UniformSampleCone, y

Average Error: 13.4 → 0.5

Time: 20.7s

Precision: binary32

Cost: 16640

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

Math

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

↓

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

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
 (cbrt
  (*
   (pow (* ux (* (- 1.0 maxCos) (+ 2.0 (* ux (+ maxCos -1.0))))) 1.5)
   (pow (sin (* PI (* 2.0 uy))) 3.0))))

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 cbrtf((powf((ux * ((1.0f - maxCos) * (2.0f + (ux * (maxCos + -1.0f))))), 1.5f) * powf(sinf((((float) M_PI) * (2.0f * uy))), 3.0f)));
}

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 cbrt(Float32((Float32(ux * Float32(Float32(Float32(1.0) - maxCos) * Float32(Float32(2.0) + Float32(ux * Float32(maxCos + Float32(-1.0)))))) ^ Float32(1.5)) * (sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) ^ Float32(3.0))))
end

Derivation

1. Initial program 13.4

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

2. Simplified 13.4

   \[\leadsto \color{blue}{\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos - 1, 1\right), ux - \mathsf{fma}\left(ux, maxCos, 1\right), 1\right)}} \]
   Proof
   (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) (sqrt.f32 (fma.f32 (fma.f32 ux (-.f32 maxCos 1) 1) (-.f32 ux (fma.f32 ux maxCos 1)) 1))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (Rewrite<= associate-*l*_binary32 (*.f32 (*.f32 uy 2) (PI.f32)))) (sqrt.f32 (fma.f32 (fma.f32 ux (-.f32 maxCos 1) 1) (-.f32 ux (fma.f32 ux maxCos 1)) 1))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 (*.f32 uy 2) (PI.f32))) (sqrt.f32 (fma.f32 (fma.f32 ux (Rewrite=> sub-neg_binary32 (+.f32 maxCos (neg.f32 1))) 1) (-.f32 ux (fma.f32 ux maxCos 1)) 1))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 (*.f32 uy 2) (PI.f32))) (sqrt.f32 (fma.f32 (fma.f32 ux (+.f32 maxCos (Rewrite=> metadata-eval -1)) 1) (-.f32 ux (fma.f32 ux maxCos 1)) 1))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 (*.f32 uy 2) (PI.f32))) (sqrt.f32 (fma.f32 (fma.f32 ux (Rewrite<= +-commutative_binary32 (+.f32 -1 maxCos)) 1) (-.f32 ux (fma.f32 ux maxCos 1)) 1))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 (*.f32 uy 2) (PI.f32))) (sqrt.f32 (fma.f32 (Rewrite<= fma-def_binary32 (+.f32 (*.f32 ux (+.f32 -1 maxCos)) 1)) (-.f32 ux (fma.f32 ux maxCos 1)) 1))): 2 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 (*.f32 uy 2) (PI.f32))) (sqrt.f32 (fma.f32 (+.f32 (Rewrite<= distribute-rgt-out_binary32 (+.f32 (*.f32 -1 ux) (*.f32 maxCos ux))) 1) (-.f32 ux (fma.f32 ux maxCos 1)) 1))): 0 points increase in error, 1 points decrease in error
   (*.f32 (sin.f32 (*.f32 (*.f32 uy 2) (PI.f32))) (sqrt.f32 (fma.f32 (+.f32 (+.f32 (Rewrite<= neg-mul-1_binary32 (neg.f32 ux)) (*.f32 maxCos ux)) 1) (-.f32 ux (fma.f32 ux maxCos 1)) 1))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 (*.f32 uy 2) (PI.f32))) (sqrt.f32 (fma.f32 (+.f32 (+.f32 (neg.f32 ux) (Rewrite<= *-commutative_binary32 (*.f32 ux maxCos))) 1) (-.f32 ux (fma.f32 ux maxCos 1)) 1))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 (*.f32 uy 2) (PI.f32))) (sqrt.f32 (fma.f32 (Rewrite<= +-commutative_binary32 (+.f32 1 (+.f32 (neg.f32 ux) (*.f32 ux maxCos)))) (-.f32 ux (fma.f32 ux maxCos 1)) 1))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 (*.f32 uy 2) (PI.f32))) (sqrt.f32 (fma.f32 (Rewrite<= associate-+l+_binary32 (+.f32 (+.f32 1 (neg.f32 ux)) (*.f32 ux maxCos))) (-.f32 ux (fma.f32 ux maxCos 1)) 1))): 3 points increase in error, 1 points decrease in error
   (*.f32 (sin.f32 (*.f32 (*.f32 uy 2) (PI.f32))) (sqrt.f32 (fma.f32 (+.f32 (Rewrite<= sub-neg_binary32 (-.f32 1 ux)) (*.f32 ux maxCos)) (-.f32 ux (fma.f32 ux maxCos 1)) 1))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 (*.f32 uy 2) (PI.f32))) (sqrt.f32 (fma.f32 (+.f32 (-.f32 1 ux) (*.f32 ux maxCos)) (-.f32 ux (Rewrite<= fma-def_binary32 (+.f32 (*.f32 ux maxCos) 1))) 1))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 (*.f32 uy 2) (PI.f32))) (sqrt.f32 (fma.f32 (+.f32 (-.f32 1 ux) (*.f32 ux maxCos)) (Rewrite<= unsub-neg_binary32 (+.f32 ux (neg.f32 (+.f32 (*.f32 ux maxCos) 1)))) 1))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 (*.f32 uy 2) (PI.f32))) (sqrt.f32 (fma.f32 (+.f32 (-.f32 1 ux) (*.f32 ux maxCos)) (+.f32 (Rewrite<= remove-double-neg_binary32 (neg.f32 (neg.f32 ux))) (neg.f32 (+.f32 (*.f32 ux maxCos) 1))) 1))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 (*.f32 uy 2) (PI.f32))) (sqrt.f32 (fma.f32 (+.f32 (-.f32 1 ux) (*.f32 ux maxCos)) (Rewrite<= distribute-neg-in_binary32 (neg.f32 (+.f32 (neg.f32 ux) (+.f32 (*.f32 ux maxCos) 1)))) 1))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 (*.f32 uy 2) (PI.f32))) (sqrt.f32 (fma.f32 (+.f32 (-.f32 1 ux) (*.f32 ux maxCos)) (neg.f32 (Rewrite<= +-commutative_binary32 (+.f32 (+.f32 (*.f32 ux maxCos) 1) (neg.f32 ux)))) 1))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 (*.f32 uy 2) (PI.f32))) (sqrt.f32 (fma.f32 (+.f32 (-.f32 1 ux) (*.f32 ux maxCos)) (neg.f32 (Rewrite<= associate-+r+_binary32 (+.f32 (*.f32 ux maxCos) (+.f32 1 (neg.f32 ux))))) 1))): 5 points increase in error, 6 points decrease in error
   (*.f32 (sin.f32 (*.f32 (*.f32 uy 2) (PI.f32))) (sqrt.f32 (fma.f32 (+.f32 (-.f32 1 ux) (*.f32 ux maxCos)) (neg.f32 (+.f32 (*.f32 ux maxCos) (Rewrite<= sub-neg_binary32 (-.f32 1 ux)))) 1))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 (*.f32 uy 2) (PI.f32))) (sqrt.f32 (fma.f32 (+.f32 (-.f32 1 ux) (*.f32 ux maxCos)) (neg.f32 (Rewrite<= +-commutative_binary32 (+.f32 (-.f32 1 ux) (*.f32 ux maxCos)))) 1))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 (*.f32 uy 2) (PI.f32))) (sqrt.f32 (Rewrite<= fma-def_binary32 (+.f32 (*.f32 (+.f32 (-.f32 1 ux) (*.f32 ux maxCos)) (neg.f32 (+.f32 (-.f32 1 ux) (*.f32 ux maxCos)))) 1)))): 98 points increase in error, 75 points decrease in error
   (*.f32 (sin.f32 (*.f32 (*.f32 uy 2) (PI.f32))) (sqrt.f32 (+.f32 (Rewrite<= *-commutative_binary32 (*.f32 (neg.f32 (+.f32 (-.f32 1 ux) (*.f32 ux maxCos))) (+.f32 (-.f32 1 ux) (*.f32 ux maxCos)))) 1))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 (*.f32 uy 2) (PI.f32))) (sqrt.f32 (Rewrite<= +-commutative_binary32 (+.f32 1 (*.f32 (neg.f32 (+.f32 (-.f32 1 ux) (*.f32 ux maxCos))) (+.f32 (-.f32 1 ux) (*.f32 ux maxCos))))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 (*.f32 uy 2) (PI.f32))) (sqrt.f32 (Rewrite<= cancel-sign-sub-inv_binary32 (-.f32 1 (*.f32 (+.f32 (-.f32 1 ux) (*.f32 ux maxCos)) (+.f32 (-.f32 1 ux) (*.f32 ux maxCos))))))): 0 points increase in error, 0 points decrease in error

3. Taylor expanded in ux around 0 0.6

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

4. Simplified 0.5

   \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(\left(-1 + maxCos\right) \cdot \left(1 - maxCos\right)\right) \cdot ux + 2 \cdot \left(1 - maxCos\right)\right)}} \]
   Proof
   (*.f32 ux (+.f32 (*.f32 (*.f32 (+.f32 -1 maxCos) (-.f32 1 maxCos)) ux) (*.f32 2 (-.f32 1 maxCos)))): 0 points increase in error, 0 points decrease in error
   (*.f32 ux (+.f32 (*.f32 (*.f32 (Rewrite<= +-commutative_binary32 (+.f32 maxCos -1)) (-.f32 1 maxCos)) ux) (*.f32 2 (-.f32 1 maxCos)))): 0 points increase in error, 0 points decrease in error
   (*.f32 ux (+.f32 (*.f32 (*.f32 (+.f32 maxCos (Rewrite<= metadata-eval (neg.f32 1))) (-.f32 1 maxCos)) ux) (*.f32 2 (-.f32 1 maxCos)))): 0 points increase in error, 0 points decrease in error
   (*.f32 ux (+.f32 (*.f32 (*.f32 (Rewrite<= sub-neg_binary32 (-.f32 maxCos 1)) (-.f32 1 maxCos)) ux) (*.f32 2 (-.f32 1 maxCos)))): 0 points increase in error, 0 points decrease in error
   (*.f32 ux (+.f32 (*.f32 (*.f32 (-.f32 maxCos 1) (-.f32 1 maxCos)) ux) (*.f32 2 (Rewrite=> sub-neg_binary32 (+.f32 1 (neg.f32 maxCos)))))): 0 points increase in error, 0 points decrease in error
   (*.f32 ux (+.f32 (*.f32 (*.f32 (-.f32 maxCos 1) (-.f32 1 maxCos)) ux) (*.f32 2 (+.f32 1 (Rewrite<= mul-1-neg_binary32 (*.f32 -1 maxCos)))))): 0 points increase in error, 0 points decrease in error
   (*.f32 ux (+.f32 (*.f32 (*.f32 (-.f32 maxCos 1) (-.f32 1 maxCos)) ux) (Rewrite<= count-2_binary32 (+.f32 (+.f32 1 (*.f32 -1 maxCos)) (+.f32 1 (*.f32 -1 maxCos)))))): 0 points increase in error, 0 points decrease in error
   (*.f32 ux (+.f32 (*.f32 (*.f32 (-.f32 maxCos 1) (-.f32 1 maxCos)) ux) (+.f32 (+.f32 1 (*.f32 -1 maxCos)) (+.f32 1 (Rewrite=> mul-1-neg_binary32 (neg.f32 maxCos)))))): 0 points increase in error, 0 points decrease in error
   (*.f32 ux (+.f32 (*.f32 (*.f32 (-.f32 maxCos 1) (-.f32 1 maxCos)) ux) (+.f32 (+.f32 1 (*.f32 -1 maxCos)) (Rewrite<= sub-neg_binary32 (-.f32 1 maxCos))))): 0 points increase in error, 0 points decrease in error
   (*.f32 ux (+.f32 (*.f32 (*.f32 (-.f32 maxCos 1) (-.f32 1 maxCos)) ux) (Rewrite<= associate--l+_binary32 (-.f32 (+.f32 (+.f32 1 (*.f32 -1 maxCos)) 1) maxCos)))): 9 points increase in error, 4 points decrease in error
   (*.f32 ux (+.f32 (*.f32 (*.f32 (-.f32 maxCos 1) (-.f32 1 maxCos)) ux) (-.f32 (+.f32 (+.f32 1 (Rewrite=> mul-1-neg_binary32 (neg.f32 maxCos))) 1) maxCos))): 0 points increase in error, 0 points decrease in error
   (*.f32 ux (+.f32 (*.f32 (*.f32 (-.f32 maxCos 1) (-.f32 1 maxCos)) ux) (-.f32 (+.f32 (Rewrite=> +-commutative_binary32 (+.f32 (neg.f32 maxCos) 1)) 1) maxCos))): 0 points increase in error, 0 points decrease in error
   (*.f32 ux (+.f32 (*.f32 (*.f32 (-.f32 maxCos 1) (-.f32 1 maxCos)) ux) (-.f32 (+.f32 (+.f32 (neg.f32 maxCos) (Rewrite<= metadata-eval (neg.f32 -1))) 1) maxCos))): 0 points increase in error, 0 points decrease in error
   (*.f32 ux (+.f32 (*.f32 (*.f32 (-.f32 maxCos 1) (-.f32 1 maxCos)) ux) (-.f32 (+.f32 (Rewrite<= distribute-neg-in_binary32 (neg.f32 (+.f32 maxCos -1))) 1) maxCos))): 0 points increase in error, 0 points decrease in error
   (*.f32 ux (+.f32 (*.f32 (*.f32 (-.f32 maxCos 1) (-.f32 1 maxCos)) ux) (-.f32 (+.f32 (neg.f32 (+.f32 maxCos (Rewrite<= metadata-eval (neg.f32 1)))) 1) maxCos))): 0 points increase in error, 0 points decrease in error
   (*.f32 ux (+.f32 (*.f32 (*.f32 (-.f32 maxCos 1) (-.f32 1 maxCos)) ux) (-.f32 (+.f32 (neg.f32 (Rewrite<= sub-neg_binary32 (-.f32 maxCos 1))) 1) maxCos))): 0 points increase in error, 0 points decrease in error
   (*.f32 ux (+.f32 (*.f32 (*.f32 (-.f32 maxCos 1) (-.f32 1 maxCos)) ux) (-.f32 (+.f32 (Rewrite<= mul-1-neg_binary32 (*.f32 -1 (-.f32 maxCos 1))) 1) maxCos))): 0 points increase in error, 0 points decrease in error
   (*.f32 ux (+.f32 (*.f32 (*.f32 (-.f32 maxCos 1) (-.f32 1 maxCos)) ux) (-.f32 (Rewrite<= +-commutative_binary32 (+.f32 1 (*.f32 -1 (-.f32 maxCos 1)))) maxCos))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-rgt-out_binary32 (+.f32 (*.f32 (*.f32 (*.f32 (-.f32 maxCos 1) (-.f32 1 maxCos)) ux) ux) (*.f32 (-.f32 (+.f32 1 (*.f32 -1 (-.f32 maxCos 1))) maxCos) ux))): 16 points increase in error, 24 points decrease in error
   (+.f32 (Rewrite<= associate-*r*_binary32 (*.f32 (*.f32 (-.f32 maxCos 1) (-.f32 1 maxCos)) (*.f32 ux ux))) (*.f32 (-.f32 (+.f32 1 (*.f32 -1 (-.f32 maxCos 1))) maxCos) ux)): 1 points increase in error, 1 points decrease in error
   (+.f32 (*.f32 (*.f32 (-.f32 maxCos 1) (-.f32 1 maxCos)) (Rewrite<= unpow2_binary32 (pow.f32 ux 2))) (*.f32 (-.f32 (+.f32 1 (*.f32 -1 (-.f32 maxCos 1))) maxCos) ux)): 0 points increase in error, 0 points decrease in error
   (+.f32 (Rewrite<= associate-*r*_binary32 (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2)))) (*.f32 (-.f32 (+.f32 1 (*.f32 -1 (-.f32 maxCos 1))) maxCos) ux)): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))) (Rewrite<= *-commutative_binary32 (*.f32 ux (-.f32 (+.f32 1 (*.f32 -1 (-.f32 maxCos 1))) maxCos)))): 0 points increase in error, 0 points decrease in error

5. Applied egg-rr 0.5

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

6. Applied egg-rr 0.5

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

7. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.5
Cost	10368

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

Alternative 2
Error	0.5
Cost	10304

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

Alternative 3
Error	0.5
Cost	10176

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

Alternative 4
Error	1.3
Cost	10052

\[\begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0006200000061653554:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(1 - maxCos\right) \cdot \left(2 + ux \cdot \left(maxCos + -1\right)\right)\right)} \cdot \left(uy \cdot \left(2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - ux\right)} \cdot \sin \left(2 \cdot \left(\pi \cdot uy\right)\right)\\ \end{array} \]

Alternative 5
Error	3.4
Cost	9988

\[\begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0026499999221414328:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(1 - maxCos\right) \cdot \left(2 + ux \cdot \left(maxCos + -1\right)\right)\right)} \cdot \left(uy \cdot \left(2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(\pi \cdot uy\right)\right) \cdot \sqrt{ux \cdot 2}\\ \end{array} \]

Alternative 6
Error	6.0
Cost	6976

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

Alternative 7
Error	7.3
Cost	6720

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

Alternative 8
Error	11.7
Cost	6656

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

Alternative 9
Error	28.0
Cost	3584

\[\left(2 \cdot \pi\right) \cdot \left(-1 + \left(1 + uy \cdot \left(ux + ux\right)\right)\right) \]

Alternative 10
Error	27.7
Cost	3584

\[-1 + \left(1 + uy \cdot \left(\pi \cdot \left(2 \cdot \left(ux + ux\right)\right)\right)\right) \]

Reproduce

herbie shell --seed 2022297 
(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)))))))