UniformSampleCone, y

Average Error: 42.73% → 1.72%

Time: 21.3s

Precision: binary32

Cost: 16672

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

Math

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

↓

\[\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt[3]{{\left(ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) - \left(1 - maxCos\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right)\right)\right)}^{1.5}} \]

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 (* uy (* 2.0 PI)))
  (cbrt
   (pow
    (* ux (- (fma -2.0 maxCos 2.0) (* (- 1.0 maxCos) (* ux (- 1.0 maxCos)))))
    1.5))))

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((uy * (2.0f * ((float) M_PI)))) * cbrtf(powf((ux * (fmaf(-2.0f, maxCos, 2.0f) - ((1.0f - maxCos) * (ux * (1.0f - maxCos))))), 1.5f));
}

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(uy * Float32(Float32(2.0) * Float32(pi)))) * cbrt((Float32(ux * Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) - Float32(Float32(Float32(1.0) - maxCos) * Float32(ux * Float32(Float32(1.0) - maxCos))))) ^ Float32(1.5))))
end

Derivation

1. Initial program 42.73

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

2. Simplified 42.74

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

   [Start] 42.73	\[ \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   associate-*l* [=>] 42.73	\[ \sin \color{blue}{\left(uy \cdot \left(2 \cdot \pi\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   cancel-sign-sub-inv [=>] 42.73	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{1 + \left(-\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}} \]
   +-commutative [=>] 42.73	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(-\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + 1}} \]
   *-commutative [=>] 42.73	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(-\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} + 1} \]
   fma-def [=>] 42.61	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\left(1 - ux\right) + ux \cdot maxCos, -\left(\left(1 - ux\right) + ux \cdot maxCos\right), 1\right)}} \]
   +-commutative [=>] 42.61	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\color{blue}{ux \cdot maxCos + \left(1 - ux\right)}, -\left(\left(1 - ux\right) + ux \cdot maxCos\right), 1\right)} \]
   associate-+r- [=>] 42.58	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\color{blue}{\left(ux \cdot maxCos + 1\right) - ux}, -\left(\left(1 - ux\right) + ux \cdot maxCos\right), 1\right)} \]
   fma-def [=>] 42.58	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(ux, maxCos, 1\right)} - ux, -\left(\left(1 - ux\right) + ux \cdot maxCos\right), 1\right)} \]
   neg-sub0 [=>] 42.58	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right)}, 1\right)} \]
   +-commutative [=>] 42.58	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, 1\right)} \]
   associate-+r- [=>] 42.73	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)}, 1\right)} \]
   associate--r- [=>] 42.73	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(ux \cdot maxCos + 1\right)\right) + ux}, 1\right)} \]
   +-commutative [=>] 42.73	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{ux + \left(0 - \left(ux \cdot maxCos + 1\right)\right)}, 1\right)} \]
   sub0-neg [=>] 42.73	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, ux + \color{blue}{\left(-\left(ux \cdot maxCos + 1\right)\right)}, 1\right)} \]
   sub-neg [<=] 42.73	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{ux - \left(ux \cdot maxCos + 1\right)}, 1\right)} \]
   fma-def [=>] 42.74	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, ux - \color{blue}{\mathsf{fma}\left(ux, maxCos, 1\right)}, 1\right)} \]

3. Taylor expanded in ux around 0 1.71

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

4. Simplified 1.68

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

   [Start] 1.71	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\left(maxCos - 1\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right) + ux \cdot \left(\left(1 + -1 \cdot \left(maxCos - 1\right)\right) - maxCos\right)} \]
   +-commutative [=>] 1.71	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(1 + -1 \cdot \left(maxCos - 1\right)\right) - maxCos\right) + \left(maxCos - 1\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right)}} \]
   fma-def [=>] 1.7	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(ux, \left(1 + -1 \cdot \left(maxCos - 1\right)\right) - maxCos, \left(maxCos - 1\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right)\right)}} \]
   associate--l+ [=>] 1.68	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, \color{blue}{1 + \left(-1 \cdot \left(maxCos - 1\right) - maxCos\right)}, \left(maxCos - 1\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right)\right)} \]
   sub-neg [=>] 1.68	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, 1 + \left(-1 \cdot \color{blue}{\left(maxCos + \left(-1\right)\right)} - maxCos\right), \left(maxCos - 1\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right)\right)} \]
   metadata-eval [=>] 1.68	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, 1 + \left(-1 \cdot \left(maxCos + \color{blue}{-1}\right) - maxCos\right), \left(maxCos - 1\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right)\right)} \]
   distribute-lft-in [=>] 1.68	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, 1 + \left(\color{blue}{\left(-1 \cdot maxCos + -1 \cdot -1\right)} - maxCos\right), \left(maxCos - 1\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right)\right)} \]
   metadata-eval [=>] 1.68	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, 1 + \left(\left(-1 \cdot maxCos + \color{blue}{1}\right) - maxCos\right), \left(maxCos - 1\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right)\right)} \]
   +-commutative [<=] 1.68	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, 1 + \left(\color{blue}{\left(1 + -1 \cdot maxCos\right)} - maxCos\right), \left(maxCos - 1\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right)\right)} \]
   mul-1-neg [=>] 1.68	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, 1 + \left(\left(1 + \color{blue}{\left(-maxCos\right)}\right) - maxCos\right), \left(maxCos - 1\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right)\right)} \]
   sub-neg [<=] 1.68	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, 1 + \left(\color{blue}{\left(1 - maxCos\right)} - maxCos\right), \left(maxCos - 1\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right)\right)} \]
   associate-*r* [=>] 1.68	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, 1 + \left(\left(1 - maxCos\right) - maxCos\right), \color{blue}{\left(\left(maxCos - 1\right) \cdot \left(1 - maxCos\right)\right) \cdot {ux}^{2}}\right)} \]
   *-commutative [=>] 1.68	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, 1 + \left(\left(1 - maxCos\right) - maxCos\right), \color{blue}{\left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)} \cdot {ux}^{2}\right)} \]
   associate-*l* [=>] 1.68	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, 1 + \left(\left(1 - maxCos\right) - maxCos\right), \color{blue}{\left(1 - maxCos\right) \cdot \left(\left(maxCos - 1\right) \cdot {ux}^{2}\right)}\right)} \]
   sub-neg [=>] 1.68	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, 1 + \left(\left(1 - maxCos\right) - maxCos\right), \left(1 - maxCos\right) \cdot \left(\color{blue}{\left(maxCos + \left(-1\right)\right)} \cdot {ux}^{2}\right)\right)} \]
   metadata-eval [=>] 1.68	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, 1 + \left(\left(1 - maxCos\right) - maxCos\right), \left(1 - maxCos\right) \cdot \left(\left(maxCos + \color{blue}{-1}\right) \cdot {ux}^{2}\right)\right)} \]
   +-commutative [=>] 1.68	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, 1 + \left(\left(1 - maxCos\right) - maxCos\right), \left(1 - maxCos\right) \cdot \left(\color{blue}{\left(-1 + maxCos\right)} \cdot {ux}^{2}\right)\right)} \]
   unpow2 [=>] 1.68	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, 1 + \left(\left(1 - maxCos\right) - maxCos\right), \left(1 - maxCos\right) \cdot \left(\left(-1 + maxCos\right) \cdot \color{blue}{\left(ux \cdot ux\right)}\right)\right)} \]

5. Applied egg-rr 3.97

   \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{{\left({\left(\mathsf{fma}\left(ux, 2 - \left(maxCos + maxCos\right), \left(1 - maxCos\right) \cdot \left(ux \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]

6. Simplified 1.71

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

   [Start] 3.97	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot {\left({\left(\mathsf{fma}\left(ux, 2 - \left(maxCos + maxCos\right), \left(1 - maxCos\right) \cdot \left(ux \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333} \]
   unpow1/3 [=>] 1.71	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(ux, 2 - \left(maxCos + maxCos\right), \left(1 - maxCos\right) \cdot \left(ux \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)}^{1.5}}} \]

7. Applied egg-rr 57.81

   \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt[3]{\color{blue}{e^{\mathsf{log1p}\left({\left(ux \cdot \left(\mathsf{fma}\left(maxCos, -2, 2\right) + ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)\right)}^{1.5}\right)} - 1}} \]

8. Simplified 1.72

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

   [Start] 57.81	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt[3]{e^{\mathsf{log1p}\left({\left(ux \cdot \left(\mathsf{fma}\left(maxCos, -2, 2\right) + ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)\right)}^{1.5}\right)} - 1} \]
   expm1-def [=>] 1.72	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt[3]{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(ux \cdot \left(\mathsf{fma}\left(maxCos, -2, 2\right) + ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)\right)}^{1.5}\right)\right)}} \]
   expm1-log1p [=>] 1.72	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt[3]{\color{blue}{{\left(ux \cdot \left(\mathsf{fma}\left(maxCos, -2, 2\right) + ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)\right)}^{1.5}}} \]
   fma-udef [=>] 1.72	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt[3]{{\left(ux \cdot \left(\color{blue}{\left(maxCos \cdot -2 + 2\right)} + ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)\right)}^{1.5}} \]
   *-commutative [<=] 1.72	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt[3]{{\left(ux \cdot \left(\left(\color{blue}{-2 \cdot maxCos} + 2\right) + ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)\right)}^{1.5}} \]
   fma-def [=>] 1.72	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt[3]{{\left(ux \cdot \left(\color{blue}{\mathsf{fma}\left(-2, maxCos, 2\right)} + ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)\right)}^{1.5}} \]
   associate-*r* [=>] 1.72	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt[3]{{\left(ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \color{blue}{\left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 - maxCos\right)}\right)\right)}^{1.5}} \]
   metadata-eval [<=] 1.72	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt[3]{{\left(ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \left(ux \cdot \left(maxCos + \color{blue}{\left(-1\right)}\right)\right) \cdot \left(1 - maxCos\right)\right)\right)}^{1.5}} \]
   sub-neg [<=] 1.72	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt[3]{{\left(ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \left(ux \cdot \color{blue}{\left(maxCos - 1\right)}\right) \cdot \left(1 - maxCos\right)\right)\right)}^{1.5}} \]
   *-commutative [=>] 1.72	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt[3]{{\left(ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \color{blue}{\left(\left(maxCos - 1\right) \cdot ux\right)} \cdot \left(1 - maxCos\right)\right)\right)}^{1.5}} \]
   associate-*r* [<=] 1.72	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt[3]{{\left(ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \color{blue}{\left(maxCos - 1\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right)}\right)\right)}^{1.5}} \]
   sub-neg [=>] 1.72	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt[3]{{\left(ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \color{blue}{\left(maxCos + \left(-1\right)\right)} \cdot \left(ux \cdot \left(1 - maxCos\right)\right)\right)\right)}^{1.5}} \]
   metadata-eval [=>] 1.72	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt[3]{{\left(ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \left(maxCos + \color{blue}{-1}\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right)\right)\right)}^{1.5}} \]

9. Final simplification 1.72

   \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt[3]{{\left(ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) - \left(1 - maxCos\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right)\right)\right)}^{1.5}} \]

Alternatives

Alternative 1
Error	4.54%
Cost	30984

\[\begin{array}{l} t_0 := \sin \left(\pi \cdot \left(uy \cdot 2\right)\right)\\ t_1 := t_0 \cdot \sqrt{1 + \left(ux \cdot maxCos + \left(1 - ux\right)\right) \cdot \left(\left(ux + -1\right) - ux \cdot maxCos\right)}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;t_0 \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{elif}\;t_1 \leq 1.13999999484804 \cdot 10^{-6}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(ux, \left(2 - maxCos\right) - maxCos, \left(1 - maxCos\right) \cdot \left(ux \cdot \left(ux \cdot maxCos - ux\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]

Alternative 2
Error	1.72%
Cost	16416

\[\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{-\mathsf{expm1}\left(2 \cdot \mathsf{log1p}\left(ux \cdot maxCos - ux\right)\right)} \]

Alternative 3
Error	1.68%
Cost	13568

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

Alternative 4
Error	1.69%
Cost	10304

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

Alternative 5
Error	4.35%
Cost	10052

\[\begin{array}{l} \mathbf{if}\;maxCos \leq 4.999999873689376 \cdot 10^{-6}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{ux \cdot \left(2 - \left(2 \cdot maxCos + \left(1 - maxCos\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right)\right)\right)} \cdot \left(uy \cdot \pi\right)\right)\\ \end{array} \]

Alternative 6
Error	10.5%
Cost	9988

\[\begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.012000000104308128:\\ \;\;\;\;2 \cdot \left(\sqrt{ux \cdot \left(2 - \left(2 \cdot maxCos + \left(1 - maxCos\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right)\right)\right)} \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]

Alternative 7
Error	4.35%
Cost	9988

\[\begin{array}{l} \mathbf{if}\;maxCos \leq 4.999999873689376 \cdot 10^{-6}:\\ \;\;\;\;\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{ux \cdot \left(2 - \left(2 \cdot maxCos + \left(1 - maxCos\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right)\right)\right)} \cdot \left(uy \cdot \pi\right)\right)\\ \end{array} \]

Alternative 8
Error	23.72%
Cost	7172

\[\begin{array}{l} \mathbf{if}\;ux \leq 0.00022000000171829015:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(ux + \left(-1 - ux \cdot maxCos\right)\right) \cdot \left(\left(1 + ux \cdot maxCos\right) - ux\right)}\right)\\ \end{array} \]

Alternative 9
Error	18.9%
Cost	7104

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

Alternative 10
Error	24.79%
Cost	6916

\[\begin{array}{l} \mathbf{if}\;ux \leq 0.00022000000171829015:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)}\right)\\ \end{array} \]

Alternative 11
Error	34.26%
Cost	6784

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

Alternative 12
Error	34.25%
Cost	6784

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

Alternative 13
Error	36.81%
Cost	6656

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

Reproduce

herbie shell --seed 2023115 
(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)))))))