UniformSampleCone, y

Average Error: 13.4 → 0.5

Time: 16.1s

Precision: binary32

Cost: 19776

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

Math

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

↓

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

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) (fma ux (- 1.0 maxCos) -2.0))) 1.5)
   (pow (sin (* PI (* uy 2.0))) 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) * fmaf(ux, (1.0f - maxCos), -2.0f))), 1.5f) * powf(sinf((((float) M_PI) * (uy * 2.0f))), 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) * fma(ux, Float32(Float32(1.0) - maxCos), Float32(-2.0)))) ^ Float32(1.5)) * (sin(Float32(Float32(pi) * Float32(uy * Float32(2.0)))) ^ 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))): 0 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))): 5 points increase in error, 3 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))): 8 points increase in error, 5 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)))): 86 points increase in error, 68 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 0.5

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

4. Simplified 0.5

   \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(\left(-1 + maxCos\right) \cdot \left(1 - maxCos\right)\right) \cdot ux - \left(-1 + maxCos\right) \cdot 2\right)}} \]
   Proof
   (*.f32 ux (-.f32 (*.f32 (*.f32 (+.f32 -1 maxCos) (-.f32 1 maxCos)) ux) (*.f32 (+.f32 -1 maxCos) 2))): 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 (+.f32 -1 maxCos) 2))): 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 (+.f32 -1 maxCos) 2))): 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 (+.f32 -1 maxCos) 2))): 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 maxCos -1)) 2))): 0 points increase in error, 0 points decrease in error
   (*.f32 ux (-.f32 (*.f32 (*.f32 (-.f32 maxCos 1) (-.f32 1 maxCos)) ux) (*.f32 (+.f32 maxCos (Rewrite<= metadata-eval (neg.f32 1))) 2))): 0 points increase in error, 0 points decrease in error
   (*.f32 ux (-.f32 (*.f32 (*.f32 (-.f32 maxCos 1) (-.f32 1 maxCos)) ux) (*.f32 (Rewrite<= sub-neg_binary32 (-.f32 maxCos 1)) 2))): 0 points increase in error, 0 points decrease in error
   (*.f32 ux (-.f32 (*.f32 (*.f32 (-.f32 maxCos 1) (-.f32 1 maxCos)) ux) (*.f32 (-.f32 maxCos 1) (Rewrite<= metadata-eval (+.f32 1 1))))): 0 points increase in error, 0 points decrease in error
   (*.f32 ux (-.f32 (*.f32 (*.f32 (-.f32 maxCos 1) (-.f32 1 maxCos)) ux) (Rewrite<= distribute-rgt-out_binary32 (+.f32 (*.f32 1 (-.f32 maxCos 1)) (*.f32 1 (-.f32 maxCos 1)))))): 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<= metadata-eval (neg.f32 -1)) (-.f32 maxCos 1)) (*.f32 1 (-.f32 maxCos 1))))): 0 points increase in error, 0 points decrease in error
   (*.f32 ux (-.f32 (*.f32 (*.f32 (-.f32 maxCos 1) (-.f32 1 maxCos)) ux) (+.f32 (Rewrite<= distribute-lft-neg-in_binary32 (neg.f32 (*.f32 -1 (-.f32 maxCos 1)))) (*.f32 1 (-.f32 maxCos 1))))): 0 points increase in error, 0 points decrease in error
   (*.f32 ux (-.f32 (*.f32 (*.f32 (-.f32 maxCos 1) (-.f32 1 maxCos)) ux) (+.f32 (Rewrite<= distribute-rgt-neg-out_binary32 (*.f32 -1 (neg.f32 (-.f32 maxCos 1)))) (*.f32 1 (-.f32 maxCos 1))))): 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 (neg.f32 (Rewrite=> sub-neg_binary32 (+.f32 maxCos (neg.f32 1))))) (*.f32 1 (-.f32 maxCos 1))))): 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 (neg.f32 (+.f32 maxCos (Rewrite=> metadata-eval -1)))) (*.f32 1 (-.f32 maxCos 1))))): 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 (neg.f32 (Rewrite=> +-commutative_binary32 (+.f32 -1 maxCos)))) (*.f32 1 (-.f32 maxCos 1))))): 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 (Rewrite=> distribute-neg-in_binary32 (+.f32 (neg.f32 -1) (neg.f32 maxCos)))) (*.f32 1 (-.f32 maxCos 1))))): 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 (Rewrite=> metadata-eval 1) (neg.f32 maxCos))) (*.f32 1 (-.f32 maxCos 1))))): 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 (Rewrite<= mul-1-neg_binary32 (*.f32 -1 maxCos)))) (*.f32 1 (-.f32 maxCos 1))))): 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 (*.f32 -1 maxCos))) (Rewrite=> *-lft-identity_binary32 (-.f32 maxCos 1))))): 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 (*.f32 -1 maxCos))) maxCos) 1)))): 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 maxCos (*.f32 -1 (+.f32 1 (*.f32 -1 maxCos))))) 1))): 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 maxCos (*.f32 -1 (+.f32 1 (*.f32 -1 maxCos)))) 1) ux))): 30 points increase in error, 25 points decrease in error
   (-.f32 (Rewrite<= associate-*r*_binary32 (*.f32 (*.f32 (-.f32 maxCos 1) (-.f32 1 maxCos)) (*.f32 ux ux))) (*.f32 (-.f32 (+.f32 maxCos (*.f32 -1 (+.f32 1 (*.f32 -1 maxCos)))) 1) ux)): 0 points increase in error, 0 points decrease in error
   (-.f32 (*.f32 (*.f32 (-.f32 maxCos 1) (-.f32 1 maxCos)) (Rewrite<= unpow2_binary32 (pow.f32 ux 2))) (*.f32 (-.f32 (+.f32 maxCos (*.f32 -1 (+.f32 1 (*.f32 -1 maxCos)))) 1) ux)): 0 points increase in error, 0 points decrease in error
   (Rewrite=> fma-neg_binary32 (fma.f32 (*.f32 (-.f32 maxCos 1) (-.f32 1 maxCos)) (pow.f32 ux 2) (neg.f32 (*.f32 (-.f32 (+.f32 maxCos (*.f32 -1 (+.f32 1 (*.f32 -1 maxCos)))) 1) ux)))): 0 points increase in error, 1 points decrease in error
   (fma.f32 (*.f32 (-.f32 maxCos 1) (-.f32 1 maxCos)) (pow.f32 ux 2) (Rewrite<= mul-1-neg_binary32 (*.f32 -1 (*.f32 (-.f32 (+.f32 maxCos (*.f32 -1 (+.f32 1 (*.f32 -1 maxCos)))) 1) ux)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary32 (+.f32 (*.f32 (*.f32 (-.f32 maxCos 1) (-.f32 1 maxCos)) (pow.f32 ux 2)) (*.f32 -1 (*.f32 (-.f32 (+.f32 maxCos (*.f32 -1 (+.f32 1 (*.f32 -1 maxCos)))) 1) ux)))): 1 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 -1 (*.f32 (-.f32 (+.f32 maxCos (*.f32 -1 (+.f32 1 (*.f32 -1 maxCos)))) 1) 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=> associate-*r*_binary32 (*.f32 (*.f32 -1 (-.f32 (+.f32 maxCos (*.f32 -1 (+.f32 1 (*.f32 -1 maxCos)))) 1)) ux))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))) (*.f32 (Rewrite=> mul-1-neg_binary32 (neg.f32 (-.f32 (+.f32 maxCos (*.f32 -1 (+.f32 1 (*.f32 -1 maxCos)))) 1))) ux)): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))) (*.f32 (neg.f32 (Rewrite=> sub-neg_binary32 (+.f32 (+.f32 maxCos (*.f32 -1 (+.f32 1 (*.f32 -1 maxCos)))) (neg.f32 1)))) ux)): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))) (*.f32 (neg.f32 (+.f32 (+.f32 maxCos (*.f32 -1 (+.f32 1 (*.f32 -1 maxCos)))) (Rewrite=> metadata-eval -1))) ux)): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))) (*.f32 (neg.f32 (Rewrite=> +-commutative_binary32 (+.f32 -1 (+.f32 maxCos (*.f32 -1 (+.f32 1 (*.f32 -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))) (*.f32 (Rewrite=> distribute-neg-in_binary32 (+.f32 (neg.f32 -1) (neg.f32 (+.f32 maxCos (*.f32 -1 (+.f32 1 (*.f32 -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))) (*.f32 (+.f32 (Rewrite=> metadata-eval 1) (neg.f32 (+.f32 maxCos (*.f32 -1 (+.f32 1 (*.f32 -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))) (*.f32 (+.f32 1 (neg.f32 (+.f32 maxCos (*.f32 -1 (+.f32 1 (Rewrite=> mul-1-neg_binary32 (neg.f32 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))) (*.f32 (+.f32 1 (neg.f32 (+.f32 maxCos (Rewrite=> distribute-lft-in_binary32 (+.f32 (*.f32 -1 1) (*.f32 -1 (neg.f32 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))) (*.f32 (+.f32 1 (neg.f32 (+.f32 maxCos (+.f32 (Rewrite=> metadata-eval -1) (*.f32 -1 (neg.f32 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))) (*.f32 (+.f32 1 (neg.f32 (Rewrite=> associate-+r+_binary32 (+.f32 (+.f32 maxCos -1) (*.f32 -1 (neg.f32 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))) (*.f32 (+.f32 1 (neg.f32 (+.f32 (+.f32 maxCos (Rewrite<= metadata-eval (neg.f32 1))) (*.f32 -1 (neg.f32 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))) (*.f32 (+.f32 1 (neg.f32 (+.f32 (Rewrite<= sub-neg_binary32 (-.f32 maxCos 1)) (*.f32 -1 (neg.f32 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))) (*.f32 (+.f32 1 (neg.f32 (+.f32 (Rewrite<= *-lft-identity_binary32 (*.f32 1 (-.f32 maxCos 1))) (*.f32 -1 (neg.f32 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))) (*.f32 (+.f32 1 (neg.f32 (+.f32 (*.f32 1 (-.f32 maxCos 1)) (*.f32 -1 (Rewrite<= mul-1-neg_binary32 (*.f32 -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))) (*.f32 (+.f32 1 (neg.f32 (+.f32 (*.f32 1 (-.f32 maxCos 1)) (Rewrite=> associate-*r*_binary32 (*.f32 (*.f32 -1 -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))) (*.f32 (+.f32 1 (neg.f32 (+.f32 (*.f32 1 (-.f32 maxCos 1)) (*.f32 (Rewrite=> metadata-eval 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))) (*.f32 (+.f32 1 (neg.f32 (Rewrite=> distribute-lft-out_binary32 (*.f32 1 (+.f32 (-.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))) (*.f32 (+.f32 1 (neg.f32 (Rewrite<= distribute-rgt-out_binary32 (+.f32 (*.f32 (-.f32 maxCos 1) 1) (*.f32 maxCos 1))))) ux)): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))) (*.f32 (+.f32 1 (neg.f32 (+.f32 (Rewrite=> *-rgt-identity_binary32 (-.f32 maxCos 1)) (*.f32 maxCos 1)))) ux)): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))) (*.f32 (+.f32 1 (neg.f32 (+.f32 (-.f32 maxCos 1) (Rewrite=> *-rgt-identity_binary32 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))) (*.f32 (+.f32 1 (Rewrite<= distribute-neg-out_binary32 (+.f32 (neg.f32 (-.f32 maxCos 1)) (neg.f32 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))) (*.f32 (+.f32 1 (+.f32 (Rewrite<= mul-1-neg_binary32 (*.f32 -1 (-.f32 maxCos 1))) (neg.f32 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))) (*.f32 (+.f32 1 (Rewrite<= sub-neg_binary32 (-.f32 (*.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))) (*.f32 (Rewrite<= associate--l+_binary32 (-.f32 (+.f32 1 (*.f32 -1 (-.f32 maxCos 1))) maxCos)) ux)): 3 points increase in error, 4 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

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

6. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.5
Cost	10304

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

Alternative 2
Error	1.3
Cost	10116

\[\begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0007999999797903001:\\ \;\;\;\;\sqrt{ux \cdot \left(ux \cdot \left(\left(-1 + maxCos\right) \cdot \left(1 - maxCos\right)\right) + \left(-1 + maxCos\right) \cdot -2\right)} \cdot \left(\pi \cdot \left(uy \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot 2 - ux \cdot ux} \cdot \sin \left(2 \cdot \left(\pi \cdot uy\right)\right)\\ \end{array} \]

Alternative 3
Error	1.3
Cost	10052

\[\begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0007999999797903001:\\ \;\;\;\;\sqrt{ux \cdot \left(ux \cdot \left(\left(-1 + maxCos\right) \cdot \left(1 - maxCos\right)\right) + \left(-1 + maxCos\right) \cdot -2\right)} \cdot \left(\pi \cdot \left(uy \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]

Alternative 4
Error	3.4
Cost	9988

\[\begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.004999999888241291:\\ \;\;\;\;\sqrt{ux \cdot \left(ux \cdot \left(\left(-1 + maxCos\right) \cdot \left(1 - maxCos\right)\right) + \left(-1 + maxCos\right) \cdot -2\right)} \cdot \left(\pi \cdot \left(uy \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(\pi \cdot uy\right)\right) \cdot \sqrt{ux \cdot 2}\\ \end{array} \]

Alternative 5
Error	5.9
Cost	7104

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

Alternative 6
Error	6.6
Cost	6852

\[\begin{array}{l} \mathbf{if}\;maxCos \leq 9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(2 - ux\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(2 + maxCos \cdot -2\right)}\right)\right)\\ \end{array} \]

Alternative 7
Error	7.3
Cost	6720

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

Alternative 8
Error	11.8
Cost	6656

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

Alternative 9
Error	11.8
Cost	6656

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

Reproduce

herbie shell --seed 2022338 
(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)))))))