UniformSampleCone, x

Average Error: 13.7 → 0.4

Time: 19.2s

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

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

↓

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

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
 (*
  (cos (* (sqrt uy) (* (* 2.0 PI) (sqrt uy))))
  (sqrt
   (*
    ux
    (+ (* ux (* (+ -1.0 maxCos) (- 1.0 maxCos))) (* 2.0 (- 1.0 maxCos)))))))

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 cosf((sqrtf(uy) * ((2.0f * ((float) M_PI)) * sqrtf(uy)))) * sqrtf((ux * ((ux * ((-1.0f + maxCos) * (1.0f - maxCos))) + (2.0f * (1.0f - maxCos)))));
}

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(cos(Float32(sqrt(uy) * Float32(Float32(Float32(2.0) * Float32(pi)) * sqrt(uy)))) * sqrt(Float32(ux * Float32(Float32(ux * Float32(Float32(Float32(-1.0) + maxCos) * Float32(Float32(1.0) - maxCos))) + Float32(Float32(2.0) * Float32(Float32(1.0) - maxCos))))))
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 = cos((sqrt(uy) * ((single(2.0) * single(pi)) * sqrt(uy)))) * sqrt((ux * ((ux * ((single(-1.0) + maxCos) * (single(1.0) - maxCos))) + (single(2.0) * (single(1.0) - maxCos)))));
end

Derivation

1. Initial program 13.7

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

2. Simplified 13.7

   \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos - 1, 1\right), ux - \mathsf{fma}\left(ux, maxCos, 1\right), 1\right)}} \]
   Proof
   (*.f32 (cos.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 (cos.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 (cos.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 (cos.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 (cos.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 (cos.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 (cos.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 (cos.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 (cos.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 (cos.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 (cos.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))): 7 points increase in error, 3 points decrease in error
   (*.f32 (cos.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 (cos.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 (cos.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 (cos.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 (cos.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 (cos.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 (cos.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 (cos.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 (cos.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 (cos.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)))): 91 points increase in error, 78 points decrease in error
   (*.f32 (cos.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 (cos.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 (cos.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 0.3

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

4. Simplified 0.3

   \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(\left(-1 + maxCos\right) \cdot \left(1 - maxCos\right)\right) \cdot ux - \left(-1 + maxCos\right) \cdot 2\right)}} \]
   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 (Rewrite=> distribute-neg-in_binary32 (+.f32 (neg.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 (+.f32 (neg.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 (Rewrite<= +-commutative_binary32 (+.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 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))): 19 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)): 1 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, 1 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))) (Rewrite=> *-commutative_binary32 (*.f32 ux (*.f32 -1 (-.f32 (+.f32 maxCos (*.f32 -1 (+.f32 1 (*.f32 -1 maxCos)))) 1))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))) (*.f32 ux (Rewrite=> mul-1-neg_binary32 (neg.f32 (-.f32 (+.f32 maxCos (*.f32 -1 (+.f32 1 (*.f32 -1 maxCos)))) 1))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))) (*.f32 ux (neg.f32 (Rewrite=> sub-neg_binary32 (+.f32 (+.f32 maxCos (*.f32 -1 (+.f32 1 (*.f32 -1 maxCos)))) (neg.f32 1)))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))) (*.f32 ux (neg.f32 (+.f32 (+.f32 maxCos (*.f32 -1 (+.f32 1 (*.f32 -1 maxCos)))) (Rewrite=> metadata-eval -1))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))) (*.f32 ux (neg.f32 (Rewrite=> +-commutative_binary32 (+.f32 -1 (+.f32 maxCos (*.f32 -1 (+.f32 1 (*.f32 -1 maxCos))))))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))) (*.f32 ux (Rewrite=> distribute-neg-in_binary32 (+.f32 (neg.f32 -1) (neg.f32 (+.f32 maxCos (*.f32 -1 (+.f32 1 (*.f32 -1 maxCos))))))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))) (*.f32 ux (+.f32 (Rewrite=> metadata-eval 1) (neg.f32 (+.f32 maxCos (*.f32 -1 (+.f32 1 (*.f32 -1 maxCos)))))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))) (*.f32 ux (+.f32 1 (neg.f32 (+.f32 maxCos (*.f32 -1 (+.f32 1 (Rewrite=> mul-1-neg_binary32 (neg.f32 maxCos))))))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))) (*.f32 ux (+.f32 1 (neg.f32 (+.f32 maxCos (Rewrite=> distribute-lft-in_binary32 (+.f32 (*.f32 -1 1) (*.f32 -1 (neg.f32 maxCos))))))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))) (*.f32 ux (+.f32 1 (neg.f32 (+.f32 maxCos (+.f32 (Rewrite=> metadata-eval -1) (*.f32 -1 (neg.f32 maxCos)))))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))) (*.f32 ux (+.f32 1 (neg.f32 (Rewrite=> associate-+r+_binary32 (+.f32 (+.f32 maxCos -1) (*.f32 -1 (neg.f32 maxCos)))))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))) (*.f32 ux (+.f32 1 (neg.f32 (+.f32 (+.f32 maxCos (Rewrite<= metadata-eval (neg.f32 1))) (*.f32 -1 (neg.f32 maxCos))))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))) (*.f32 ux (+.f32 1 (neg.f32 (+.f32 (Rewrite<= sub-neg_binary32 (-.f32 maxCos 1)) (*.f32 -1 (neg.f32 maxCos))))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))) (*.f32 ux (+.f32 1 (neg.f32 (+.f32 (Rewrite<= *-lft-identity_binary32 (*.f32 1 (-.f32 maxCos 1))) (*.f32 -1 (neg.f32 maxCos))))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))) (*.f32 ux (+.f32 1 (neg.f32 (+.f32 (*.f32 1 (-.f32 maxCos 1)) (*.f32 -1 (Rewrite<= mul-1-neg_binary32 (*.f32 -1 maxCos)))))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))) (*.f32 ux (+.f32 1 (neg.f32 (+.f32 (*.f32 1 (-.f32 maxCos 1)) (Rewrite=> associate-*r*_binary32 (*.f32 (*.f32 -1 -1) maxCos))))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))) (*.f32 ux (+.f32 1 (neg.f32 (+.f32 (*.f32 1 (-.f32 maxCos 1)) (*.f32 (Rewrite=> metadata-eval 1) maxCos)))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))) (*.f32 ux (+.f32 1 (neg.f32 (Rewrite=> distribute-lft-out_binary32 (*.f32 1 (+.f32 (-.f32 maxCos 1) maxCos))))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))) (*.f32 ux (+.f32 1 (neg.f32 (Rewrite<= distribute-rgt-out_binary32 (+.f32 (*.f32 (-.f32 maxCos 1) 1) (*.f32 maxCos 1))))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))) (*.f32 ux (+.f32 1 (neg.f32 (+.f32 (Rewrite=> *-rgt-identity_binary32 (-.f32 maxCos 1)) (*.f32 maxCos 1)))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))) (*.f32 ux (+.f32 1 (neg.f32 (+.f32 (-.f32 maxCos 1) (Rewrite=> *-rgt-identity_binary32 maxCos)))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))) (*.f32 ux (+.f32 1 (Rewrite<= distribute-neg-out_binary32 (+.f32 (neg.f32 (-.f32 maxCos 1)) (neg.f32 maxCos)))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))) (*.f32 ux (+.f32 1 (+.f32 (Rewrite<= mul-1-neg_binary32 (*.f32 -1 (-.f32 maxCos 1))) (neg.f32 maxCos))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))) (*.f32 ux (+.f32 1 (Rewrite<= sub-neg_binary32 (-.f32 (*.f32 -1 (-.f32 maxCos 1)) maxCos))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))) (*.f32 ux (Rewrite<= associate--l+_binary32 (-.f32 (+.f32 1 (*.f32 -1 (-.f32 maxCos 1))) maxCos)))): 6 points increase in error, 7 points decrease in error

5. Applied egg-rr 0.4

   \[\leadsto \cos \color{blue}{\left({\left({\left(uy \cdot \left(2 \cdot \pi\right)\right)}^{3}\right)}^{0.3333333333333333}\right)} \cdot \sqrt{ux \cdot \left(\left(\left(-1 + maxCos\right) \cdot \left(1 - maxCos\right)\right) \cdot ux - \left(-1 + maxCos\right) \cdot 2\right)} \]

6. Applied egg-rr 0.4

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

7. Final simplification 0.4

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

Alternatives

Alternative 1
Error	3.4
Cost	16420

\[\begin{array}{l} t_0 := \cos \left(\pi \cdot \left(2 \cdot uy\right)\right)\\ \mathbf{if}\;t_0 \leq 0.9998859763145447:\\ \;\;\;\;t_0 \cdot \sqrt{2 \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot ux + maxCos \cdot \left(ux \cdot \left(-2 + 2 \cdot ux\right)\right)\right) + \left(ux \cdot ux\right) \cdot \left(-1 - maxCos \cdot maxCos\right)}\\ \end{array} \]

Alternative 2
Error	0.3
Cost	10368

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

Alternative 3
Error	0.3
Cost	10304

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

Alternative 4
Error	0.6
Cost	10240

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

Alternative 5
Error	0.6
Cost	10240

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

Alternative 6
Error	0.8
Cost	10112

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

Alternative 7
Error	1.3
Cost	9988

\[\begin{array}{l} \mathbf{if}\;maxCos \leq 9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;\cos \left(\left(2 \cdot \pi\right) \cdot uy\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot ux + maxCos \cdot \left(ux \cdot \left(-2 + 2 \cdot ux\right)\right)\right) + \left(ux \cdot ux\right) \cdot \left(-1 - maxCos \cdot maxCos\right)}\\ \end{array} \]

Alternative 8
Error	6.5
Cost	3936

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

Alternative 9
Error	6.6
Cost	3872

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

Alternative 10
Error	6.6
Cost	3808

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

Alternative 11
Error	6.6
Cost	3744

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

Alternative 12
Error	8.0
Cost	3360

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

Alternative 13
Error	25.6
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2022295 
(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)))))))