UniformSampleCone, x

Average Error: 13.6 → 0.3

Time: 21.5s

Precision: binary32

Cost: 20096

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(1 - maxCos\right) \cdot ux\\ \sqrt[3]{\left(\frac{\cos \left(\pi \cdot \left(uy \cdot 2\right)\right)}{2} \cdot \left(1 + \cos \left(uy \cdot \left(\pi \cdot 4\right)\right)\right)\right) \cdot {\left(t_0 \cdot \left(2 - t_0\right)\right)}^{1.5}} \end{array} \]

FPCore

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

↓

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (- 1.0 maxCos) ux)))
   (cbrt
    (*
     (* (/ (cos (* PI (* uy 2.0))) 2.0) (+ 1.0 (cos (* uy (* PI 4.0)))))
     (pow (* t_0 (- 2.0 t_0)) 1.5)))))

C

float code(float ux, float uy, float maxCos) {
	return cosf(((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) {
	float t_0 = (1.0f - maxCos) * ux;
	return cbrtf((((cosf((((float) M_PI) * (uy * 2.0f))) / 2.0f) * (1.0f + cosf((uy * (((float) M_PI) * 4.0f))))) * powf((t_0 * (2.0f - t_0)), 1.5f)));
}

Julia

function code(ux, uy, maxCos)
	return Float32(cos(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)
	t_0 = Float32(Float32(Float32(1.0) - maxCos) * ux)
	return cbrt(Float32(Float32(Float32(cos(Float32(Float32(pi) * Float32(uy * Float32(2.0)))) / Float32(2.0)) * Float32(Float32(1.0) + cos(Float32(uy * Float32(Float32(pi) * Float32(4.0)))))) * (Float32(t_0 * Float32(Float32(2.0) - t_0)) ^ Float32(1.5))))
end

Derivation

1. Initial program 13.6

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

2. Simplified 13.6

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

   [Start] 13.6	\[ \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   associate-*l* [=>] 13.6	\[ \cos \color{blue}{\left(uy \cdot \left(2 \cdot \pi\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   cancel-sign-sub-inv [=>] 13.6	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{1 + \left(-\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}} \]
   +-commutative [=>] 13.6	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(-\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + 1}} \]
   *-commutative [=>] 13.6	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(-\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} + 1} \]
   fma-def [=>] 13.6	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\left(1 - ux\right) + ux \cdot maxCos, -\left(\left(1 - ux\right) + ux \cdot maxCos\right), 1\right)}} \]
   +-commutative [=>] 13.6	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\color{blue}{ux \cdot maxCos + \left(1 - ux\right)}, -\left(\left(1 - ux\right) + ux \cdot maxCos\right), 1\right)} \]
   associate-+r- [=>] 13.6	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\color{blue}{\left(ux \cdot maxCos + 1\right) - ux}, -\left(\left(1 - ux\right) + ux \cdot maxCos\right), 1\right)} \]
   fma-def [=>] 13.6	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(ux, maxCos, 1\right)} - ux, -\left(\left(1 - ux\right) + ux \cdot maxCos\right), 1\right)} \]
   neg-sub0 [=>] 13.6	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right)}, 1\right)} \]
   +-commutative [=>] 13.6	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, 1\right)} \]
   associate-+r- [=>] 13.6	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)}, 1\right)} \]
   associate--r- [=>] 13.6	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(ux \cdot maxCos + 1\right)\right) + ux}, 1\right)} \]
   +-commutative [=>] 13.6	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{ux + \left(0 - \left(ux \cdot maxCos + 1\right)\right)}, 1\right)} \]
   sub0-neg [=>] 13.6	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, ux + \color{blue}{\left(-\left(ux \cdot maxCos + 1\right)\right)}, 1\right)} \]
   sub-neg [<=] 13.6	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{ux - \left(ux \cdot maxCos + 1\right)}, 1\right)} \]
   fma-def [=>] 13.6	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, ux - \color{blue}{\mathsf{fma}\left(ux, maxCos, 1\right)}, 1\right)} \]

3. Taylor expanded in ux around -inf 0.3

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

4. Simplified 0.3

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

   [Start] 0.3	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{-1 \cdot \left({ux}^{2} \cdot {\left(1 + -1 \cdot maxCos\right)}^{2}\right) + 2 \cdot \left(ux \cdot \left(1 + -1 \cdot maxCos\right)\right)} \]
   +-commutative [=>] 0.3	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{2 \cdot \left(ux \cdot \left(1 + -1 \cdot maxCos\right)\right) + -1 \cdot \left({ux}^{2} \cdot {\left(1 + -1 \cdot maxCos\right)}^{2}\right)}} \]
   mul-1-neg [=>] 0.3	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot \left(ux \cdot \left(1 + -1 \cdot maxCos\right)\right) + \color{blue}{\left(-{ux}^{2} \cdot {\left(1 + -1 \cdot maxCos\right)}^{2}\right)}} \]
   unsub-neg [=>] 0.3	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{2 \cdot \left(ux \cdot \left(1 + -1 \cdot maxCos\right)\right) - {ux}^{2} \cdot {\left(1 + -1 \cdot maxCos\right)}^{2}}} \]
   *-commutative [=>] 0.3	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux \cdot \left(1 + -1 \cdot maxCos\right)\right) \cdot 2} - {ux}^{2} \cdot {\left(1 + -1 \cdot maxCos\right)}^{2}} \]
   mul-1-neg [=>] 0.3	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\left(ux \cdot \left(1 + \color{blue}{\left(-maxCos\right)}\right)\right) \cdot 2 - {ux}^{2} \cdot {\left(1 + -1 \cdot maxCos\right)}^{2}} \]
   sub-neg [<=] 0.3	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\left(ux \cdot \color{blue}{\left(1 - maxCos\right)}\right) \cdot 2 - {ux}^{2} \cdot {\left(1 + -1 \cdot maxCos\right)}^{2}} \]
   associate-*l* [=>] 0.3	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(1 - maxCos\right) \cdot 2\right)} - {ux}^{2} \cdot {\left(1 + -1 \cdot maxCos\right)}^{2}} \]
   unpow2 [=>] 0.3	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(1 - maxCos\right) \cdot 2\right) - \color{blue}{\left(ux \cdot ux\right)} \cdot {\left(1 + -1 \cdot maxCos\right)}^{2}} \]
   associate-*l* [=>] 0.3	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(1 - maxCos\right) \cdot 2\right) - \color{blue}{ux \cdot \left(ux \cdot {\left(1 + -1 \cdot maxCos\right)}^{2}\right)}} \]
   mul-1-neg [=>] 0.3	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(1 - maxCos\right) \cdot 2\right) - ux \cdot \left(ux \cdot {\left(1 + \color{blue}{\left(-maxCos\right)}\right)}^{2}\right)} \]
   sub-neg [<=] 0.3	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(1 - maxCos\right) \cdot 2\right) - ux \cdot \left(ux \cdot {\color{blue}{\left(1 - maxCos\right)}}^{2}\right)} \]

5. Applied egg-rr 0.3

   \[\leadsto \color{blue}{\sqrt[3]{{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)}^{3} \cdot {\left(2 \cdot \left(ux \cdot \left(1 - maxCos\right)\right) - {\left(ux \cdot \left(1 - maxCos\right)\right)}^{2}\right)}^{1.5}}} \]

6. Simplified 0.3

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

   [Start] 0.3	\[ \sqrt[3]{{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)}^{3} \cdot {\left(2 \cdot \left(ux \cdot \left(1 - maxCos\right)\right) - {\left(ux \cdot \left(1 - maxCos\right)\right)}^{2}\right)}^{1.5}} \]
   *-commutative [=>] 0.3	\[ \sqrt[3]{{\cos \left(uy \cdot \color{blue}{\left(\pi \cdot 2\right)}\right)}^{3} \cdot {\left(2 \cdot \left(ux \cdot \left(1 - maxCos\right)\right) - {\left(ux \cdot \left(1 - maxCos\right)\right)}^{2}\right)}^{1.5}} \]
   associate-*r* [=>] 0.3	\[ \sqrt[3]{{\cos \color{blue}{\left(\left(uy \cdot \pi\right) \cdot 2\right)}}^{3} \cdot {\left(2 \cdot \left(ux \cdot \left(1 - maxCos\right)\right) - {\left(ux \cdot \left(1 - maxCos\right)\right)}^{2}\right)}^{1.5}} \]
   *-commutative [<=] 0.3	\[ \sqrt[3]{{\cos \color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right)}}^{3} \cdot {\left(2 \cdot \left(ux \cdot \left(1 - maxCos\right)\right) - {\left(ux \cdot \left(1 - maxCos\right)\right)}^{2}\right)}^{1.5}} \]
   unpow2 [=>] 0.3	\[ \sqrt[3]{{\cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}^{3} \cdot {\left(2 \cdot \left(ux \cdot \left(1 - maxCos\right)\right) - \color{blue}{\left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right)}\right)}^{1.5}} \]
   distribute-rgt-out-- [=>] 0.3	\[ \sqrt[3]{{\cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}^{3} \cdot {\color{blue}{\left(\left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(2 - ux \cdot \left(1 - maxCos\right)\right)\right)}}^{1.5}} \]
   *-commutative [=>] 0.3	\[ \sqrt[3]{{\cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}^{3} \cdot {\left(\color{blue}{\left(\left(1 - maxCos\right) \cdot ux\right)} \cdot \left(2 - ux \cdot \left(1 - maxCos\right)\right)\right)}^{1.5}} \]
   *-commutative [=>] 0.3	\[ \sqrt[3]{{\cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}^{3} \cdot {\left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(2 - \color{blue}{\left(1 - maxCos\right) \cdot ux}\right)\right)}^{1.5}} \]

7. Applied egg-rr 0.3

   \[\leadsto \sqrt[3]{\color{blue}{\frac{\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \left(\cos \left(2 \cdot \left(uy \cdot \pi\right) + 2 \cdot \left(uy \cdot \pi\right)\right) + \cos \left(2 \cdot \left(uy \cdot \pi\right) - 2 \cdot \left(uy \cdot \pi\right)\right)\right)}{2}} \cdot {\left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(2 - \left(1 - maxCos\right) \cdot ux\right)\right)}^{1.5}} \]

8. Simplified 0.3

   \[\leadsto \sqrt[3]{\color{blue}{\left(\frac{\cos \left(\pi \cdot \left(uy \cdot 2\right)\right)}{2} \cdot \left(1 + \cos \left(uy \cdot \left(\pi \cdot 4\right)\right)\right)\right)} \cdot {\left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(2 - \left(1 - maxCos\right) \cdot ux\right)\right)}^{1.5}} \]
   Proof

   [Start] 0.3	\[ \sqrt[3]{\frac{\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \left(\cos \left(2 \cdot \left(uy \cdot \pi\right) + 2 \cdot \left(uy \cdot \pi\right)\right) + \cos \left(2 \cdot \left(uy \cdot \pi\right) - 2 \cdot \left(uy \cdot \pi\right)\right)\right)}{2} \cdot {\left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(2 - \left(1 - maxCos\right) \cdot ux\right)\right)}^{1.5}} \]
   associate-/l* [=>] 0.3	\[ \sqrt[3]{\color{blue}{\frac{\cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{\frac{2}{\cos \left(2 \cdot \left(uy \cdot \pi\right) + 2 \cdot \left(uy \cdot \pi\right)\right) + \cos \left(2 \cdot \left(uy \cdot \pi\right) - 2 \cdot \left(uy \cdot \pi\right)\right)}}} \cdot {\left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(2 - \left(1 - maxCos\right) \cdot ux\right)\right)}^{1.5}} \]
   associate-/r/ [=>] 0.3	\[ \sqrt[3]{\color{blue}{\left(\frac{\cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{2} \cdot \left(\cos \left(2 \cdot \left(uy \cdot \pi\right) + 2 \cdot \left(uy \cdot \pi\right)\right) + \cos \left(2 \cdot \left(uy \cdot \pi\right) - 2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \cdot {\left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(2 - \left(1 - maxCos\right) \cdot ux\right)\right)}^{1.5}} \]
   associate-*r* [=>] 0.3	\[ \sqrt[3]{\left(\frac{\cos \color{blue}{\left(\left(2 \cdot uy\right) \cdot \pi\right)}}{2} \cdot \left(\cos \left(2 \cdot \left(uy \cdot \pi\right) + 2 \cdot \left(uy \cdot \pi\right)\right) + \cos \left(2 \cdot \left(uy \cdot \pi\right) - 2 \cdot \left(uy \cdot \pi\right)\right)\right)\right) \cdot {\left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(2 - \left(1 - maxCos\right) \cdot ux\right)\right)}^{1.5}} \]
   *-commutative [=>] 0.3	\[ \sqrt[3]{\left(\frac{\cos \color{blue}{\left(\pi \cdot \left(2 \cdot uy\right)\right)}}{2} \cdot \left(\cos \left(2 \cdot \left(uy \cdot \pi\right) + 2 \cdot \left(uy \cdot \pi\right)\right) + \cos \left(2 \cdot \left(uy \cdot \pi\right) - 2 \cdot \left(uy \cdot \pi\right)\right)\right)\right) \cdot {\left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(2 - \left(1 - maxCos\right) \cdot ux\right)\right)}^{1.5}} \]
   *-commutative [=>] 0.3	\[ \sqrt[3]{\left(\frac{\cos \left(\pi \cdot \color{blue}{\left(uy \cdot 2\right)}\right)}{2} \cdot \left(\cos \left(2 \cdot \left(uy \cdot \pi\right) + 2 \cdot \left(uy \cdot \pi\right)\right) + \cos \left(2 \cdot \left(uy \cdot \pi\right) - 2 \cdot \left(uy \cdot \pi\right)\right)\right)\right) \cdot {\left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(2 - \left(1 - maxCos\right) \cdot ux\right)\right)}^{1.5}} \]
   +-commutative [=>] 0.3	\[ \sqrt[3]{\left(\frac{\cos \left(\pi \cdot \left(uy \cdot 2\right)\right)}{2} \cdot \color{blue}{\left(\cos \left(2 \cdot \left(uy \cdot \pi\right) - 2 \cdot \left(uy \cdot \pi\right)\right) + \cos \left(2 \cdot \left(uy \cdot \pi\right) + 2 \cdot \left(uy \cdot \pi\right)\right)\right)}\right) \cdot {\left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(2 - \left(1 - maxCos\right) \cdot ux\right)\right)}^{1.5}} \]
   +-inverses [=>] 0.3	\[ \sqrt[3]{\left(\frac{\cos \left(\pi \cdot \left(uy \cdot 2\right)\right)}{2} \cdot \left(\cos \color{blue}{0} + \cos \left(2 \cdot \left(uy \cdot \pi\right) + 2 \cdot \left(uy \cdot \pi\right)\right)\right)\right) \cdot {\left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(2 - \left(1 - maxCos\right) \cdot ux\right)\right)}^{1.5}} \]
   cos-0 [=>] 0.3	\[ \sqrt[3]{\left(\frac{\cos \left(\pi \cdot \left(uy \cdot 2\right)\right)}{2} \cdot \left(\color{blue}{1} + \cos \left(2 \cdot \left(uy \cdot \pi\right) + 2 \cdot \left(uy \cdot \pi\right)\right)\right)\right) \cdot {\left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(2 - \left(1 - maxCos\right) \cdot ux\right)\right)}^{1.5}} \]
   distribute-rgt-out [=>] 0.3	\[ \sqrt[3]{\left(\frac{\cos \left(\pi \cdot \left(uy \cdot 2\right)\right)}{2} \cdot \left(1 + \cos \color{blue}{\left(\left(uy \cdot \pi\right) \cdot \left(2 + 2\right)\right)}\right)\right) \cdot {\left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(2 - \left(1 - maxCos\right) \cdot ux\right)\right)}^{1.5}} \]
   metadata-eval [=>] 0.3	\[ \sqrt[3]{\left(\frac{\cos \left(\pi \cdot \left(uy \cdot 2\right)\right)}{2} \cdot \left(1 + \cos \left(\left(uy \cdot \pi\right) \cdot \color{blue}{4}\right)\right)\right) \cdot {\left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(2 - \left(1 - maxCos\right) \cdot ux\right)\right)}^{1.5}} \]

9. Final simplification 0.3

   \[\leadsto \sqrt[3]{\left(\frac{\cos \left(\pi \cdot \left(uy \cdot 2\right)\right)}{2} \cdot \left(1 + \cos \left(uy \cdot \left(\pi \cdot 4\right)\right)\right)\right) \cdot {\left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(2 - \left(1 - maxCos\right) \cdot ux\right)\right)}^{1.5}} \]

Alternatives

Alternative 1
Error	0.3
Cost	16640

\[\begin{array}{l} t_0 := \left(1 - maxCos\right) \cdot ux\\ \sqrt[3]{{\left(t_0 \cdot \left(2 - t_0\right)\right)}^{1.5} \cdot {\cos \left(2 \cdot \left(\pi \cdot uy\right)\right)}^{3}} \end{array} \]

Alternative 2
Error	0.3
Cost	13312

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

Alternative 3
Error	0.3
Cost	10368

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

Alternative 4
Error	1.2
Cost	10244

\[\begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0001694999955361709:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(1 - maxCos\right) \cdot \mathsf{fma}\left(ux, maxCos + -1, 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(2 \cdot \left(\pi \cdot uy\right)\right) \cdot \sqrt{ux \cdot \frac{4 - ux \cdot ux}{2 + ux}}\\ \end{array} \]

Alternative 5
Error	0.3
Cost	10176

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

Alternative 6
Error	0.3
Cost	10176

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

Alternative 7
Error	1.2
Cost	10052

\[\begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0001694999955361709:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(1 - maxCos\right) \cdot \mathsf{fma}\left(ux, maxCos + -1, 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]

Alternative 8
Error	3.4
Cost	9988

\[\begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0044999998062849045:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(1 - maxCos\right) \cdot \mathsf{fma}\left(ux, maxCos + -1, 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]

Alternative 9
Error	6.4
Cost	6752

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

Alternative 10
Error	6.5
Cost	3616

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

Alternative 11
Error	7.8
Cost	3360

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

Alternative 12
Error	12.2
Cost	3296

\[\sqrt{2 \cdot ux} \]

Alternative 13
Error	29.9
Cost	3232

\[\sqrt{0} \]

Reproduce

herbie shell --seed 2023060 
(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)))))))