UniformSampleCone 2

Percentage Accurate: 98.9% → 99.0%

Time: 14.9s

Alternatives: 1

Speedup: N/A×

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, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\\ \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(-\pi, uy \cdot 2, \frac{\pi}{2}\right)\right) \cdot \sin \cos^{-1} t\_0, xi, \mathsf{fma}\left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right), \sqrt{\left({ux}^{-4} - \mathsf{fma}\left(\frac{maxCos \cdot maxCos}{ux}, -2, \mathsf{fma}\left(maxCos, maxCos, \frac{maxCos}{ux} \cdot \frac{maxCos}{ux}\right)\right)\right) \cdot {ux}^{4}} \cdot yi, t\_0 \cdot zi\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux)))
   (fma
    (* (sin (fma (- PI) (* uy 2.0) (/ PI 2.0))) (sin (acos t_0)))
    xi
    (fma
     (sin (* (* uy 2.0) PI))
     (*
      (sqrt
       (*
        (-
         (pow ux -4.0)
         (fma
          (/ (* maxCos maxCos) ux)
          -2.0
          (fma maxCos maxCos (* (/ maxCos ux) (/ maxCos ux)))))
        (pow ux 4.0)))
      yi)
     (* t_0 zi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((1.0f - ux) * maxCos) * ux;
	return fmaf((sinf(fmaf(-((float) M_PI), (uy * 2.0f), (((float) M_PI) / 2.0f))) * sinf(acosf(t_0))), xi, fmaf(sinf(((uy * 2.0f) * ((float) M_PI))), (sqrtf(((powf(ux, -4.0f) - fmaf(((maxCos * maxCos) / ux), -2.0f, fmaf(maxCos, maxCos, ((maxCos / ux) * (maxCos / ux))))) * powf(ux, 4.0f))) * yi), (t_0 * zi)));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux)
	return fma(Float32(sin(fma(Float32(-Float32(pi)), Float32(uy * Float32(2.0)), Float32(Float32(pi) / Float32(2.0)))) * sin(acos(t_0))), xi, fma(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))), Float32(sqrt(Float32(Float32((ux ^ Float32(-4.0)) - fma(Float32(Float32(maxCos * maxCos) / ux), Float32(-2.0), fma(maxCos, maxCos, Float32(Float32(maxCos / ux) * Float32(maxCos / ux))))) * (ux ^ Float32(4.0)))) * yi), Float32(t_0 * zi)))
end

Derivation

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 ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 99.2%

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

7. Applied rewrites 99.2%

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

   1. lift-cos.f32 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)} \cdot \sin \cos^{-1} \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right), xi, \mathsf{fma}\left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right), \sqrt{\left({ux}^{-4} - \mathsf{fma}\left(\frac{maxCos \cdot maxCos}{ux}, -2, \mathsf{fma}\left(maxCos, maxCos, \frac{maxCos}{ux} \cdot \frac{maxCos}{ux}\right)\right)\right) \cdot {ux}^{4}} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

   2. cos-neg-rev N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(uy \cdot 2\right) \cdot \pi\right)\right)} \cdot \sin \cos^{-1} \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right), xi, \mathsf{fma}\left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right), \sqrt{\left({ux}^{-4} - \mathsf{fma}\left(\frac{maxCos \cdot maxCos}{ux}, -2, \mathsf{fma}\left(maxCos, maxCos, \frac{maxCos}{ux} \cdot \frac{maxCos}{ux}\right)\right)\right) \cdot {ux}^{4}} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

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

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

   4. lower-sin.f32 N/A

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

   5. lower-+.f32 N/A

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

   6. lower-neg.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

   15. lower-/.f32 N/A

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

   16. lift-PI.f32 99.2

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

9. Applied rewrites 99.2%

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

    1. lift-+.f32 N/A

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

    2. lift-neg.f32 N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\color{blue}{\left(\mathsf{neg}\left(\pi \cdot \left(2 \cdot uy\right)\right)\right)} + \frac{\pi}{2}\right) \cdot \sin \cos^{-1} \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right), xi, \mathsf{fma}\left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right), \sqrt{\left({ux}^{-4} - \mathsf{fma}\left(\frac{maxCos \cdot maxCos}{ux}, -2, \mathsf{fma}\left(maxCos, maxCos, \frac{maxCos}{ux} \cdot \frac{maxCos}{ux}\right)\right)\right) \cdot {ux}^{4}} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

    3. lift-PI.f32 N/A

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

    4. lift-*.f32 N/A

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

    5. lift-*.f32 N/A

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

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

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

    7. lower-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(\sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right), 2 \cdot uy, \frac{\pi}{2}\right)\right)} \cdot \sin \cos^{-1} \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right), xi, \mathsf{fma}\left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right), \sqrt{\left({ux}^{-4} - \mathsf{fma}\left(\frac{maxCos \cdot maxCos}{ux}, -2, \mathsf{fma}\left(maxCos, maxCos, \frac{maxCos}{ux} \cdot \frac{maxCos}{ux}\right)\right)\right) \cdot {ux}^{4}} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

    8. lower-neg.f32 N/A

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

    9. lift-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(-\color{blue}{\pi}, 2 \cdot uy, \frac{\pi}{2}\right)\right) \cdot \sin \cos^{-1} \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right), xi, \mathsf{fma}\left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right), \sqrt{\left({ux}^{-4} - \mathsf{fma}\left(\frac{maxCos \cdot maxCos}{ux}, -2, \mathsf{fma}\left(maxCos, maxCos, \frac{maxCos}{ux} \cdot \frac{maxCos}{ux}\right)\right)\right) \cdot {ux}^{4}} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

    10. *-commutative N/A

        \[\leadsto \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(-\pi, \color{blue}{uy \cdot 2}, \frac{\pi}{2}\right)\right) \cdot \sin \cos^{-1} \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right), xi, \mathsf{fma}\left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right), \sqrt{\left({ux}^{-4} - \mathsf{fma}\left(\frac{maxCos \cdot maxCos}{ux}, -2, \mathsf{fma}\left(maxCos, maxCos, \frac{maxCos}{ux} \cdot \frac{maxCos}{ux}\right)\right)\right) \cdot {ux}^{4}} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

    11. lift-*.f32 99.3

        \[\leadsto \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(-\pi, \color{blue}{uy \cdot 2}, \frac{\pi}{2}\right)\right) \cdot \sin \cos^{-1} \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right), xi, \mathsf{fma}\left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right), \sqrt{\left({ux}^{-4} - \mathsf{fma}\left(\frac{maxCos \cdot maxCos}{ux}, -2, \mathsf{fma}\left(maxCos, maxCos, \frac{maxCos}{ux} \cdot \frac{maxCos}{ux}\right)\right)\right) \cdot {ux}^{4}} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

11. Applied rewrites 99.3%

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

Reproduce

herbie shell --seed 2025062 -o reduce:regimes -o reduce:simplify
(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)))