UniformSampleCone, y

Percentage Accurate: 58.0% → 98.3%

Time: 16.0s

Precision: binary32

Cost: 13504

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

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Derivation

1. Initial program 53.8%

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

2. Simplified 53.9%

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

   [Start] 53.8%	\[ \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   associate-*l* [=>] 53.8%	\[ \sin \color{blue}{\left(uy \cdot \left(2 \cdot \pi\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   sub-neg [=>] 53.8%	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{1 + \left(-\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}} \]
   +-commutative [=>] 53.8%	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(-\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + 1}} \]
   distribute-rgt-neg-in [=>] 53.8%	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(-\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} + 1} \]
   fma-def [=>] 53.9%	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\left(1 - ux\right) + ux \cdot maxCos, -\left(\left(1 - ux\right) + ux \cdot maxCos\right), 1\right)}} \]
   +-commutative [=>] 53.9%	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\color{blue}{ux \cdot maxCos + \left(1 - ux\right)}, -\left(\left(1 - ux\right) + ux \cdot maxCos\right), 1\right)} \]
   associate-+r- [=>] 54.1%	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\color{blue}{\left(ux \cdot maxCos + 1\right) - ux}, -\left(\left(1 - ux\right) + ux \cdot maxCos\right), 1\right)} \]
   fma-def [=>] 54.1%	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(ux, maxCos, 1\right)} - ux, -\left(\left(1 - ux\right) + ux \cdot maxCos\right), 1\right)} \]
   neg-sub0 [=>] 54.1%	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right)}, 1\right)} \]
   +-commutative [=>] 54.1%	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, 1\right)} \]
   associate-+r- [=>] 53.9%	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)}, 1\right)} \]
   associate--r- [=>] 53.9%	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(ux \cdot maxCos + 1\right)\right) + ux}, 1\right)} \]
   neg-sub0 [<=] 53.9%	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(-\left(ux \cdot maxCos + 1\right)\right)} + ux, 1\right)} \]
   +-commutative [=>] 53.9%	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{ux + \left(-\left(ux \cdot maxCos + 1\right)\right)}, 1\right)} \]
   sub-neg [<=] 53.9%	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{ux - \left(ux \cdot maxCos + 1\right)}, 1\right)} \]
   fma-def [=>] 53.9%	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, ux - \color{blue}{\mathsf{fma}\left(ux, maxCos, 1\right)}, 1\right)} \]

3. Taylor expanded in ux around 0 98.4%

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

4. Simplified 98.4%

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

   [Start] 98.4%	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\left(maxCos - 1\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right) + ux \cdot \left(\left(1 + -1 \cdot \left(maxCos - 1\right)\right) - maxCos\right)} \]
   fma-def [=>] 98.4%	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(maxCos - 1, \left(1 - maxCos\right) \cdot {ux}^{2}, ux \cdot \left(\left(1 + -1 \cdot \left(maxCos - 1\right)\right) - maxCos\right)\right)}} \]
   sub-neg [=>] 98.4%	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\color{blue}{maxCos + \left(-1\right)}, \left(1 - maxCos\right) \cdot {ux}^{2}, ux \cdot \left(\left(1 + -1 \cdot \left(maxCos - 1\right)\right) - maxCos\right)\right)} \]
   metadata-eval [=>] 98.4%	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(maxCos + \color{blue}{-1}, \left(1 - maxCos\right) \cdot {ux}^{2}, ux \cdot \left(\left(1 + -1 \cdot \left(maxCos - 1\right)\right) - maxCos\right)\right)} \]
   *-commutative [=>] 98.4%	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(maxCos + -1, \color{blue}{{ux}^{2} \cdot \left(1 - maxCos\right)}, ux \cdot \left(\left(1 + -1 \cdot \left(maxCos - 1\right)\right) - maxCos\right)\right)} \]
   unpow2 [=>] 98.4%	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(maxCos + -1, \color{blue}{\left(ux \cdot ux\right)} \cdot \left(1 - maxCos\right), ux \cdot \left(\left(1 + -1 \cdot \left(maxCos - 1\right)\right) - maxCos\right)\right)} \]
   associate--l+ [=>] 98.4%	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(maxCos + -1, \left(ux \cdot ux\right) \cdot \left(1 - maxCos\right), ux \cdot \color{blue}{\left(1 + \left(-1 \cdot \left(maxCos - 1\right) - maxCos\right)\right)}\right)} \]
   mul-1-neg [=>] 98.4%	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(maxCos + -1, \left(ux \cdot ux\right) \cdot \left(1 - maxCos\right), ux \cdot \left(1 + \left(\color{blue}{\left(-\left(maxCos - 1\right)\right)} - maxCos\right)\right)\right)} \]
   sub-neg [=>] 98.4%	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(maxCos + -1, \left(ux \cdot ux\right) \cdot \left(1 - maxCos\right), ux \cdot \left(1 + \left(\left(-\color{blue}{\left(maxCos + \left(-1\right)\right)}\right) - maxCos\right)\right)\right)} \]
   metadata-eval [=>] 98.4%	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(maxCos + -1, \left(ux \cdot ux\right) \cdot \left(1 - maxCos\right), ux \cdot \left(1 + \left(\left(-\left(maxCos + \color{blue}{-1}\right)\right) - maxCos\right)\right)\right)} \]

5. Applied egg-rr 98.3%

   \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{\sqrt[3]{\mathsf{fma}\left(maxCos + -1, ux \cdot \left(ux \cdot \left(1 - maxCos\right)\right), ux \cdot \left(1 + \left(\left(\left(-maxCos\right) + 1\right) - maxCos\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(maxCos + -1, ux \cdot \left(ux \cdot \left(1 - maxCos\right)\right), ux \cdot \left(1 + \left(\left(\left(-maxCos\right) + 1\right) - maxCos\right)\right)\right)}}} \]
   Step-by-step derivation

   [Start] 98.4%	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(maxCos + -1, \left(ux \cdot ux\right) \cdot \left(1 - maxCos\right), ux \cdot \left(1 + \left(\left(-\left(maxCos + -1\right)\right) - maxCos\right)\right)\right)} \]
   add-cbrt-cube [=>] 98.4%	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{\sqrt[3]{\left(\sqrt{\mathsf{fma}\left(maxCos + -1, \left(ux \cdot ux\right) \cdot \left(1 - maxCos\right), ux \cdot \left(1 + \left(\left(-\left(maxCos + -1\right)\right) - maxCos\right)\right)\right)} \cdot \sqrt{\mathsf{fma}\left(maxCos + -1, \left(ux \cdot ux\right) \cdot \left(1 - maxCos\right), ux \cdot \left(1 + \left(\left(-\left(maxCos + -1\right)\right) - maxCos\right)\right)\right)}\right) \cdot \sqrt{\mathsf{fma}\left(maxCos + -1, \left(ux \cdot ux\right) \cdot \left(1 - maxCos\right), ux \cdot \left(1 + \left(\left(-\left(maxCos + -1\right)\right) - maxCos\right)\right)\right)}}} \]
   add-sqr-sqrt [<=] 98.3%	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt[3]{\color{blue}{\mathsf{fma}\left(maxCos + -1, \left(ux \cdot ux\right) \cdot \left(1 - maxCos\right), ux \cdot \left(1 + \left(\left(-\left(maxCos + -1\right)\right) - maxCos\right)\right)\right)} \cdot \sqrt{\mathsf{fma}\left(maxCos + -1, \left(ux \cdot ux\right) \cdot \left(1 - maxCos\right), ux \cdot \left(1 + \left(\left(-\left(maxCos + -1\right)\right) - maxCos\right)\right)\right)}} \]
   associate-*l* [=>] 98.3%	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt[3]{\mathsf{fma}\left(maxCos + -1, \color{blue}{ux \cdot \left(ux \cdot \left(1 - maxCos\right)\right)}, ux \cdot \left(1 + \left(\left(-\left(maxCos + -1\right)\right) - maxCos\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(maxCos + -1, \left(ux \cdot ux\right) \cdot \left(1 - maxCos\right), ux \cdot \left(1 + \left(\left(-\left(maxCos + -1\right)\right) - maxCos\right)\right)\right)}} \]
   distribute-neg-in [=>] 98.3%	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt[3]{\mathsf{fma}\left(maxCos + -1, ux \cdot \left(ux \cdot \left(1 - maxCos\right)\right), ux \cdot \left(1 + \left(\color{blue}{\left(\left(-maxCos\right) + \left(--1\right)\right)} - maxCos\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(maxCos + -1, \left(ux \cdot ux\right) \cdot \left(1 - maxCos\right), ux \cdot \left(1 + \left(\left(-\left(maxCos + -1\right)\right) - maxCos\right)\right)\right)}} \]
   metadata-eval [=>] 98.3%	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt[3]{\mathsf{fma}\left(maxCos + -1, ux \cdot \left(ux \cdot \left(1 - maxCos\right)\right), ux \cdot \left(1 + \left(\left(\left(-maxCos\right) + \color{blue}{1}\right) - maxCos\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(maxCos + -1, \left(ux \cdot ux\right) \cdot \left(1 - maxCos\right), ux \cdot \left(1 + \left(\left(-\left(maxCos + -1\right)\right) - maxCos\right)\right)\right)}} \]

6. Simplified 98.3%

   \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(1 - maxCos, \left(maxCos + -1\right) \cdot \left(ux \cdot ux\right), ux \cdot \left(\left(2 + \left(-maxCos\right)\right) - maxCos\right)\right)\right)}^{1.5}}} \]
   Step-by-step derivation

   [Start] 98.3%	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt[3]{\mathsf{fma}\left(maxCos + -1, ux \cdot \left(ux \cdot \left(1 - maxCos\right)\right), ux \cdot \left(1 + \left(\left(\left(-maxCos\right) + 1\right) - maxCos\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(maxCos + -1, ux \cdot \left(ux \cdot \left(1 - maxCos\right)\right), ux \cdot \left(1 + \left(\left(\left(-maxCos\right) + 1\right) - maxCos\right)\right)\right)}} \]
   *-commutative [=>] 98.3%	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt[3]{\color{blue}{\sqrt{\mathsf{fma}\left(maxCos + -1, ux \cdot \left(ux \cdot \left(1 - maxCos\right)\right), ux \cdot \left(1 + \left(\left(\left(-maxCos\right) + 1\right) - maxCos\right)\right)\right)} \cdot \mathsf{fma}\left(maxCos + -1, ux \cdot \left(ux \cdot \left(1 - maxCos\right)\right), ux \cdot \left(1 + \left(\left(\left(-maxCos\right) + 1\right) - maxCos\right)\right)\right)}} \]
   unpow1/2 [<=] 98.3%	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt[3]{\color{blue}{{\left(\mathsf{fma}\left(maxCos + -1, ux \cdot \left(ux \cdot \left(1 - maxCos\right)\right), ux \cdot \left(1 + \left(\left(\left(-maxCos\right) + 1\right) - maxCos\right)\right)\right)\right)}^{0.5}} \cdot \mathsf{fma}\left(maxCos + -1, ux \cdot \left(ux \cdot \left(1 - maxCos\right)\right), ux \cdot \left(1 + \left(\left(\left(-maxCos\right) + 1\right) - maxCos\right)\right)\right)} \]
   pow-plus [=>] 98.3%	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt[3]{\color{blue}{{\left(\mathsf{fma}\left(maxCos + -1, ux \cdot \left(ux \cdot \left(1 - maxCos\right)\right), ux \cdot \left(1 + \left(\left(\left(-maxCos\right) + 1\right) - maxCos\right)\right)\right)\right)}^{\left(0.5 + 1\right)}}} \]

7. Taylor expanded in uy around inf 98.4%

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

8. Simplified 98.4%

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

   [Start] 98.4%	\[ \sqrt{\left(maxCos - 1\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right) + ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \]
   *-commutative [=>] 98.4%	\[ \color{blue}{\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{\left(maxCos - 1\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right) + ux \cdot \left(2 - 2 \cdot maxCos\right)}} \]
   associate-*r* [=>] 98.4%	\[ \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(\left(maxCos - 1\right) \cdot \left(1 - maxCos\right)\right) \cdot {ux}^{2}} + ux \cdot \left(2 - 2 \cdot maxCos\right)} \]
   *-commutative [=>] 98.4%	\[ \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)} \cdot {ux}^{2} + ux \cdot \left(2 - 2 \cdot maxCos\right)} \]
   associate-*r* [<=] 98.4%	\[ \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(1 - maxCos\right) \cdot \left(\left(maxCos - 1\right) \cdot {ux}^{2}\right)} + ux \cdot \left(2 - 2 \cdot maxCos\right)} \]
   cancel-sign-sub-inv [=>] 98.4%	\[ \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{\left(1 - maxCos\right) \cdot \left(\left(maxCos - 1\right) \cdot {ux}^{2}\right) + ux \cdot \color{blue}{\left(2 + \left(-2\right) \cdot maxCos\right)}} \]
   metadata-eval [=>] 98.4%	\[ \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{\left(1 - maxCos\right) \cdot \left(\left(maxCos - 1\right) \cdot {ux}^{2}\right) + ux \cdot \left(2 + \color{blue}{-2} \cdot maxCos\right)} \]
   metadata-eval [<=] 98.4%	\[ \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{\left(1 - maxCos\right) \cdot \left(\left(maxCos - 1\right) \cdot {ux}^{2}\right) + ux \cdot \left(2 + \color{blue}{\left(-2\right)} \cdot maxCos\right)} \]
   cancel-sign-sub-inv [<=] 98.4%	\[ \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{\left(1 - maxCos\right) \cdot \left(\left(maxCos - 1\right) \cdot {ux}^{2}\right) + ux \cdot \color{blue}{\left(2 - 2 \cdot maxCos\right)}} \]
   count-2 [<=] 98.4%	\[ \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{\left(1 - maxCos\right) \cdot \left(\left(maxCos - 1\right) \cdot {ux}^{2}\right) + ux \cdot \left(2 - \color{blue}{\left(maxCos + maxCos\right)}\right)} \]
   associate--l- [<=] 98.4%	\[ \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{\left(1 - maxCos\right) \cdot \left(\left(maxCos - 1\right) \cdot {ux}^{2}\right) + ux \cdot \color{blue}{\left(\left(2 - maxCos\right) - maxCos\right)}} \]
   fma-def [=>] 98.4%	\[ \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot {ux}^{2}, ux \cdot \left(\left(2 - maxCos\right) - maxCos\right)\right)}} \]

9. Final simplification 98.4%

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

Alternatives

Alternative 1
Accuracy	98.3%
Cost	13504

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

Alternative 2
Accuracy	95.7%
Cost	10436

\[\begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 9.200000204145908 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(ux \cdot ux\right) \cdot \left(maxCos + -1\right), ux \cdot \left(2 + maxCos \cdot -2\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \end{array} \]

Alternative 3
Accuracy	97.7%
Cost	10432

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

Alternative 4
Accuracy	92.5%
Cost	9984

\[\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux} \]

Alternative 5
Accuracy	85.6%
Cost	9924

\[\begin{array}{l} \mathbf{if}\;uy \leq 0.001500000013038516:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]

Alternative 6
Accuracy	92.5%
Cost	9920

\[\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)} \]

Alternative 7
Accuracy	77.3%
Cost	6784

\[2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux}\right) \]

Alternative 8
Accuracy	77.3%
Cost	6720

\[\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)} \]

Alternative 9
Accuracy	63.2%
Cost	6656

\[\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux} \]

Alternative 10
Accuracy	7.1%
Cost	6592

\[2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{0}\right) \]

Reproduce

herbie shell --seed 2023167 
(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)))))))