UniformSampleCone, y

Average Error: 13.6 → 0.5

Time: 17.2s

Precision: binary32

Cost: 10304

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

Math

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

↓

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

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

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

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 Float32(sqrt(Float32(ux * Float32(Float32(Float32(2.0) + Float32(maxCos * Float32(-2.0))) + Float32(Float32(Float32(1.0) - maxCos) * Float32(ux * Float32(maxCos + Float32(-1.0))))))) * sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sin(((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 = sqrt((ux * ((single(2.0) + (maxCos * single(-2.0))) + ((single(1.0) - maxCos) * (ux * (maxCos + single(-1.0))))))) * sin((single(2.0) * (uy * single(pi))));
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

   \[\leadsto \color{blue}{\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, ux - \mathsf{fma}\left(ux, maxCos, 1\right), 1\right)}} \]
   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))): 17 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, 17 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))): 17 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 (-.f32 1 ux) (*.f32 ux maxCos))) (-.f32 ux (fma.f32 ux maxCos 1)) 1))): 0 points increase in error, 17 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))): 17 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=> sub-neg_binary32 (+.f32 ux (neg.f32 (+.f32 (*.f32 ux maxCos) 1)))) 1))): 0 points increase in error, 17 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, 17 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))): 17 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, 17 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))): 0 points increase in error, 17 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))): 17 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, 17 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)))): 17 points increase in error, 0 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))): 17 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, 17 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, 17 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 0.6

   \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(ux, \left(2 + \left(-maxCos\right)\right) - maxCos, \left(1 - maxCos\right) \cdot \left(\left(-1 + maxCos\right) \cdot \left(ux \cdot ux\right)\right)\right)}} \]
   Proof
   (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) (sqrt.f32 (fma.f32 ux (-.f32 (+.f32 2 (neg.f32 maxCos)) maxCos) (*.f32 (-.f32 1 maxCos) (*.f32 (+.f32 -1 maxCos) (*.f32 ux ux)))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) (sqrt.f32 (fma.f32 ux (-.f32 (+.f32 (Rewrite<= metadata-eval (+.f32 1 1)) (neg.f32 maxCos)) maxCos) (*.f32 (-.f32 1 maxCos) (*.f32 (+.f32 -1 maxCos) (*.f32 ux ux)))))): 0 points increase in error, 18 points decrease in error
   (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) (sqrt.f32 (fma.f32 ux (-.f32 (+.f32 (+.f32 1 1) (Rewrite<= mul-1-neg_binary32 (*.f32 -1 maxCos))) maxCos) (*.f32 (-.f32 1 maxCos) (*.f32 (+.f32 -1 maxCos) (*.f32 ux ux)))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) (sqrt.f32 (fma.f32 ux (-.f32 (Rewrite<= associate-+r+_binary32 (+.f32 1 (+.f32 1 (*.f32 -1 maxCos)))) maxCos) (*.f32 (-.f32 1 maxCos) (*.f32 (+.f32 -1 maxCos) (*.f32 ux ux)))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) (sqrt.f32 (fma.f32 ux (-.f32 (+.f32 1 (Rewrite=> +-commutative_binary32 (+.f32 (*.f32 -1 maxCos) 1))) maxCos) (*.f32 (-.f32 1 maxCos) (*.f32 (+.f32 -1 maxCos) (*.f32 ux ux)))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) (sqrt.f32 (fma.f32 ux (-.f32 (+.f32 1 (+.f32 (*.f32 -1 maxCos) (Rewrite<= metadata-eval (*.f32 -1 -1)))) maxCos) (*.f32 (-.f32 1 maxCos) (*.f32 (+.f32 -1 maxCos) (*.f32 ux ux)))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) (sqrt.f32 (fma.f32 ux (-.f32 (+.f32 1 (Rewrite<= distribute-lft-in_binary32 (*.f32 -1 (+.f32 maxCos -1)))) maxCos) (*.f32 (-.f32 1 maxCos) (*.f32 (+.f32 -1 maxCos) (*.f32 ux ux)))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) (sqrt.f32 (fma.f32 ux (-.f32 (+.f32 1 (*.f32 -1 (+.f32 maxCos (Rewrite<= metadata-eval (neg.f32 1))))) maxCos) (*.f32 (-.f32 1 maxCos) (*.f32 (+.f32 -1 maxCos) (*.f32 ux ux)))))): 18 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) (sqrt.f32 (fma.f32 ux (-.f32 (+.f32 1 (*.f32 -1 (Rewrite<= sub-neg_binary32 (-.f32 maxCos 1)))) maxCos) (*.f32 (-.f32 1 maxCos) (*.f32 (+.f32 -1 maxCos) (*.f32 ux ux)))))): 0 points increase in error, 18 points decrease in error
   (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) (sqrt.f32 (fma.f32 ux (-.f32 (+.f32 1 (*.f32 -1 (-.f32 maxCos 1))) maxCos) (*.f32 (-.f32 1 maxCos) (*.f32 (Rewrite<= +-commutative_binary32 (+.f32 maxCos -1)) (*.f32 ux ux)))))): 18 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) (sqrt.f32 (fma.f32 ux (-.f32 (+.f32 1 (*.f32 -1 (-.f32 maxCos 1))) maxCos) (*.f32 (-.f32 1 maxCos) (*.f32 (+.f32 maxCos (Rewrite<= metadata-eval (neg.f32 1))) (*.f32 ux ux)))))): 0 points increase in error, 18 points decrease in error
   (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) (sqrt.f32 (fma.f32 ux (-.f32 (+.f32 1 (*.f32 -1 (-.f32 maxCos 1))) maxCos) (*.f32 (-.f32 1 maxCos) (*.f32 (Rewrite<= sub-neg_binary32 (-.f32 maxCos 1)) (*.f32 ux ux)))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) (sqrt.f32 (fma.f32 ux (-.f32 (+.f32 1 (*.f32 -1 (-.f32 maxCos 1))) maxCos) (*.f32 (-.f32 1 maxCos) (*.f32 (-.f32 maxCos 1) (Rewrite<= unpow2_binary32 (pow.f32 ux 2))))))): 18 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) (sqrt.f32 (fma.f32 ux (-.f32 (+.f32 1 (*.f32 -1 (-.f32 maxCos 1))) maxCos) (Rewrite<= associate-*l*_binary32 (*.f32 (*.f32 (-.f32 1 maxCos) (-.f32 maxCos 1)) (pow.f32 ux 2)))))): 0 points increase in error, 18 points decrease in error
   (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) (sqrt.f32 (fma.f32 ux (-.f32 (+.f32 1 (*.f32 -1 (-.f32 maxCos 1))) maxCos) (*.f32 (Rewrite<= *-commutative_binary32 (*.f32 (-.f32 maxCos 1) (-.f32 1 maxCos))) (pow.f32 ux 2))))): 18 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) (sqrt.f32 (fma.f32 ux (-.f32 (+.f32 1 (*.f32 -1 (-.f32 maxCos 1))) maxCos) (Rewrite<= associate-*r*_binary32 (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))))))): 0 points increase in error, 18 points decrease in error
   (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) (sqrt.f32 (Rewrite<= fma-def_binary32 (+.f32 (*.f32 ux (-.f32 (+.f32 1 (*.f32 -1 (-.f32 maxCos 1))) maxCos)) (*.f32 (-.f32 maxCos 1) (*.f32 (-.f32 1 maxCos) (pow.f32 ux 2))))))): 18 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) (sqrt.f32 (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, 18 points decrease in error

5. Taylor expanded in uy around inf 0.5

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

6. Simplified 0.5

   \[\leadsto \color{blue}{\sqrt{ux \cdot \left(\left(2 + maxCos \cdot -2\right) + \left(1 - maxCos\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)} \]
   Proof
   (*.f32 (sqrt.f32 (*.f32 ux (+.f32 (+.f32 2 (*.f32 maxCos -2)) (*.f32 (-.f32 1 maxCos) (*.f32 ux (+.f32 maxCos -1)))))) (sin.f32 (*.f32 2 (*.f32 uy (PI.f32))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sqrt.f32 (*.f32 ux (+.f32 (+.f32 2 (Rewrite<= *-commutative_binary32 (*.f32 -2 maxCos))) (*.f32 (-.f32 1 maxCos) (*.f32 ux (+.f32 maxCos -1)))))) (sin.f32 (*.f32 2 (*.f32 uy (PI.f32))))): 0 points increase in error, 6 points decrease in error
   (*.f32 (sqrt.f32 (*.f32 ux (+.f32 (+.f32 2 (*.f32 (Rewrite<= metadata-eval (neg.f32 2)) maxCos)) (*.f32 (-.f32 1 maxCos) (*.f32 ux (+.f32 maxCos -1)))))) (sin.f32 (*.f32 2 (*.f32 uy (PI.f32))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sqrt.f32 (*.f32 ux (+.f32 (Rewrite<= cancel-sign-sub-inv_binary32 (-.f32 2 (*.f32 2 maxCos))) (*.f32 (-.f32 1 maxCos) (*.f32 ux (+.f32 maxCos -1)))))) (sin.f32 (*.f32 2 (*.f32 uy (PI.f32))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sqrt.f32 (*.f32 ux (+.f32 (-.f32 2 (*.f32 2 maxCos)) (*.f32 (-.f32 1 maxCos) (*.f32 ux (+.f32 maxCos (Rewrite<= metadata-eval (neg.f32 1)))))))) (sin.f32 (*.f32 2 (*.f32 uy (PI.f32))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sqrt.f32 (*.f32 ux (+.f32 (-.f32 2 (*.f32 2 maxCos)) (*.f32 (-.f32 1 maxCos) (*.f32 ux (Rewrite<= sub-neg_binary32 (-.f32 maxCos 1))))))) (sin.f32 (*.f32 2 (*.f32 uy (PI.f32))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sqrt.f32 (*.f32 ux (+.f32 (-.f32 2 (*.f32 2 maxCos)) (*.f32 (-.f32 1 maxCos) (Rewrite<= *-commutative_binary32 (*.f32 (-.f32 maxCos 1) ux)))))) (sin.f32 (*.f32 2 (*.f32 uy (PI.f32))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sqrt.f32 (*.f32 ux (+.f32 (-.f32 2 (*.f32 2 maxCos)) (*.f32 (-.f32 1 maxCos) (*.f32 (Rewrite=> sub-neg_binary32 (+.f32 maxCos (neg.f32 1))) ux))))) (sin.f32 (*.f32 2 (*.f32 uy (PI.f32))))): 15 points increase in error, 0 points decrease in error
   (*.f32 (sqrt.f32 (*.f32 ux (+.f32 (-.f32 2 (*.f32 2 maxCos)) (*.f32 (-.f32 1 maxCos) (*.f32 (+.f32 maxCos (Rewrite=> metadata-eval -1)) ux))))) (sin.f32 (*.f32 2 (*.f32 uy (PI.f32))))): 0 points increase in error, 15 points decrease in error
   (*.f32 (sqrt.f32 (Rewrite<= distribute-rgt-out_binary32 (+.f32 (*.f32 (-.f32 2 (*.f32 2 maxCos)) ux) (*.f32 (*.f32 (-.f32 1 maxCos) (*.f32 (+.f32 maxCos -1) ux)) ux)))) (sin.f32 (*.f32 2 (*.f32 uy (PI.f32))))): 15 points increase in error, 0 points decrease in error
   (*.f32 (sqrt.f32 (+.f32 (*.f32 (-.f32 2 (*.f32 2 maxCos)) ux) (Rewrite<= associate-*r*_binary32 (*.f32 (-.f32 1 maxCos) (*.f32 (*.f32 (+.f32 maxCos -1) ux) ux))))) (sin.f32 (*.f32 2 (*.f32 uy (PI.f32))))): 0 points increase in error, 15 points decrease in error
   (*.f32 (sqrt.f32 (+.f32 (*.f32 (-.f32 2 (*.f32 2 maxCos)) ux) (*.f32 (-.f32 1 maxCos) (Rewrite<= associate-*r*_binary32 (*.f32 (+.f32 maxCos -1) (*.f32 ux ux)))))) (sin.f32 (*.f32 2 (*.f32 uy (PI.f32))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sqrt.f32 (+.f32 (*.f32 (-.f32 2 (*.f32 2 maxCos)) ux) (*.f32 (-.f32 1 maxCos) (*.f32 (+.f32 maxCos (Rewrite<= metadata-eval (neg.f32 1))) (*.f32 ux ux))))) (sin.f32 (*.f32 2 (*.f32 uy (PI.f32))))): 15 points increase in error, 0 points decrease in error
   (*.f32 (sqrt.f32 (+.f32 (*.f32 (-.f32 2 (*.f32 2 maxCos)) ux) (*.f32 (-.f32 1 maxCos) (*.f32 (Rewrite<= sub-neg_binary32 (-.f32 maxCos 1)) (*.f32 ux ux))))) (sin.f32 (*.f32 2 (*.f32 uy (PI.f32))))): 0 points increase in error, 15 points decrease in error
   (*.f32 (sqrt.f32 (+.f32 (*.f32 (-.f32 2 (*.f32 2 maxCos)) ux) (*.f32 (-.f32 1 maxCos) (*.f32 (-.f32 maxCos 1) (Rewrite<= unpow2_binary32 (pow.f32 ux 2)))))) (sin.f32 (*.f32 2 (*.f32 uy (PI.f32))))): 15 points increase in error, 0 points decrease in error

7. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.5
Cost	10176

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

Alternative 2
Error	1.5
Cost	10052

\[\begin{array}{l} \mathbf{if}\;maxCos \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{\frac{ux}{\frac{1}{2 - ux}}}\\ \mathbf{else}:\\ \;\;\;\;\left(uy \cdot \pi\right) \cdot \left(2 \cdot \sqrt{ux \cdot \left(\left(2 + maxCos \cdot -2\right) + \left(1 - maxCos\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)}\right)\\ \end{array} \]

Alternative 3
Error	3.3
Cost	9988

\[\begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.012000000104308128:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(\left(1 - maxCos\right) \cdot \left(2 + ux \cdot \left(maxCos + -1\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot 2}\\ \end{array} \]

Alternative 4
Error	1.5
Cost	9988

\[\begin{array}{l} \mathbf{if}\;maxCos \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(uy \cdot \pi\right) \cdot \left(2 \cdot \sqrt{ux \cdot \left(\left(2 + maxCos \cdot -2\right) + \left(1 - maxCos\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)}\right)\\ \end{array} \]

Alternative 5
Error	5.9
Cost	7104

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

Alternative 6
Error	5.9
Cost	6976

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

Alternative 7
Error	5.9
Cost	6976

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

Alternative 8
Error	7.3
Cost	6784

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

Alternative 9
Error	7.3
Cost	6784

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

Alternative 10
Error	7.3
Cost	6720

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

Alternative 11
Error	7.3
Cost	6720

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

Alternative 12
Error	11.8
Cost	6656

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

Alternative 13
Error	29.7
Cost	6592

\[2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{0}\right) \]

Reproduce

herbie shell --seed 2022343 
(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)))))))