UniformSampleCone, y

Average Error: 13.5 → 0.6

Time: 16.4s

Precision: binary32

Cost: 16800

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

Math

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

↓

\[\begin{array}{l} t_0 := ux - ux \cdot maxCos\\ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\frac{t_0}{\frac{\mathsf{fma}\left(ux, 1 - maxCos, 2\right)}{4 - {t_0}^{2}}}} \end{array} \]

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
 (let* ((t_0 (- ux (* ux maxCos))))
   (*
    (sin (* uy (* 2.0 PI)))
    (sqrt (/ t_0 (/ (fma ux (- 1.0 maxCos) 2.0) (- 4.0 (pow t_0 2.0))))))))

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) {
	float t_0 = ux - (ux * maxCos);
	return sinf((uy * (2.0f * ((float) M_PI)))) * sqrtf((t_0 / (fmaf(ux, (1.0f - maxCos), 2.0f) / (4.0f - powf(t_0, 2.0f)))));
}

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)
	t_0 = Float32(ux - Float32(ux * maxCos))
	return Float32(sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(t_0 / Float32(fma(ux, Float32(Float32(1.0) - maxCos), Float32(2.0)) / Float32(Float32(4.0) - (t_0 ^ Float32(2.0)))))))
end

Derivation

1. Initial program 13.5

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

2. Simplified 13.5

   \[\leadsto \color{blue}{\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, ux - \mathsf{fma}\left(ux, maxCos, 1\right), 1\right)}} \]
   Proof
   (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) (sqrt.f32 (fma.f32 (-.f32 (fma.f32 ux maxCos 1) ux) (-.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 (-.f32 (fma.f32 ux maxCos 1) ux) (-.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<= fma-def_binary32 (+.f32 (*.f32 ux maxCos) 1)) ux) (-.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-+r-_binary32 (+.f32 (*.f32 ux maxCos) (-.f32 1 ux))) (-.f32 ux (fma.f32 ux maxCos 1)) 1))): 1 points increase in error, 12 points decrease in error
   (*.f32 (sin.f32 (*.f32 (*.f32 uy 2) (PI.f32))) (sqrt.f32 (fma.f32 (Rewrite<= +-commutative_binary32 (+.f32 (-.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))): 2 points increase in error, 7 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)))): 85 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 0.5

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

4. Simplified 0.5

   \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux - ux \cdot maxCos\right) \cdot \left(2 - \left(ux - ux \cdot maxCos\right)\right)}} \]
   Proof
   (*.f32 (-.f32 ux (*.f32 ux maxCos)) (-.f32 2 (-.f32 ux (*.f32 ux maxCos)))): 0 points increase in error, 0 points decrease in error
   (*.f32 (-.f32 (Rewrite<= *-rgt-identity_binary32 (*.f32 ux 1)) (*.f32 ux maxCos)) (-.f32 2 (-.f32 ux (*.f32 ux maxCos)))): 0 points increase in error, 0 points decrease in error
   (*.f32 (Rewrite=> distribute-lft-out--_binary32 (*.f32 ux (-.f32 1 maxCos))) (-.f32 2 (-.f32 ux (*.f32 ux maxCos)))): 7 points increase in error, 3 points decrease in error
   (*.f32 (*.f32 ux (Rewrite=> sub-neg_binary32 (+.f32 1 (neg.f32 maxCos)))) (-.f32 2 (-.f32 ux (*.f32 ux maxCos)))): 0 points increase in error, 0 points decrease in error
   (*.f32 (*.f32 ux (+.f32 1 (Rewrite<= mul-1-neg_binary32 (*.f32 -1 maxCos)))) (-.f32 2 (-.f32 ux (*.f32 ux maxCos)))): 0 points increase in error, 0 points decrease in error
   (*.f32 (*.f32 ux (+.f32 1 (*.f32 -1 maxCos))) (-.f32 2 (-.f32 (Rewrite<= *-rgt-identity_binary32 (*.f32 ux 1)) (*.f32 ux maxCos)))): 0 points increase in error, 0 points decrease in error
   (*.f32 (*.f32 ux (+.f32 1 (*.f32 -1 maxCos))) (-.f32 2 (Rewrite=> distribute-lft-out--_binary32 (*.f32 ux (-.f32 1 maxCos))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (*.f32 ux (+.f32 1 (*.f32 -1 maxCos))) (-.f32 2 (*.f32 ux (Rewrite=> sub-neg_binary32 (+.f32 1 (neg.f32 maxCos)))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (*.f32 ux (+.f32 1 (*.f32 -1 maxCos))) (-.f32 2 (*.f32 ux (+.f32 1 (Rewrite<= mul-1-neg_binary32 (*.f32 -1 maxCos)))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-rgt-out--_binary32 (-.f32 (*.f32 2 (*.f32 ux (+.f32 1 (*.f32 -1 maxCos)))) (*.f32 (*.f32 ux (+.f32 1 (*.f32 -1 maxCos))) (*.f32 ux (+.f32 1 (*.f32 -1 maxCos)))))): 23 points increase in error, 22 points decrease in error
   (-.f32 (*.f32 2 (*.f32 ux (+.f32 1 (*.f32 -1 maxCos)))) (*.f32 (*.f32 ux (+.f32 1 (Rewrite=> mul-1-neg_binary32 (neg.f32 maxCos)))) (*.f32 ux (+.f32 1 (*.f32 -1 maxCos))))): 0 points increase in error, 0 points decrease in error
   (-.f32 (*.f32 2 (*.f32 ux (+.f32 1 (*.f32 -1 maxCos)))) (*.f32 (*.f32 ux (Rewrite<= sub-neg_binary32 (-.f32 1 maxCos))) (*.f32 ux (+.f32 1 (*.f32 -1 maxCos))))): 0 points increase in error, 0 points decrease in error
   (-.f32 (*.f32 2 (*.f32 ux (+.f32 1 (*.f32 -1 maxCos)))) (*.f32 (*.f32 ux (-.f32 1 maxCos)) (*.f32 ux (+.f32 1 (Rewrite=> mul-1-neg_binary32 (neg.f32 maxCos)))))): 0 points increase in error, 0 points decrease in error
   (-.f32 (*.f32 2 (*.f32 ux (+.f32 1 (*.f32 -1 maxCos)))) (*.f32 (*.f32 ux (-.f32 1 maxCos)) (*.f32 ux (Rewrite<= sub-neg_binary32 (-.f32 1 maxCos))))): 0 points increase in error, 0 points decrease in error
   (-.f32 (*.f32 2 (*.f32 ux (+.f32 1 (*.f32 -1 maxCos)))) (Rewrite<= unswap-sqr_binary32 (*.f32 (*.f32 ux ux) (*.f32 (-.f32 1 maxCos) (-.f32 1 maxCos))))): 1 points increase in error, 1 points decrease in error
   (-.f32 (*.f32 2 (*.f32 ux (+.f32 1 (*.f32 -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
   (-.f32 (*.f32 2 (*.f32 ux (+.f32 1 (*.f32 -1 maxCos)))) (*.f32 (pow.f32 ux 2) (Rewrite<= unpow2_binary32 (pow.f32 (-.f32 1 maxCos) 2)))): 0 points increase in error, 0 points decrease in error
   (-.f32 (*.f32 2 (*.f32 ux (+.f32 1 (*.f32 -1 maxCos)))) (*.f32 (pow.f32 ux 2) (pow.f32 (Rewrite=> sub-neg_binary32 (+.f32 1 (neg.f32 maxCos))) 2))): 0 points increase in error, 0 points decrease in error
   (-.f32 (*.f32 2 (*.f32 ux (+.f32 1 (*.f32 -1 maxCos)))) (*.f32 (pow.f32 ux 2) (pow.f32 (+.f32 1 (Rewrite<= mul-1-neg_binary32 (*.f32 -1 maxCos))) 2))): 0 points increase in error, 0 points decrease in error
   (-.f32 (*.f32 2 (*.f32 ux (+.f32 1 (*.f32 -1 maxCos)))) (Rewrite=> *-commutative_binary32 (*.f32 (pow.f32 (+.f32 1 (*.f32 -1 maxCos)) 2) (pow.f32 ux 2)))): 0 points increase in error, 0 points decrease in error
   (-.f32 (*.f32 2 (*.f32 ux (+.f32 1 (*.f32 -1 maxCos)))) (*.f32 (pow.f32 (+.f32 1 (Rewrite=> mul-1-neg_binary32 (neg.f32 maxCos))) 2) (pow.f32 ux 2))): 0 points increase in error, 0 points decrease in error
   (-.f32 (*.f32 2 (*.f32 ux (+.f32 1 (*.f32 -1 maxCos)))) (*.f32 (pow.f32 (Rewrite<= sub-neg_binary32 (-.f32 1 maxCos)) 2) (pow.f32 ux 2))): 0 points increase in error, 0 points decrease in error
   (-.f32 (*.f32 2 (*.f32 ux (+.f32 1 (*.f32 -1 maxCos)))) (*.f32 (Rewrite=> unpow2_binary32 (*.f32 (-.f32 1 maxCos) (-.f32 1 maxCos))) (pow.f32 ux 2))): 0 points increase in error, 0 points decrease in error
   (-.f32 (*.f32 2 (*.f32 ux (+.f32 1 (*.f32 -1 maxCos)))) (Rewrite=> associate-*l*_binary32 (*.f32 (-.f32 1 maxCos) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))))): 1 points increase in error, 0 points decrease in error
   (-.f32 (*.f32 2 (*.f32 ux (+.f32 1 (*.f32 -1 maxCos)))) (*.f32 (Rewrite=> sub-neg_binary32 (+.f32 1 (neg.f32 maxCos))) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2)))): 0 points increase in error, 0 points decrease in error
   (-.f32 (*.f32 2 (*.f32 ux (+.f32 1 (*.f32 -1 maxCos)))) (*.f32 (Rewrite=> +-commutative_binary32 (+.f32 (neg.f32 maxCos) 1)) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2)))): 0 points increase in error, 0 points decrease in error
   (-.f32 (*.f32 2 (*.f32 ux (+.f32 1 (*.f32 -1 maxCos)))) (*.f32 (+.f32 (neg.f32 maxCos) (Rewrite<= metadata-eval (neg.f32 -1))) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2)))): 0 points increase in error, 0 points decrease in error
   (-.f32 (*.f32 2 (*.f32 ux (+.f32 1 (*.f32 -1 maxCos)))) (*.f32 (Rewrite<= distribute-neg-in_binary32 (neg.f32 (+.f32 maxCos -1))) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2)))): 0 points increase in error, 0 points decrease in error
   (-.f32 (*.f32 2 (*.f32 ux (+.f32 1 (*.f32 -1 maxCos)))) (*.f32 (neg.f32 (+.f32 maxCos (Rewrite<= metadata-eval (neg.f32 1)))) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2)))): 0 points increase in error, 0 points decrease in error
   (-.f32 (*.f32 2 (*.f32 ux (+.f32 1 (*.f32 -1 maxCos)))) (*.f32 (neg.f32 (Rewrite<= sub-neg_binary32 (-.f32 maxCos 1))) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2)))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> cancel-sign-sub_binary32 (+.f32 (*.f32 2 (*.f32 ux (+.f32 1 (*.f32 -1 maxCos)))) (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 2 (Rewrite=> *-commutative_binary32 (*.f32 (+.f32 1 (*.f32 -1 maxCos)) ux))) (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2)))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 2 (*.f32 (+.f32 1 (Rewrite=> mul-1-neg_binary32 (neg.f32 maxCos))) ux)) (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2)))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 2 (*.f32 (Rewrite<= sub-neg_binary32 (-.f32 1 maxCos)) ux)) (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2)))): 0 points increase in error, 0 points decrease in error
   (+.f32 (Rewrite=> associate-*r*_binary32 (*.f32 (*.f32 2 (-.f32 1 maxCos)) ux)) (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2)))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (Rewrite<= count-2_binary32 (+.f32 (-.f32 1 maxCos) (-.f32 1 maxCos))) ux) (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2)))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (Rewrite<= associate--l+_binary32 (-.f32 (+.f32 (-.f32 1 maxCos) 1) maxCos)) ux) (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2)))): 4 points increase in error, 4 points decrease in error
   (+.f32 (*.f32 (-.f32 (+.f32 (Rewrite=> sub-neg_binary32 (+.f32 1 (neg.f32 maxCos))) 1) maxCos) ux) (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2)))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 (+.f32 (Rewrite=> +-commutative_binary32 (+.f32 (neg.f32 maxCos) 1)) 1) maxCos) ux) (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2)))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 (+.f32 (+.f32 (neg.f32 maxCos) (Rewrite<= metadata-eval (neg.f32 -1))) 1) maxCos) ux) (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2)))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 (+.f32 (Rewrite<= distribute-neg-in_binary32 (neg.f32 (+.f32 maxCos -1))) 1) maxCos) ux) (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2)))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 (+.f32 (neg.f32 (+.f32 maxCos (Rewrite<= metadata-eval (neg.f32 1)))) 1) maxCos) ux) (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2)))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 (+.f32 (neg.f32 (Rewrite<= sub-neg_binary32 (-.f32 maxCos 1))) 1) maxCos) ux) (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2)))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 (+.f32 (Rewrite<= mul-1-neg_binary32 (*.f32 -1 (-.f32 maxCos 1))) 1) maxCos) ux) (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2)))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 (Rewrite<= +-commutative_binary32 (+.f32 1 (*.f32 -1 (-.f32 maxCos 1)))) maxCos) ux) (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2)))): 0 points increase in error, 0 points decrease in error
   (+.f32 (Rewrite<= *-commutative_binary32 (*.f32 ux (-.f32 (+.f32 1 (*.f32 -1 (-.f32 maxCos 1))) maxCos))) (*.f32 (-.f32 maxCos 1) (*.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 0.6

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

6. Simplified 0.6

   \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\frac{ux - ux \cdot maxCos}{\frac{\mathsf{fma}\left(ux, 1 - maxCos, 2\right)}{4 - {\left(ux - ux \cdot maxCos\right)}^{2}}}}} \]
   Proof
   (/.f32 (-.f32 ux (*.f32 ux maxCos)) (/.f32 (fma.f32 ux (-.f32 1 maxCos) 2) (-.f32 4 (pow.f32 (-.f32 ux (*.f32 ux maxCos)) 2)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/l*_binary32 (/.f32 (*.f32 (-.f32 ux (*.f32 ux maxCos)) (-.f32 4 (pow.f32 (-.f32 ux (*.f32 ux maxCos)) 2))) (fma.f32 ux (-.f32 1 maxCos) 2))): 22 points increase in error, 10 points decrease in error
   (/.f32 (Rewrite<= *-commutative_binary32 (*.f32 (-.f32 4 (pow.f32 (-.f32 ux (*.f32 ux maxCos)) 2)) (-.f32 ux (*.f32 ux maxCos)))) (fma.f32 ux (-.f32 1 maxCos) 2)): 0 points increase in error, 0 points decrease in error

7. Final simplification 0.6

   \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\frac{ux - ux \cdot maxCos}{\frac{\mathsf{fma}\left(ux, 1 - maxCos, 2\right)}{4 - {\left(ux - ux \cdot maxCos\right)}^{2}}}} \]

Alternatives

Alternative 1
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 2
Error	0.5
Cost	10176

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

Alternative 3
Error	3.3
Cost	9988

\[\begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.006399999838322401:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(\left(1 - maxCos\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right) + 2 \cdot \left(1 - maxCos\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]

Alternative 4
Error	2.4
Cost	9920

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

Alternative 5
Error	5.9
Cost	7104

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

Alternative 6
Error	5.9
Cost	6976

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

Alternative 7
Error	5.9
Cost	6976

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

Alternative 8
Error	7.2
Cost	6720

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

Alternative 9
Error	11.7
Cost	6656

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

Reproduce

herbie shell --seed 2022339 
(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)))))))