

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

   5. Applied rewrites 93.9%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.019500000402331352:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.3333333333333333, \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot yi\right) \cdot uy, \left(\left(\pi \cdot \pi\right) \cdot xi\right) \cdot -2\right) \cdot \sqrt{1 - \left(maxCos \cdot maxCos\right) \cdot \left(\left(\left(1 - ux\right) \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot ux\right)\right)}, uy, \left(\left(\pi \cdot yi\right) \cdot 2\right) \cdot \sqrt{1 - \left(maxCos \cdot maxCos\right) \cdot \left(\left(\left(1 - ux\right) \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot ux\right)\right)}\right), uy, \mathsf{fma}\left(xi, \sqrt{1 - \left(maxCos \cdot maxCos\right) \cdot \left(\left(\left(1 - ux\right) \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot ux\right)\right)}, \left(\left(\left(1 - ux\right) \cdot zi\right) \cdot ux\right) \cdot maxCos\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(\left(uy \cdot \pi\right) \cdot 2\right), xi, \sin \left(\left(uy \cdot \pi\right) \cdot 2\right) \cdot yi\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

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

   2. associate-+l+ N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-cos.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f32 N/A

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

5. Applied rewrites 97.0%

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

6. Final simplification 97.0%

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

Alternative 8: 89.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in xi around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 98.0%

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

6. Taylor expanded in uy around 0

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

8. Applied rewrites 89.6%

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

10. Applied rewrites 90.3%

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

11. Final simplification 90.3%

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

Alternative 9: 89.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in xi around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 98.0%

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

6. Taylor expanded in uy around 0

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

8. Applied rewrites 89.6%

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

9. Final simplification 89.6%

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

Alternative 10: 89.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in xi around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 98.0%

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

6. Taylor expanded in uy around 0

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

8. Applied rewrites 89.6%

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

9. Taylor expanded in maxCos around 0

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

11. Applied rewrites 89.4%

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

12. Final simplification 89.4%

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

Alternative 11: 88.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) \cdot ux\\ \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\left(uy \cdot yi\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{xi}, -1.3333333333333333, -2 \cdot \left(\pi \cdot \pi\right)\right), uy, \frac{\pi \cdot yi}{xi} \cdot 2\right), uy, 1\right) \cdot \sqrt{1 - \left(maxCos \cdot maxCos\right) \cdot \left(t\_0 \cdot t\_0\right)}\right) \cdot xi - \left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (- 1.0 ux) ux)))
   (-
    (*
     (*
      (fma
       (fma
        (fma
         (/ (* (* uy yi) (* (* PI PI) PI)) xi)
         -1.3333333333333333
         (* -2.0 (* PI PI)))
        uy
        (* (/ (* PI yi) xi) 2.0))
       uy
       1.0)
      (sqrt (- 1.0 (* (* maxCos maxCos) (* t_0 t_0)))))
     xi)
    (* (* (* (- ux 1.0) maxCos) ux) zi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) * ux;
	return ((fmaf(fmaf(fmaf((((uy * yi) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))) / xi), -1.3333333333333333f, (-2.0f * (((float) M_PI) * ((float) M_PI)))), uy, (((((float) M_PI) * yi) / xi) * 2.0f)), uy, 1.0f) * sqrtf((1.0f - ((maxCos * maxCos) * (t_0 * t_0))))) * xi) - ((((ux - 1.0f) * maxCos) * ux) * zi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) * ux)
	return Float32(Float32(Float32(fma(fma(fma(Float32(Float32(Float32(uy * yi) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))) / xi), Float32(-1.3333333333333333), Float32(Float32(-2.0) * Float32(Float32(pi) * Float32(pi)))), uy, Float32(Float32(Float32(Float32(pi) * yi) / xi) * Float32(2.0))), uy, Float32(1.0)) * sqrt(Float32(Float32(1.0) - Float32(Float32(maxCos * maxCos) * Float32(t_0 * t_0))))) * xi) - Float32(Float32(Float32(Float32(ux - Float32(1.0)) * maxCos) * ux) * zi))
end

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in xi around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 98.0%

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

6. Taylor expanded in uy around 0

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

8. Applied rewrites 88.2%

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

9. Final simplification 88.2%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\left(uy \cdot yi\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{xi}, -1.3333333333333333, -2 \cdot \left(\pi \cdot \pi\right)\right), uy, \frac{\pi \cdot yi}{xi} \cdot 2\right), uy, 1\right) \cdot \sqrt{1 - \left(maxCos \cdot maxCos\right) \cdot \left(\left(\left(1 - ux\right) \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot ux\right)\right)}\right) \cdot xi - \left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
10. Add Preprocessing

Alternative 12: 85.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in uy around 0

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

5. Applied rewrites 86.4%

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

6. Final simplification 86.4%

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

Alternative 13: 84.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) \cdot ux\\ \mathbf{if}\;uy \cdot 2 \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{1 - \left(maxCos \cdot maxCos\right) \cdot \left(t\_0 \cdot t\_0\right)}, \mathsf{fma}\left(uy \cdot 2, \pi \cdot yi, xi\right), \left(\left(\left(1 - ux\right) \cdot zi\right) \cdot ux\right) \cdot maxCos\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(uy \cdot \pi\right) \cdot 2\right) \cdot xi\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (- 1.0 ux) ux)))
   (if (<= (* uy 2.0) 0.10000000149011612)
     (fma
      (sqrt (- 1.0 (* (* maxCos maxCos) (* t_0 t_0))))
      (fma (* uy 2.0) (* PI yi) xi)
      (* (* (* (- 1.0 ux) zi) ux) maxCos))
     (* (cos (* (* uy PI) 2.0)) xi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) * ux;
	float tmp;
	if ((uy * 2.0f) <= 0.10000000149011612f) {
		tmp = fmaf(sqrtf((1.0f - ((maxCos * maxCos) * (t_0 * t_0)))), fmaf((uy * 2.0f), (((float) M_PI) * yi), xi), ((((1.0f - ux) * zi) * ux) * maxCos));
	} else {
		tmp = cosf(((uy * ((float) M_PI)) * 2.0f)) * xi;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) * ux)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.10000000149011612))
		tmp = fma(sqrt(Float32(Float32(1.0) - Float32(Float32(maxCos * maxCos) * Float32(t_0 * t_0)))), fma(Float32(uy * Float32(2.0)), Float32(Float32(pi) * yi), xi), Float32(Float32(Float32(Float32(Float32(1.0) - ux) * zi) * ux) * maxCos));
	else
		tmp = Float32(cos(Float32(Float32(uy * Float32(pi)) * Float32(2.0))) * xi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.100000001

   1. Initial program 99.1%

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

   3. Taylor expanded in zi around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 11.4

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

   5. Applied rewrites 11.4%

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

   6. Taylor expanded in ux around inf

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

   8. Applied rewrites 8.5%

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

   9. Taylor expanded in uy around 0

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

       1. +-commutative N/A

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

       2. associate-+r+ N/A

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

       3. associate-*r* N/A

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

       4. distribute-rgt-out N/A

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

   11. Applied rewrites 90.3%

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

   if 0.100000001 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 97.1%

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

   3. Taylor expanded in yi around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in maxCos around 0

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

   8. Applied rewrites 52.0%

      \[\leadsto \cos \left(\left(\pi \cdot uy\right) \cdot 2\right) \cdot \color{blue}{xi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{1 - \left(maxCos \cdot maxCos\right) \cdot \left(\left(\left(1 - ux\right) \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot ux\right)\right)}, \mathsf{fma}\left(uy \cdot 2, \pi \cdot yi, xi\right), \left(\left(\left(1 - ux\right) \cdot zi\right) \cdot ux\right) \cdot maxCos\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(uy \cdot \pi\right) \cdot 2\right) \cdot xi\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 84.9% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in xi around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 98.0%

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

6. Taylor expanded in uy around 0

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

8. Applied rewrites 85.0%

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

9. Final simplification 85.0%

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

Alternative 15: 81.5% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in zi around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 11.1

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

5. Applied rewrites 11.1%

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

6. Taylor expanded in ux around inf

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

8. Applied rewrites 8.5%

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

9. Taylor expanded in uy around 0

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

    1. +-commutative N/A

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

    2. associate-+r+ N/A

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

    3. associate-*r* N/A

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

    4. distribute-rgt-out N/A

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

11. Applied rewrites 82.2%

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

12. Final simplification 82.2%

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

Alternative 16: 81.5% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

5. Applied rewrites 82.2%

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

6. Final simplification 82.2%

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

Alternative 17: 54.5% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in yi around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 54.7%

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

6. Taylor expanded in uy around 0

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

8. Applied rewrites 50.6%

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

9. Taylor expanded in ux around 0

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

11. Applied rewrites 50.6%

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

12. Final simplification 50.6%

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

Alternative 18: 54.5% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in yi around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 54.7%

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

6. Taylor expanded in uy around 0

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

8. Applied rewrites 50.6%

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

9. Taylor expanded in maxCos around 0

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

11. Applied rewrites 50.6%

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

12. Final simplification 50.6%

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

Alternative 19: 54.5% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in yi around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 54.7%

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

6. Taylor expanded in uy around 0

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

8. Applied rewrites 50.6%

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

10. Applied rewrites 50.6%

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

11. Taylor expanded in maxCos around 0

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \left(\mathsf{PI}\left(\right) \cdot xi\right) \cdot \mathsf{PI}\left(\right), xi\right), \sqrt{1}, \left(\left(zi \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot maxCos\right) \]
12. Step-by-step derivation

13. Applied rewrites 50.6%

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

14. Final simplification 50.6%

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

Alternative 20: 13.6% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(xi, yi, zi, ux, uy, maxcos)
    real(4), intent (in) :: xi
    real(4), intent (in) :: yi
    real(4), intent (in) :: zi
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = (maxcos * ux) * ((1.0e0 - ux) * zi)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in zi around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 11.1

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

5. Applied rewrites 11.1%

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

7. Applied rewrites 11.1%

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

8. Final simplification 11.1%

   \[\leadsto \left(maxCos \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot zi\right) \]
9. Add Preprocessing

Alternative 21: 12.1% accurate, 32.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(xi, yi, zi, ux, uy, maxcos)
    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 = (ux * zi) * maxcos
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in zi around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 11.1

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

5. Applied rewrites 11.1%

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

6. Taylor expanded in ux around 0

   \[\leadsto \left(ux \cdot zi\right) \cdot maxCos \]
7. Step-by-step derivation

8. Applied rewrites 10.5%

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

9. Final simplification 10.5%

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

Reproduce

herbie shell --seed 2024235 
(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)))