UniformSampleCone, y

Average Error: 13.6 → 0.5

Time: 18.9s

Precision: binary32

Cost: 16768

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

Math

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

↓

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

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
     (- (- 2.0 (+ maxCos maxCos)) (* ux (* (+ maxCos -1.0) (+ 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 * ((2.0f - (maxCos + maxCos)) - (ux * ((maxCos + -1.0f) * (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(2.0) - Float32(maxCos + maxCos)) - Float32(ux * Float32(Float32(maxCos + Float32(-1.0)) * 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 13.6

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

2. Simplified 13.6

   \[\leadsto \color{blue}{\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos - 1, 1\right), ux - \mathsf{fma}\left(ux, maxCos, 1\right), 1\right)}} \]
   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, 0 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))): 2 points increase in error, 5 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, 4 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)))): 74 points increase in error, 76 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

   \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(ux, 1 + \left(\left(1 - maxCos\right) - maxCos\right), \left(ux \cdot ux\right) \cdot \left(\left(-1 + maxCos\right) \cdot \left(1 - maxCos\right)\right)\right)}} \]
   Proof
   (fma.f32 ux (+.f32 1 (-.f32 (-.f32 1 maxCos) maxCos)) (*.f32 (*.f32 ux ux) (*.f32 (+.f32 -1 maxCos) (-.f32 1 maxCos)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 ux (+.f32 1 (-.f32 (Rewrite=> sub-neg_binary32 (+.f32 1 (neg.f32 maxCos))) maxCos)) (*.f32 (*.f32 ux ux) (*.f32 (+.f32 -1 maxCos) (-.f32 1 maxCos)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 ux (+.f32 1 (-.f32 (+.f32 (Rewrite<= metadata-eval (neg.f32 -1)) (neg.f32 maxCos)) maxCos)) (*.f32 (*.f32 ux ux) (*.f32 (+.f32 -1 maxCos) (-.f32 1 maxCos)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 ux (+.f32 1 (-.f32 (Rewrite<= distribute-neg-in_binary32 (neg.f32 (+.f32 -1 maxCos))) maxCos)) (*.f32 (*.f32 ux ux) (*.f32 (+.f32 -1 maxCos) (-.f32 1 maxCos)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 ux (+.f32 1 (-.f32 (neg.f32 (Rewrite<= +-commutative_binary32 (+.f32 maxCos -1))) maxCos)) (*.f32 (*.f32 ux ux) (*.f32 (+.f32 -1 maxCos) (-.f32 1 maxCos)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 ux (+.f32 1 (-.f32 (neg.f32 (+.f32 maxCos (Rewrite<= metadata-eval (neg.f32 1)))) maxCos)) (*.f32 (*.f32 ux ux) (*.f32 (+.f32 -1 maxCos) (-.f32 1 maxCos)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 ux (+.f32 1 (-.f32 (neg.f32 (Rewrite<= sub-neg_binary32 (-.f32 maxCos 1))) maxCos)) (*.f32 (*.f32 ux ux) (*.f32 (+.f32 -1 maxCos) (-.f32 1 maxCos)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 ux (+.f32 1 (-.f32 (Rewrite<= mul-1-neg_binary32 (*.f32 -1 (-.f32 maxCos 1))) maxCos)) (*.f32 (*.f32 ux ux) (*.f32 (+.f32 -1 maxCos) (-.f32 1 maxCos)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 ux (Rewrite<= associate--l+_binary32 (-.f32 (+.f32 1 (*.f32 -1 (-.f32 maxCos 1))) maxCos)) (*.f32 (*.f32 ux ux) (*.f32 (+.f32 -1 maxCos) (-.f32 1 maxCos)))): 5 points increase in error, 9 points decrease in error
   (fma.f32 ux (-.f32 (+.f32 1 (*.f32 -1 (-.f32 maxCos 1))) maxCos) (*.f32 (Rewrite<= unpow2_binary32 (pow.f32 ux 2)) (*.f32 (+.f32 -1 maxCos) (-.f32 1 maxCos)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 ux (-.f32 (+.f32 1 (*.f32 -1 (-.f32 maxCos 1))) maxCos) (*.f32 (pow.f32 ux 2) (*.f32 (Rewrite<= +-commutative_binary32 (+.f32 maxCos -1)) (-.f32 1 maxCos)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 ux (-.f32 (+.f32 1 (*.f32 -1 (-.f32 maxCos 1))) maxCos) (*.f32 (pow.f32 ux 2) (*.f32 (+.f32 maxCos (Rewrite<= metadata-eval (neg.f32 1))) (-.f32 1 maxCos)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 ux (-.f32 (+.f32 1 (*.f32 -1 (-.f32 maxCos 1))) maxCos) (*.f32 (pow.f32 ux 2) (*.f32 (Rewrite<= sub-neg_binary32 (-.f32 maxCos 1)) (-.f32 1 maxCos)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 ux (-.f32 (+.f32 1 (*.f32 -1 (-.f32 maxCos 1))) maxCos) (*.f32 (pow.f32 ux 2) (*.f32 (-.f32 maxCos 1) (Rewrite=> sub-neg_binary32 (+.f32 1 (neg.f32 maxCos)))))): 0 points increase in error, 0 points decrease in error
   (fma.f32 ux (-.f32 (+.f32 1 (*.f32 -1 (-.f32 maxCos 1))) maxCos) (*.f32 (pow.f32 ux 2) (*.f32 (-.f32 maxCos 1) (+.f32 1 (Rewrite<= mul-1-neg_binary32 (*.f32 -1 maxCos)))))): 0 points increase in error, 0 points decrease in error
   (fma.f32 ux (-.f32 (+.f32 1 (*.f32 -1 (-.f32 maxCos 1))) maxCos) (*.f32 (pow.f32 ux 2) (Rewrite=> *-commutative_binary32 (*.f32 (+.f32 1 (*.f32 -1 maxCos)) (-.f32 maxCos 1))))): 0 points increase in error, 0 points decrease in error
   (fma.f32 ux (-.f32 (+.f32 1 (*.f32 -1 (-.f32 maxCos 1))) maxCos) (Rewrite<= associate-*l*_binary32 (*.f32 (*.f32 (pow.f32 ux 2) (+.f32 1 (*.f32 -1 maxCos))) (-.f32 maxCos 1)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 ux (-.f32 (+.f32 1 (*.f32 -1 (-.f32 maxCos 1))) maxCos) (Rewrite<= *-commutative_binary32 (*.f32 (-.f32 maxCos 1) (*.f32 (pow.f32 ux 2) (+.f32 1 (*.f32 -1 maxCos)))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary32 (+.f32 (*.f32 ux (-.f32 (+.f32 1 (*.f32 -1 (-.f32 maxCos 1))) maxCos)) (*.f32 (-.f32 maxCos 1) (*.f32 (pow.f32 ux 2) (+.f32 1 (*.f32 -1 maxCos)))))): 5 points increase in error, 4 points decrease in error
   (+.f32 (*.f32 ux (-.f32 (+.f32 1 (*.f32 -1 (-.f32 maxCos 1))) maxCos)) (*.f32 (-.f32 maxCos 1) (*.f32 (pow.f32 ux 2) (+.f32 1 (Rewrite=> mul-1-neg_binary32 (neg.f32 maxCos)))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 ux (-.f32 (+.f32 1 (*.f32 -1 (-.f32 maxCos 1))) maxCos)) (*.f32 (-.f32 maxCos 1) (*.f32 (pow.f32 ux 2) (Rewrite<= sub-neg_binary32 (-.f32 1 maxCos))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 ux (-.f32 (+.f32 1 (*.f32 -1 (-.f32 maxCos 1))) maxCos)) (*.f32 (-.f32 maxCos 1) (Rewrite<= *-commutative_binary32 (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-commutative_binary32 (+.f32 (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))) (*.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

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

6. Final simplification 0.5

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

Alternatives

Alternative 1
Error	1.6
Cost	23240

\[\begin{array}{l} t_0 := \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\\ t_1 := \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{if}\;t_0 \leq -0.4000000059604645:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 0.0005000000237487257:\\ \;\;\;\;2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{\mathsf{fma}\left(ux, 1 - maxCos, -2\right) \cdot \left(ux \cdot maxCos - ux\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	1.6
Cost	23240

\[\begin{array}{l} t_0 := \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\\ t_1 := uy \cdot \left(2 \cdot \pi\right)\\ t_2 := \sin t_1 \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{if}\;t_0 \leq -0.4000000059604645:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_0 \leq 0.0005000000237487257:\\ \;\;\;\;t_1 \cdot \sqrt{\left(maxCos + -1\right) \cdot \left(ux \cdot \mathsf{fma}\left(ux, 1 - maxCos, -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	0.5
Cost	13568

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

Alternative 4
Error	0.5
Cost	10304

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

Alternative 5
Error	1.4
Cost	10052

\[\begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0001500000071246177:\\ \;\;\;\;\left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right) + 2 \cdot \left(1 - maxCos\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]

Alternative 6
Error	3.3
Cost	9988

\[\begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.002199999988079071:\\ \;\;\;\;\left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right) + 2 \cdot \left(1 - maxCos\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(\pi \cdot uy\right)\right) \cdot \sqrt{ux \cdot 2}\\ \end{array} \]

Alternative 7
Error	5.9
Cost	7104

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

Alternative 8
Error	7.3
Cost	6720

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

Alternative 9
Error	11.8
Cost	6656

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

Alternative 10
Error	11.8
Cost	6656

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

Alternative 11
Error	29.7
Cost	32

\[0 \]

Reproduce

herbie shell --seed 2022329 
(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)))))))