?

Average Accuracy: 56.8% → 99.0%
Time: 23.7s
Precision: binary32
Cost: 10304

?

\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]
\[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
↓
\[\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(2 \cdot \left(1 - maxCos\right) + \left(1 - maxCos\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)} \]
(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 (* PI (* uy 2.0)))
  (sqrt
   (*
    ux
    (+ (* 2.0 (- 1.0 maxCos)) (* (- 1.0 maxCos) (* ux (+ 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((((float) M_PI) * (uy * 2.0f))) * sqrtf((ux * ((2.0f * (1.0f - maxCos)) + ((1.0f - maxCos) * (ux * (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(Float32(Float32(pi) * Float32(uy * Float32(2.0)))) * sqrt(Float32(ux * Float32(Float32(Float32(2.0) * Float32(Float32(1.0) - maxCos)) + Float32(Float32(Float32(1.0) - maxCos) * Float32(ux * Float32(maxCos + Float32(-1.0))))))))
end

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((single(pi) * (uy * single(2.0)))) * sqrt((ux * ((single(2.0) * (single(1.0) - maxCos)) + ((single(1.0) - maxCos) * (ux * (maxCos + single(-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(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(2 \cdot \left(1 - maxCos\right) + \left(1 - maxCos\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 56.8%

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

  2. Simplified56.7%

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

    [Start]56.8

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

    associate-*l* [=>]56.8

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

    cancel-sign-sub-inv [=>]56.8

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

    +-commutative [=>]56.8

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

    *-commutative [=>]56.8

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

    fma-def [=>]56.8

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

    +-commutative [=>]56.8

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

    associate-+r- [=>]56.8

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

    fma-def [=>]56.9

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

    neg-sub0 [=>]56.9

    \[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right)}, 1\right)} \]

    +-commutative [=>]56.9

    \[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, 1\right)} \]

    associate-+r- [=>]56.7

    \[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)}, 1\right)} \]

    associate--r- [=>]56.7

    \[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(ux \cdot maxCos + 1\right)\right) + ux}, 1\right)} \]

    +-commutative [=>]56.7

    \[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{ux + \left(0 - \left(ux \cdot maxCos + 1\right)\right)}, 1\right)} \]

    sub0-neg [=>]56.7

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

    sub-neg [<=]56.7

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

    fma-def [=>]56.7

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

  3. Taylor expanded in ux around 0 99.0%

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

  4. Simplified99.0%

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

    [Start]99.0

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

    +-commutative [=>]99.0

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

    fma-def [=>]99.0

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

    mul-1-neg [=>]99.0

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

    unsub-neg [=>]99.0

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

    associate-+l- [<=]99.0

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

    +-commutative [<=]99.0

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

    associate-+r- [=>]99.0

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

    metadata-eval [=>]99.0

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

    associate-*r* [=>]99.0

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

    *-commutative [=>]99.0

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

    associate-*l* [=>]99.0

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

    sub-neg [=>]99.0

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

    metadata-eval [=>]99.0

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

    +-commutative [=>]99.0

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

    unpow2 [=>]99.0

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

  5. Applied egg-rr97.5%

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

    [Start]99.0

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

    add-log-exp [=>]97.5

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

  6. Taylor expanded in uy around inf 99.1%

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

  7. Simplified99.0%

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

    [Start]99.1

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

    associate-*r* [=>]99.1

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

    *-commutative [=>]99.1

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

    *-commutative [=>]99.1

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

    *-commutative [=>]99.1

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

    unpow2 [=>]99.1

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

    sub-neg [=>]99.1

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

    metadata-eval [=>]99.1

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

    associate-*r* [<=]99.1

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

    associate-*r* [=>]99.1

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

    *-commutative [=>]99.1

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

    metadata-eval [<=]99.1

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

    sub-neg [<=]99.1

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

    associate-*r* [=>]99.1

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

    distribute-rgt-out [=>]99.0

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

  8. Final simplification99.0%

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

Alternatives

Alternative 1
Accuracy96.4%
Cost16484
\[\begin{array}{l} t_0 := \cos \left(\pi \cdot \left(uy \cdot 2\right)\right)\\ \mathbf{if}\;t_0 \leq 0.9999992251396179:\\ \;\;\;\;t_0 \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2 - \left(maxCos + maxCos\right), ux, \left(1 - maxCos\right) \cdot \left(ux \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)}\\ \end{array} \]
Alternative 2
Accuracy99.0%
Cost10176
\[\sqrt{\left(maxCos + -1\right) \cdot \left(ux \cdot \left(ux \cdot \left(1 - maxCos\right) + -2\right)\right)} \cdot \cos \left(uy \cdot \left(\pi \cdot 2\right)\right) \]
Alternative 3
Accuracy95.5%
Cost10052
\[\begin{array}{l} \mathbf{if}\;maxCos \leq 0.00039999998989515007:\\ \;\;\;\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(\left(2 - maxCos\right) - maxCos\right)}\\ \end{array} \]
Alternative 4
Accuracy80.0%
Cost6944
\[\sqrt{\mathsf{fma}\left(2 - \left(maxCos + maxCos\right), ux, \left(1 - maxCos\right) \cdot \left(ux \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)} \]
Alternative 5
Accuracy79.9%
Cost6912
\[\sqrt{ux \cdot \left(2 \cdot \left(1 - maxCos\right)\right) - {\left(1 - maxCos\right)}^{2} \cdot \left(ux \cdot ux\right)} \]
Alternative 6
Accuracy79.9%
Cost3616
\[\sqrt{\left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(1 - maxCos\right) + -2\right)} \]
Alternative 7
Accuracy77.7%
Cost3556
\[\begin{array}{l} \mathbf{if}\;maxCos \leq 6.000000212225132 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{2 \cdot ux - ux \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot ux + -2 \cdot \left(ux \cdot maxCos\right)}\\ \end{array} \]
Alternative 8
Accuracy78.9%
Cost3552
\[\sqrt{ux \cdot \left(2 \cdot \left(1 - maxCos\right)\right) - ux \cdot ux} \]
Alternative 9
Accuracy77.7%
Cost3492
\[\begin{array}{l} \mathbf{if}\;maxCos \leq 6.000000212225132 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{2 \cdot ux - ux \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 + maxCos \cdot -2\right)}\\ \end{array} \]
Alternative 10
Accuracy64.9%
Cost3424
\[\sqrt{ux \cdot \left(2 + maxCos \cdot -2\right)} \]
Alternative 11
Accuracy62.2%
Cost3296
\[\sqrt{2 \cdot ux} \]
Alternative 12
Accuracy6.6%
Cost3232
\[\sqrt{0} \]

Error

Reproduce?

herbie shell --seed 2023135 
(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)))))))