UniformSampleCone, x

Average Error: 13.6 → 0.3

Time: 22.4s

Precision: binary32

Cost: 37056

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

Math

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

↓

\[\frac{2 \cdot \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) - \left(\cos \left(\frac{\pi + \left(\left(-\pi \cdot \left(0.5 - \left(uy + uy\right)\right)\right) + \pi \cdot \left(\left(1 + \frac{uy}{-0.25}\right) - \left(\left(uy + uy\right) - 0.5\right)\right)\right)}{2}\right) + \cos \left(\frac{\pi - \pi \cdot \left(1 - uy \cdot 4\right)}{2}\right)\right)}{2} \cdot \sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux + \left(-{\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}\right)} \]

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
 (*
  (/
   (-
    (* 2.0 (cos (* uy (* 2.0 PI))))
    (+
     (cos
      (/
       (+
        PI
        (+
         (- (* PI (- 0.5 (+ uy uy))))
         (* PI (- (+ 1.0 (/ uy -0.25)) (- (+ uy uy) 0.5)))))
       2.0))
     (cos (/ (- PI (* PI (- 1.0 (* uy 4.0)))) 2.0))))
   2.0)
  (sqrt
   (+ (* (- 2.0 (* 2.0 maxCos)) ux) (- (pow (* ux (+ maxCos -1.0)) 2.0))))))

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) {
	return (((2.0f * cosf((uy * (2.0f * ((float) M_PI))))) - (cosf(((((float) M_PI) + (-(((float) M_PI) * (0.5f - (uy + uy))) + (((float) M_PI) * ((1.0f + (uy / -0.25f)) - ((uy + uy) - 0.5f))))) / 2.0f)) + cosf(((((float) M_PI) - (((float) M_PI) * (1.0f - (uy * 4.0f)))) / 2.0f)))) / 2.0f) * sqrtf((((2.0f - (2.0f * maxCos)) * ux) + -powf((ux * (maxCos + -1.0f)), 2.0f)));
}

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)
	return Float32(Float32(Float32(Float32(Float32(2.0) * cos(Float32(uy * Float32(Float32(2.0) * Float32(pi))))) - Float32(cos(Float32(Float32(Float32(pi) + Float32(Float32(-Float32(Float32(pi) * Float32(Float32(0.5) - Float32(uy + uy)))) + Float32(Float32(pi) * Float32(Float32(Float32(1.0) + Float32(uy / Float32(-0.25))) - Float32(Float32(uy + uy) - Float32(0.5)))))) / Float32(2.0))) + cos(Float32(Float32(Float32(pi) - Float32(Float32(pi) * Float32(Float32(1.0) - Float32(uy * Float32(4.0))))) / Float32(2.0))))) / Float32(2.0)) * sqrt(Float32(Float32(Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)) * ux) + Float32(-(Float32(ux * Float32(maxCos + Float32(-1.0))) ^ Float32(2.0))))))
end

MATLAB

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

↓

function tmp = code(ux, uy, maxCos)
	tmp = (((single(2.0) * cos((uy * (single(2.0) * single(pi))))) - (cos(((single(pi) + (-(single(pi) * (single(0.5) - (uy + uy))) + (single(pi) * ((single(1.0) + (uy / single(-0.25))) - ((uy + uy) - single(0.5)))))) / single(2.0))) + cos(((single(pi) - (single(pi) * (single(1.0) - (uy * single(4.0))))) / single(2.0))))) / single(2.0)) * sqrt((((single(2.0) - (single(2.0) * maxCos)) * ux) + -((ux * (maxCos + single(-1.0))) ^ single(2.0))));
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

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

   [Start] 13.6	\[ \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   rational_best-simplify-59 [=>] 13.6	\[ \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \color{blue}{\left(ux \cdot maxCos - \left(-\left(1 - ux\right)\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   rational_best-simplify-59 [=>] 13.6	\[ \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(ux \cdot maxCos - \left(-\left(1 - ux\right)\right)\right) \cdot \color{blue}{\left(ux \cdot maxCos - \left(-\left(1 - ux\right)\right)\right)}} \]
   rational_best-simplify-14 [=>] 13.6	\[ \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(ux \cdot maxCos - \color{blue}{\left(0 - \left(1 - ux\right)\right)}\right) \cdot \left(ux \cdot maxCos - \left(-\left(1 - ux\right)\right)\right)} \]
   rational_best-simplify-51 [=>] 13.6	\[ \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(ux \cdot maxCos - \color{blue}{\left(ux - \left(1 - 0\right)\right)}\right) \cdot \left(ux \cdot maxCos - \left(-\left(1 - ux\right)\right)\right)} \]
   metadata-eval [=>] 13.6	\[ \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(ux \cdot maxCos - \left(ux - \color{blue}{1}\right)\right) \cdot \left(ux \cdot maxCos - \left(-\left(1 - ux\right)\right)\right)} \]
   rational_best-simplify-18 [=>] 13.6	\[ \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(ux \cdot maxCos - \color{blue}{\left(ux + -1\right)}\right) \cdot \left(ux \cdot maxCos - \left(-\left(1 - ux\right)\right)\right)} \]
   rational_best-simplify-57 [=>] 13.5	\[ \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \color{blue}{\left(\left(ux \cdot maxCos - ux\right) + \left(--1\right)\right)} \cdot \left(ux \cdot maxCos - \left(-\left(1 - ux\right)\right)\right)} \]
   metadata-eval [=>] 13.5	\[ \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(ux \cdot maxCos - ux\right) + \color{blue}{1}\right) \cdot \left(ux \cdot maxCos - \left(-\left(1 - ux\right)\right)\right)} \]
   rational_best-simplify-3 [=>] 13.5	\[ \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \color{blue}{\left(1 + \left(ux \cdot maxCos - ux\right)\right)} \cdot \left(ux \cdot maxCos - \left(-\left(1 - ux\right)\right)\right)} \]
   rational_best-simplify-14 [=>] 13.5	\[ \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(1 + \left(ux \cdot maxCos - ux\right)\right) \cdot \left(ux \cdot maxCos - \color{blue}{\left(0 - \left(1 - ux\right)\right)}\right)} \]
   rational_best-simplify-51 [=>] 13.5	\[ \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(1 + \left(ux \cdot maxCos - ux\right)\right) \cdot \left(ux \cdot maxCos - \color{blue}{\left(ux - \left(1 - 0\right)\right)}\right)} \]
   metadata-eval [=>] 13.5	\[ \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(1 + \left(ux \cdot maxCos - ux\right)\right) \cdot \left(ux \cdot maxCos - \left(ux - \color{blue}{1}\right)\right)} \]
   rational_best-simplify-18 [=>] 13.5	\[ \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(1 + \left(ux \cdot maxCos - ux\right)\right) \cdot \left(ux \cdot maxCos - \color{blue}{\left(ux + -1\right)}\right)} \]
   rational_best-simplify-57 [=>] 13.6	\[ \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(1 + \left(ux \cdot maxCos - ux\right)\right) \cdot \color{blue}{\left(\left(ux \cdot maxCos - ux\right) + \left(--1\right)\right)}} \]
   metadata-eval [=>] 13.6	\[ \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(1 + \left(ux \cdot maxCos - ux\right)\right) \cdot \left(\left(ux \cdot maxCos - ux\right) + \color{blue}{1}\right)} \]
   rational_best-simplify-3 [=>] 13.6	\[ \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(1 + \left(ux \cdot maxCos - ux\right)\right) \cdot \color{blue}{\left(1 + \left(ux \cdot maxCos - ux\right)\right)}} \]

3. Taylor expanded in ux around 0 0.3

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

4. Simplified 0.3

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

   [Start] 0.3	\[ \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux + -1 \cdot \left({\left(maxCos - 1\right)}^{2} \cdot {ux}^{2}\right)} \]
   rational_best-simplify-1 [=>] 0.3	\[ \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux + \color{blue}{\left({\left(maxCos - 1\right)}^{2} \cdot {ux}^{2}\right) \cdot -1}} \]
   rational_best-simplify-10 [=>] 0.3	\[ \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux + \color{blue}{\left(-{\left(maxCos - 1\right)}^{2} \cdot {ux}^{2}\right)}} \]
   exponential-simplify-28 [=>] 0.3	\[ \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux + \left(-\color{blue}{{\left(\left(maxCos - 1\right) \cdot ux\right)}^{2}}\right)} \]
   rational_best-simplify-1 [=>] 0.3	\[ \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux + \left(-{\color{blue}{\left(ux \cdot \left(maxCos - 1\right)\right)}}^{2}\right)} \]
   rational_best-simplify-18 [=>] 0.3	\[ \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux + \left(-{\left(ux \cdot \color{blue}{\left(maxCos + -1\right)}\right)}^{2}\right)} \]

5. Applied egg-rr 0.3

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

6. Simplified 0.3

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

   [Start] 0.3	\[ \frac{2 \cdot \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) - \left(\cos \left(\frac{\pi}{2} + \frac{\pi \cdot \left(1 - uy \cdot 4\right)}{2}\right) + \cos \left(\frac{\pi - \pi \cdot \left(1 - uy \cdot 4\right)}{2}\right)\right)}{2} \cdot \sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux + \left(-{\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}\right)} \]
   rational_best-simplify-57 [=>] 0.4	\[ \frac{\color{blue}{\left(2 \cdot \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) - \cos \left(\frac{\pi}{2} + \frac{\pi \cdot \left(1 - uy \cdot 4\right)}{2}\right)\right) + \left(-\cos \left(\frac{\pi - \pi \cdot \left(1 - uy \cdot 4\right)}{2}\right)\right)}}{2} \cdot \sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux + \left(-{\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}\right)} \]
   rational_best-simplify-65 [=>] 0.4	\[ \color{blue}{\left(\frac{2 \cdot \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) - \cos \left(\frac{\pi}{2} + \frac{\pi \cdot \left(1 - uy \cdot 4\right)}{2}\right)}{2} + \frac{-\cos \left(\frac{\pi - \pi \cdot \left(1 - uy \cdot 4\right)}{2}\right)}{2}\right)} \cdot \sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux + \left(-{\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}\right)} \]

7. Applied egg-rr 0.3

   \[\leadsto \frac{2 \cdot \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) - \left(\cos \left(\frac{\pi + \color{blue}{\left(\pi \cdot \left(1 + \frac{uy}{-0.25}\right) - \left(\left(\pi \cdot \left(uy + uy\right) - \pi \cdot 0.5\right) + \pi \cdot \left(0.5 - \left(uy + uy\right)\right)\right)\right)}}{2}\right) + \cos \left(\frac{\pi - \pi \cdot \left(1 - uy \cdot 4\right)}{2}\right)\right)}{2} \cdot \sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux + \left(-{\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}\right)} \]

8. Simplified 0.3

   \[\leadsto \frac{2 \cdot \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) - \left(\cos \left(\frac{\pi + \color{blue}{\left(\left(-\pi \cdot \left(0.5 - \left(uy + uy\right)\right)\right) + \pi \cdot \left(\left(1 + \frac{uy}{-0.25}\right) - \left(\left(uy + uy\right) - 0.5\right)\right)\right)}}{2}\right) + \cos \left(\frac{\pi - \pi \cdot \left(1 - uy \cdot 4\right)}{2}\right)\right)}{2} \cdot \sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux + \left(-{\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}\right)} \]
   Proof

   [Start] 0.3	\[ \frac{2 \cdot \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) - \left(\cos \left(\frac{\pi + \left(\pi \cdot \left(1 + \frac{uy}{-0.25}\right) - \left(\left(\pi \cdot \left(uy + uy\right) - \pi \cdot 0.5\right) + \pi \cdot \left(0.5 - \left(uy + uy\right)\right)\right)\right)}{2}\right) + \cos \left(\frac{\pi - \pi \cdot \left(1 - uy \cdot 4\right)}{2}\right)\right)}{2} \cdot \sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux + \left(-{\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}\right)} \]
   rational_best-simplify-57 [=>] 0.3	\[ \frac{2 \cdot \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) - \left(\cos \left(\frac{\pi + \color{blue}{\left(\left(\pi \cdot \left(1 + \frac{uy}{-0.25}\right) - \left(\pi \cdot \left(uy + uy\right) - \pi \cdot 0.5\right)\right) + \left(-\pi \cdot \left(0.5 - \left(uy + uy\right)\right)\right)\right)}}{2}\right) + \cos \left(\frac{\pi - \pi \cdot \left(1 - uy \cdot 4\right)}{2}\right)\right)}{2} \cdot \sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux + \left(-{\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}\right)} \]
   rational_best-simplify-3 [=>] 0.3	\[ \frac{2 \cdot \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) - \left(\cos \left(\frac{\pi + \color{blue}{\left(\left(-\pi \cdot \left(0.5 - \left(uy + uy\right)\right)\right) + \left(\pi \cdot \left(1 + \frac{uy}{-0.25}\right) - \left(\pi \cdot \left(uy + uy\right) - \pi \cdot 0.5\right)\right)\right)}}{2}\right) + \cos \left(\frac{\pi - \pi \cdot \left(1 - uy \cdot 4\right)}{2}\right)\right)}{2} \cdot \sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux + \left(-{\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}\right)} \]
   rational_best-simplify-1 [=>] 0.3	\[ \frac{2 \cdot \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) - \left(\cos \left(\frac{\pi + \left(\left(-\pi \cdot \left(0.5 - \left(uy + uy\right)\right)\right) + \left(\color{blue}{\left(1 + \frac{uy}{-0.25}\right) \cdot \pi} - \left(\pi \cdot \left(uy + uy\right) - \pi \cdot 0.5\right)\right)\right)}{2}\right) + \cos \left(\frac{\pi - \pi \cdot \left(1 - uy \cdot 4\right)}{2}\right)\right)}{2} \cdot \sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux + \left(-{\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}\right)} \]
   rational_best-simplify-1 [=>] 0.3	\[ \frac{2 \cdot \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) - \left(\cos \left(\frac{\pi + \left(\left(-\pi \cdot \left(0.5 - \left(uy + uy\right)\right)\right) + \left(\left(1 + \frac{uy}{-0.25}\right) \cdot \pi - \left(\color{blue}{\left(uy + uy\right) \cdot \pi} - \pi \cdot 0.5\right)\right)\right)}{2}\right) + \cos \left(\frac{\pi - \pi \cdot \left(1 - uy \cdot 4\right)}{2}\right)\right)}{2} \cdot \sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux + \left(-{\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}\right)} \]
   rational_best-simplify-62 [=>] 0.3	\[ \frac{2 \cdot \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) - \left(\cos \left(\frac{\pi + \left(\left(-\pi \cdot \left(0.5 - \left(uy + uy\right)\right)\right) + \left(\left(1 + \frac{uy}{-0.25}\right) \cdot \pi - \color{blue}{\pi \cdot \left(\left(uy + uy\right) - 0.5\right)}\right)\right)}{2}\right) + \cos \left(\frac{\pi - \pi \cdot \left(1 - uy \cdot 4\right)}{2}\right)\right)}{2} \cdot \sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux + \left(-{\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}\right)} \]
   rational_best-simplify-62 [=>] 0.3	\[ \frac{2 \cdot \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) - \left(\cos \left(\frac{\pi + \left(\left(-\pi \cdot \left(0.5 - \left(uy + uy\right)\right)\right) + \color{blue}{\pi \cdot \left(\left(1 + \frac{uy}{-0.25}\right) - \left(\left(uy + uy\right) - 0.5\right)\right)}\right)}{2}\right) + \cos \left(\frac{\pi - \pi \cdot \left(1 - uy \cdot 4\right)}{2}\right)\right)}{2} \cdot \sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux + \left(-{\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}\right)} \]

9. Final simplification 0.3

   \[\leadsto \frac{2 \cdot \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) - \left(\cos \left(\frac{\pi + \left(\left(-\pi \cdot \left(0.5 - \left(uy + uy\right)\right)\right) + \pi \cdot \left(\left(1 + \frac{uy}{-0.25}\right) - \left(\left(uy + uy\right) - 0.5\right)\right)\right)}{2}\right) + \cos \left(\frac{\pi - \pi \cdot \left(1 - uy \cdot 4\right)}{2}\right)\right)}{2} \cdot \sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux + \left(-{\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}\right)} \]

Alternatives

Alternative 1
Error	0.3
Cost	30112

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

Alternative 2
Error	1.1
Cost	19716

\[\begin{array}{l} \mathbf{if}\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \leq 0.9999997019767761:\\ \;\;\;\;\sqrt{2 \cdot ux - {ux}^{2}} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux - {\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}}\\ \end{array} \]

Alternative 3
Error	3.4
Cost	16420

\[\begin{array}{l} t_0 := \cos \left(\left(uy \cdot 2\right) \cdot \pi\right)\\ \mathbf{if}\;t_0 \leq 0.9999845027923584:\\ \;\;\;\;t_0 \cdot \sqrt{2 \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux - {\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}}\\ \end{array} \]

Alternative 4
Error	0.3
Cost	13408

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

Alternative 5
Error	0.8
Cost	13280

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

Alternative 6
Error	2.7
Cost	10884

\[\begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ t_1 := 1 + \left(ux \cdot maxCos - ux\right)\\ t_2 := \cos \left(\left(uy \cdot 2\right) \cdot \pi\right)\\ \mathbf{if}\;1 - t_0 \cdot t_0 \leq 0.00022000000171829015:\\ \;\;\;\;t_2 \cdot \sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \sqrt{1 - t_1 \cdot t_1}\\ \end{array} \]

Alternative 7
Error	2.8
Cost	10372

\[\begin{array}{l} t_0 := \cos \left(\left(uy \cdot 2\right) \cdot \pi\right)\\ \mathbf{if}\;ux \leq 0.00011000000085914508:\\ \;\;\;\;t_0 \cdot \sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \sqrt{1 + \left(maxCos \cdot ux - \left(-1 + ux\right)\right) \cdot \left(ux + \left(-1 - maxCos \cdot ux\right)\right)}\\ \end{array} \]

Alternative 8
Error	2.7
Cost	10372

\[\begin{array}{l} t_0 := \cos \left(\left(uy \cdot 2\right) \cdot \pi\right)\\ \mathbf{if}\;ux \leq 0.00011000000085914508:\\ \;\;\;\;t_0 \cdot \sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \sqrt{\left(1 + \left(ux \cdot maxCos - ux\right)\right) \cdot \left(ux + \left(-1 - ux \cdot maxCos\right)\right) + 1}\\ \end{array} \]

Alternative 9
Error	3.1
Cost	10116

\[\begin{array}{l} t_0 := \left(2 - 2 \cdot maxCos\right) \cdot ux\\ \mathbf{if}\;uy \cdot 2 \leq 0.0017500000540167093:\\ \;\;\;\;\sqrt{t_0 - {\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{t_0}\\ \end{array} \]

Alternative 10
Error	6.5
Cost	6848

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

Alternative 11
Error	8.1
Cost	6788

\[\begin{array}{l} \mathbf{if}\;ux \leq 0.00011000000085914508:\\ \;\;\;\;\sqrt{ux \cdot \left(2 \cdot maxCos - \left(maxCos \cdot 4 + -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - {\left(1 - \left(ux - maxCos \cdot ux\right)\right)}^{2}}\\ \end{array} \]

Alternative 12
Error	8.5
Cost	6660

\[\begin{array}{l} \mathbf{if}\;ux \leq 0.00011000000085914508:\\ \;\;\;\;\sqrt{ux \cdot \left(2 \cdot maxCos - \left(maxCos \cdot 4 + -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - {\left(1 - ux\right)}^{2}}\\ \end{array} \]

Alternative 13
Error	11.4
Cost	3552

\[\sqrt{ux \cdot \left(2 \cdot maxCos - \left(maxCos \cdot 4 + -2\right)\right)} \]

Alternative 14
Error	11.4
Cost	3424

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

Alternative 15
Error	12.1
Cost	3296

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

Reproduce

herbie shell --seed 2023100 
(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)))))))