Average Error: 13.8 → 0.3
Time: 17.9s
Precision: binary32
Cost: 23232
\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]
\[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
↓
\[\cos \left(\sqrt[3]{{\left(2 \cdot \pi\right)}^{3} \cdot {uy}^{3}}\right) \cdot \sqrt{\mathsf{fma}\left(ux, \left(1 - maxCos\right) + \left(1 - maxCos\right), \left(ux \cdot ux\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)\right)} \]
(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 (cbrt (* (pow (* 2.0 PI) 3.0) (pow uy 3.0))))
  (sqrt
   (fma
    ux
    (+ (- 1.0 maxCos) (- 1.0 maxCos))
    (* (* ux ux) (* (- 1.0 maxCos) (+ maxCos -1.0)))))))
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(cbrtf((powf((2.0f * ((float) M_PI)), 3.0f) * powf(uy, 3.0f)))) * sqrtf(fmaf(ux, ((1.0f - maxCos) + (1.0f - maxCos)), ((ux * ux) * ((1.0f - maxCos) * (maxCos + -1.0f)))));
}

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(cbrt(Float32((Float32(Float32(2.0) * Float32(pi)) ^ Float32(3.0)) * (uy ^ Float32(3.0))))) * sqrt(fma(ux, Float32(Float32(Float32(1.0) - maxCos) + Float32(Float32(1.0) - maxCos)), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos + Float32(-1.0)))))))
end

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

Error

Derivation

  1. Initial program 13.8

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

  2. Simplified13.8

    \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos - 1, 1\right), ux - \mathsf{fma}\left(ux, maxCos, 1\right), 1\right)}} \]
    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))): 5 points increase in error, 1 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))): 4 points increase in error, 1 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, 69 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. Simplified0.3

    \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(ux, \left(1 - maxCos\right) + \left(1 - maxCos\right), \left(ux \cdot ux\right) \cdot \left(\left(-1 + maxCos\right) \cdot \left(1 - maxCos\right)\right)\right)}} \]
    Proof
    (fma.f32 ux (+.f32 (-.f32 1 maxCos) (-.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 (Rewrite=> sub-neg_binary32 (+.f32 1 (neg.f32 maxCos))) (-.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 (+.f32 1 (Rewrite<= mul-1-neg_binary32 (*.f32 -1 maxCos))) (-.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 (Rewrite<= associate--l+_binary32 (-.f32 (+.f32 (+.f32 1 (*.f32 -1 maxCos)) 1) maxCos)) (*.f32 (*.f32 ux ux) (*.f32 (+.f32 -1 maxCos) (-.f32 1 maxCos)))): 2 points increase in error, 6 points decrease in error
    (fma.f32 ux (-.f32 (+.f32 (+.f32 (Rewrite<= metadata-eval (*.f32 -1 -1)) (*.f32 -1 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 (+.f32 (Rewrite<= distribute-lft-in_binary32 (*.f32 -1 (+.f32 -1 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 (+.f32 (*.f32 -1 (Rewrite<= +-commutative_binary32 (+.f32 maxCos -1))) 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 (+.f32 (*.f32 -1 (+.f32 maxCos (Rewrite<= metadata-eval (neg.f32 1)))) 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 (+.f32 (*.f32 -1 (Rewrite<= sub-neg_binary32 (-.f32 maxCos 1))) 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 (Rewrite<= +-commutative_binary32 (+.f32 1 (*.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 (-.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)))))): 3 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-rr0.3

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

  6. Final simplification0.3

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

Alternatives

Alternative 1
Error0.3
Cost23008
\[\sqrt[3]{{\left(ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right) - {\left(ux \cdot \left(1 - maxCos\right)\right)}^{2}\right)}^{1.5} \cdot {\cos \left(2 \cdot \left(\pi \cdot uy\right)\right)}^{3}} \]
Alternative 2
Error0.3
Cost13568
\[\sqrt{\mathsf{fma}\left(ux, \left(1 - maxCos\right) + \left(1 - maxCos\right), \left(ux \cdot ux\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot uy\right) \]
Alternative 3
Error0.3
Cost10304
\[\cos \left(\left(2 \cdot \pi\right) \cdot uy\right) \cdot \sqrt{ux \cdot \left(ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right) + 2 \cdot \left(1 - maxCos\right)\right)} \]
Alternative 4
Error2.3
Cost9984
\[\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux} \]
Alternative 5
Error2.3
Cost9920
\[\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)} \]
Alternative 6
Error6.4
Cost6944
\[\sqrt{\mathsf{fma}\left(2 + maxCos \cdot -2, ux, \left(1 - maxCos\right) \cdot \left(ux \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)} \]
Alternative 7
Error11.2
Cost3424
\[\sqrt{ux \cdot \left(2 + maxCos \cdot -2\right)} \]
Alternative 8
Error7.8
Cost3424
\[\sqrt{2 \cdot ux - ux \cdot ux} \]
Alternative 9
Error12.1
Cost3296
\[\sqrt{2 \cdot ux} \]

Error

Reproduce

herbie shell --seed 2022325 
(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)))))))