?

Average Accuracy: 56.9% → 98.4%
Time: 22.4s
Precision: binary32
Cost: 16768

?

\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]
\[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
↓
\[\sqrt[3]{{\sin \left(\pi \cdot \left(2 \cdot uy\right)\right)}^{3} \cdot {\left(ux \cdot \left(\left(2 + maxCos \cdot -2\right) + \left(maxCos + -1\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right)\right)\right)}^{1.5}} \]
(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
 (cbrt
  (*
   (pow (sin (* PI (* 2.0 uy))) 3.0)
   (pow
    (*
     ux
     (+ (+ 2.0 (* maxCos -2.0)) (* (+ maxCos -1.0) (* ux (- 1.0 maxCos)))))
    1.5))))
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 cbrtf((powf(sinf((((float) M_PI) * (2.0f * uy))), 3.0f) * powf((ux * ((2.0f + (maxCos * -2.0f)) + ((maxCos + -1.0f) * (ux * (1.0f - maxCos))))), 1.5f)));
}

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 cbrt(Float32((sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) ^ Float32(3.0)) * (Float32(ux * Float32(Float32(Float32(2.0) + Float32(maxCos * Float32(-2.0))) + Float32(Float32(maxCos + Float32(-1.0)) * Float32(ux * Float32(Float32(1.0) - maxCos))))) ^ Float32(1.5))))
end

\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[3]{{\sin \left(\pi \cdot \left(2 \cdot uy\right)\right)}^{3} \cdot {\left(ux \cdot \left(\left(2 + maxCos \cdot -2\right) + \left(maxCos + -1\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right)\right)\right)}^{1.5}}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 56.9%

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

  2. Simplified56.8%

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

    [Start]56.9

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

    associate-*l* [=>]56.9

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

    cancel-sign-sub-inv [=>]56.9

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

    +-commutative [=>]56.9

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

    *-commutative [=>]56.9

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

    fma-def [=>]56.9

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

    +-commutative [=>]56.9

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

    associate-+r- [=>]57.0

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

    fma-def [=>]57.0

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

    neg-sub0 [=>]57.0

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

    +-commutative [=>]57.0

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

    associate-+r- [=>]56.8

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

    associate--r- [=>]56.8

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

    +-commutative [=>]56.8

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

    sub0-neg [=>]56.8

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

    sub-neg [<=]56.8

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

    fma-def [=>]56.8

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

  3. Taylor expanded in ux around 0 98.3%

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

  4. Simplified98.3%

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

    [Start]98.3

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

    +-commutative [=>]98.3

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

    fma-def [=>]98.3

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

    mul-1-neg [=>]98.3

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

    unsub-neg [=>]98.3

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

    associate-+l- [<=]98.3

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

    +-commutative [<=]98.3

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

    associate-+r- [=>]98.3

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

    metadata-eval [=>]98.3

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

    sub-neg [=>]98.3

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

    metadata-eval [=>]98.3

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

    +-commutative [=>]98.3

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

    *-commutative [=>]98.3

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

    unpow2 [=>]98.3

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

    associate-*l* [=>]98.3

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

  5. Applied egg-rr98.4%

    \[\leadsto \color{blue}{\sqrt[3]{{\sin \left(uy \cdot \left(2 \cdot \pi\right)\right)}^{3} \cdot {\left(\mathsf{fma}\left(ux, 2 - \left(maxCos + maxCos\right), \left(maxCos + -1\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(ux \cdot ux\right)\right)\right)\right)}^{1.5}}} \]
    Proof

    [Start]98.3

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

    add-cbrt-cube [=>]98.3

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

    add-cbrt-cube [=>]98.3

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

    cbrt-unprod [=>]98.2

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

  6. Simplified98.4%

    \[\leadsto \color{blue}{\sqrt[3]{{\sin \left(\pi \cdot \left(2 \cdot uy\right)\right)}^{3} \cdot {\left(ux \cdot \left(\left(2 + maxCos \cdot -2\right) + \left(maxCos + -1\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right)\right)\right)}^{1.5}}} \]
    Proof

    [Start]98.4

    \[ \sqrt[3]{{\sin \left(uy \cdot \left(2 \cdot \pi\right)\right)}^{3} \cdot {\left(\mathsf{fma}\left(ux, 2 - \left(maxCos + maxCos\right), \left(maxCos + -1\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(ux \cdot ux\right)\right)\right)\right)}^{1.5}} \]

    associate-*r* [=>]98.4

    \[ \sqrt[3]{{\sin \color{blue}{\left(\left(uy \cdot 2\right) \cdot \pi\right)}}^{3} \cdot {\left(\mathsf{fma}\left(ux, 2 - \left(maxCos + maxCos\right), \left(maxCos + -1\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(ux \cdot ux\right)\right)\right)\right)}^{1.5}} \]

    *-commutative [=>]98.4

    \[ \sqrt[3]{{\sin \color{blue}{\left(\pi \cdot \left(uy \cdot 2\right)\right)}}^{3} \cdot {\left(\mathsf{fma}\left(ux, 2 - \left(maxCos + maxCos\right), \left(maxCos + -1\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(ux \cdot ux\right)\right)\right)\right)}^{1.5}} \]

    *-commutative [=>]98.4

    \[ \sqrt[3]{{\sin \left(\pi \cdot \color{blue}{\left(2 \cdot uy\right)}\right)}^{3} \cdot {\left(\mathsf{fma}\left(ux, 2 - \left(maxCos + maxCos\right), \left(maxCos + -1\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(ux \cdot ux\right)\right)\right)\right)}^{1.5}} \]

  7. Final simplification98.4%

    \[\leadsto \sqrt[3]{{\sin \left(\pi \cdot \left(2 \cdot uy\right)\right)}^{3} \cdot {\left(ux \cdot \left(\left(2 + maxCos \cdot -2\right) + \left(maxCos + -1\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right)\right)\right)}^{1.5}} \]

Alternatives

Alternative 1
Accuracy95.6%
Cost23049
\[\begin{array}{l} t_0 := \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\\ \mathbf{if}\;t_0 \leq 2.800000009628434 \cdot 10^{-9} \lor \neg \left(t_0 \leq 0.004000000189989805\right):\\ \;\;\;\;t_0 \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;uy \cdot \left(2 \cdot \left(\pi \cdot \sqrt{ux \cdot \left(\left(2 + maxCos \cdot -2\right) + \left(maxCos + -1\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right)\right)}\right)\right)\\ \end{array} \]
Alternative 2
Accuracy98.3%
Cost13504
\[\sin \left(uy \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, \left(2 - maxCos\right) - maxCos, \left(maxCos + -1\right) \cdot \left(ux \cdot \left(ux \cdot \left(1 - maxCos\right)\right)\right)\right)} \]
Alternative 3
Accuracy98.3%
Cost10432
\[\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{\left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) \cdot \left(ux \cdot ux\right) + ux \cdot \left(1 + \left(\left(1 - maxCos\right) - maxCos\right)\right)} \]
Alternative 4
Accuracy98.3%
Cost10304
\[\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(\left(2 + maxCos \cdot -2\right) + \left(maxCos + -1\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right)\right)} \]
Alternative 5
Accuracy96.8%
Cost10176
\[\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{\left(ux \cdot \left(maxCos \cdot -2\right) - ux \cdot ux\right) + 2 \cdot ux} \]
Alternative 6
Accuracy94.9%
Cost10052
\[\begin{array}{l} \mathbf{if}\;maxCos \leq 0.00039999998989515007:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(\left(2 - maxCos\right) - maxCos\right)}\\ \end{array} \]
Alternative 7
Accuracy94.9%
Cost10052
\[\begin{array}{l} \mathbf{if}\;maxCos \leq 0.00039999998989515007:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(\left(2 - maxCos\right) - maxCos\right)}\\ \end{array} \]
Alternative 8
Accuracy94.9%
Cost10052
\[\begin{array}{l} \mathbf{if}\;maxCos \leq 0.00039999998989515007:\\ \;\;\;\;\sqrt{\frac{ux}{\frac{1}{2 - ux}}} \cdot \sin \left(2 \cdot \left(\pi \cdot uy\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(\left(2 - maxCos\right) - maxCos\right)}\\ \end{array} \]
Alternative 9
Accuracy89.1%
Cost9988
\[\begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.019999999552965164:\\ \;\;\;\;uy \cdot \left(2 \cdot \left(\pi \cdot \sqrt{ux \cdot \left(\left(2 + maxCos \cdot -2\right) + \left(maxCos + -1\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(\pi \cdot uy\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]
Alternative 10
Accuracy81.4%
Cost7104
\[uy \cdot \left(2 \cdot \left(\pi \cdot \sqrt{ux \cdot \left(\left(2 + maxCos \cdot -2\right) + \left(maxCos + -1\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right)\right)}\right)\right) \]
Alternative 11
Accuracy79.1%
Cost6916
\[\begin{array}{l} \mathbf{if}\;maxCos \leq 0.00039999998989515007:\\ \;\;\;\;2 \cdot \left(\sqrt{\frac{ux}{\frac{1}{2 - ux}}} \cdot \left(\pi \cdot uy\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{ux \cdot \left(\left(1 - maxCos\right) + \left(1 - maxCos\right)\right)}\right)\\ \end{array} \]
Alternative 12
Accuracy79.1%
Cost6852
\[\begin{array}{l} \mathbf{if}\;maxCos \leq 0.00039999998989515007:\\ \;\;\;\;uy \cdot \left(2 \cdot \left(\pi \cdot \sqrt{ux \cdot \left(2 - ux\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{ux \cdot \left(\left(2 - maxCos\right) - maxCos\right)}\right)\\ \end{array} \]
Alternative 13
Accuracy79.1%
Cost6852
\[\begin{array}{l} \mathbf{if}\;maxCos \leq 0.00039999998989515007:\\ \;\;\;\;2 \cdot \left(\sqrt{\frac{ux}{\frac{1}{2 - ux}}} \cdot \left(\pi \cdot uy\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{ux \cdot \left(\left(2 - maxCos\right) - maxCos\right)}\right)\\ \end{array} \]
Alternative 14
Accuracy76.9%
Cost6720
\[2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\right) \]
Alternative 15
Accuracy76.9%
Cost6720
\[uy \cdot \left(2 \cdot \left(\pi \cdot \sqrt{ux \cdot \left(2 - ux\right)}\right)\right) \]
Alternative 16
Accuracy63.4%
Cost6656
\[2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{2 \cdot ux}\right) \]

Error

Reproduce?

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