UniformSampleCone, x

Percentage Accurate: 57.5% → 98.8%

Time: 15.8s

Alternatives: 17

Speedup: 9.8×

Specification

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

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* t_0 t_0))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	return cosf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((1.0f - (t_0 * t_0)));
}

Julia

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

MATLAB

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

Initial Program: 57.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* t_0 t_0))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	return cosf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((1.0f - (t_0 * t_0)));
}

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (cos (* uy PI))))
   (*
    (- (* t_0 t_0) (* (sin (* uy (* PI (log E)))) (sin (* uy PI))))
    (sqrt
     (*
      ux
      (fma maxCos -2.0 (fma (- 1.0 maxCos) (fma ux maxCos (- ux)) 2.0)))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = cosf((uy * ((float) M_PI)));
	return ((t_0 * t_0) - (sinf((uy * (((float) M_PI) * logf(((float) M_E))))) * sinf((uy * ((float) M_PI))))) * sqrtf((ux * fmaf(maxCos, -2.0f, fmaf((1.0f - maxCos), fmaf(ux, maxCos, -ux), 2.0f))));
}

Julia

function code(ux, uy, maxCos)
	t_0 = cos(Float32(uy * Float32(pi)))
	return Float32(Float32(Float32(t_0 * t_0) - Float32(sin(Float32(uy * Float32(Float32(pi) * log(Float32(exp(1)))))) * sin(Float32(uy * Float32(pi))))) * sqrt(Float32(ux * fma(maxCos, Float32(-2.0), fma(Float32(Float32(1.0) - maxCos), fma(ux, maxCos, Float32(-ux)), Float32(2.0))))))
end

Derivation

1. Initial program 57.5%

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

3. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f32 N/A

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

   7. +-commutative N/A

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

   8. mul-1-neg N/A

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

   9. *-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. *-commutative N/A

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

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

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

   14. neg-sub0 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, \color{blue}{\left(0 - \left(maxCos - 1\right)\right)} \cdot \left(ux \cdot \left(maxCos - 1\right)\right) + 2\right)} \]

   15. associate-+l- N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, \color{blue}{\left(\left(0 - maxCos\right) + 1\right)} \cdot \left(ux \cdot \left(maxCos - 1\right)\right) + 2\right)} \]

   16. neg-sub0 N/A

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

   17. mul-1-neg N/A

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

   18. +-commutative N/A

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

5. Applied rewrites 98.8%

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

   1. lift-cos.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-PI.f32 N/A

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

   7. associate-*l* N/A

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

   8. cos-2 N/A

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

   9. lower--.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-cos.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-cos.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lower-*.f32 N/A

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

   16. lower-sin.f32 N/A

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

   17. lower-*.f32 N/A

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

   18. lower-sin.f32 N/A

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

   19. lower-*.f32 98.7

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

7. Applied rewrites 98.7%

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

   1. lift-PI.f32 N/A

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

   2. add-log-exp N/A

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

   3. *-un-lft-identity N/A

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

   4. lift-PI.f32 N/A

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

   5. exp-prod N/A

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

   6. log-pow N/A

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

   7. lower-*.f32 N/A

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

   8. lower-log.f32 N/A

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

   9. exp-1-e N/A

      \[\leadsto \left(\cos \left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \cos \left(uy \cdot \mathsf{PI}\left(\right)\right) - \sin \left(uy \cdot \left(\mathsf{PI}\left(\right) \cdot \log \color{blue}{\mathsf{E}\left(\right)}\right)\right) \cdot \sin \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, \mathsf{fma}\left(1 - maxCos, \mathsf{fma}\left(ux, maxCos, \mathsf{neg}\left(ux\right)\right), 2\right)\right)} \]

   10. lower-E.f32 98.9

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

9. Applied rewrites 98.9%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.5%

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

3. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f32 N/A

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

   7. +-commutative N/A

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

   8. mul-1-neg N/A

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

   9. *-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. *-commutative N/A

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

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

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

   14. neg-sub0 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, \color{blue}{\left(0 - \left(maxCos - 1\right)\right)} \cdot \left(ux \cdot \left(maxCos - 1\right)\right) + 2\right)} \]

   15. associate-+l- N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, \color{blue}{\left(\left(0 - maxCos\right) + 1\right)} \cdot \left(ux \cdot \left(maxCos - 1\right)\right) + 2\right)} \]

   16. neg-sub0 N/A

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

   17. mul-1-neg N/A

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

   18. +-commutative N/A

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

5. Applied rewrites 98.9%

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

7. Applied rewrites 99.0%

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

8. Final simplification 99.0%

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

Reproduce

herbie shell --seed 2024228 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, x"
  :precision binary32
  :pre (and (and (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) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))