UniformSampleCone, y

Percentage Accurate: 57.4% → 98.3%

Time: 17.6s

Alternatives: 16

Speedup: 5.0×

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

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (sin (* (* 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 sinf(((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(sin(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 = sin(((uy * single(2.0)) * single(pi))) * sqrt((single(1.0) - (t_0 * t_0)));
end

Initial Program: 57.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (sin (* (* 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 sinf(((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(sin(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 = sin(((uy * single(2.0)) * single(pi))) * sqrt((single(1.0) - (t_0 * t_0)));
end

Alternative 1: 98.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.2%

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

3. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

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

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. associate-+l+ N/A

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

   6. mul-1-neg N/A

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

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

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

   8. lower-fma.f32 N/A

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

5. Simplified 98.4%

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

7. Applied egg-rr 98.6%

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

8. Taylor expanded in maxCos around 0

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\frac{\left(\mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot \color{blue}{\left(\left(-1 \cdot ux + maxCos \cdot \left(2 + \left(-1 \cdot \left(maxCos \cdot ux\right) + 2 \cdot ux\right)\right)\right) - 2\right)}\right) \cdot ux}{\mathsf{fma}\left(maxCos + -1, \mathsf{fma}\left(maxCos, \mathsf{neg}\left(ux\right), ux\right), \mathsf{neg}\left(\mathsf{fma}\left(maxCos, -2, 2\right)\right)\right)}} \]
9. Step-by-step derivation

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. lower-+.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. +-commutative N/A

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

   8. associate-*r* N/A

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

   9. distribute-rgt-out N/A

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

   10. lower-fma.f32 N/A

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

   11. lower-+.f32 N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f32 N/A

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

   14. mul-1-neg N/A

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

   15. lower-neg.f32 98.6

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

10. Simplified 98.6%

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

11. Final simplification 98.6%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.4%

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

3. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

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

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. associate-+l+ N/A

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

   6. mul-1-neg N/A

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

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

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

   8. lower-fma.f32 N/A

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

5. Simplified 98.3%

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

Reproduce

herbie shell --seed 2024218 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, y"
  :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)))
  (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))