?

Average Accuracy: 57.8% → 98.3%
Time: 17.3s
Precision: binary32
Cost: 13408

?

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

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

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 = sin((uy * (single(2.0) * single(pi)))) * sqrt((((single(2.0) * ux) * (single(1.0) - maxCos)) - ((ux * (single(1.0) - maxCos)) ^ single(2.0))));
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)}
↓
\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\left(2 \cdot ux\right) \cdot \left(1 - maxCos\right) - {\left(ux \cdot \left(1 - maxCos\right)\right)}^{2}}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 57.8%

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

  2. Simplified57.7%

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

    [Start]57.8

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

    associate-*l* [=>]57.8

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

    cancel-sign-sub-inv [=>]57.8

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

    +-commutative [=>]57.8

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

    *-commutative [=>]57.8

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

    fma-def [=>]57.9

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

    +-commutative [=>]57.9

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

    associate-+r- [=>]57.9

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

    fma-def [=>]57.9

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

    neg-sub0 [=>]57.9

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

    +-commutative [=>]57.9

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

    associate-+r- [=>]57.7

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

    associate--r- [=>]57.7

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

    +-commutative [=>]57.7

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

    sub0-neg [=>]57.7

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

    sub-neg [<=]57.7

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

    fma-def [=>]57.7

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

  3. Taylor expanded in ux around -inf 98.3%

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

  4. Simplified98.3%

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

    [Start]98.3

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

    +-commutative [=>]98.3

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

    mul-1-neg [=>]98.3

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

    unsub-neg [=>]98.3

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

    *-commutative [=>]98.3

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

    mul-1-neg [=>]98.3

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

    sub-neg [<=]98.3

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

    associate-*l* [=>]98.3

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

    unpow2 [=>]98.3

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

    associate-*l* [=>]98.3

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

    mul-1-neg [=>]98.3

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

    sub-neg [<=]98.3

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

  5. Applied egg-rr81.3%

    \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(ux \cdot \left(1 - maxCos\right)\right) - {\left(ux \cdot \left(1 - maxCos\right)\right)}^{2}}\right)} - 1\right)} \]
    Proof

    [Start]98.3

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

    expm1-log1p-u [=>]98.2

    \[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{ux \cdot \left(\left(1 - maxCos\right) \cdot 2\right) - ux \cdot \left(ux \cdot {\left(1 - maxCos\right)}^{2}\right)}\right)\right)} \]

    expm1-udef [=>]81.3

    \[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{ux \cdot \left(\left(1 - maxCos\right) \cdot 2\right) - ux \cdot \left(ux \cdot {\left(1 - maxCos\right)}^{2}\right)}\right)} - 1\right)} \]

  6. Simplified98.3%

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

    [Start]81.3

    \[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(ux \cdot \left(1 - maxCos\right)\right) - {\left(ux \cdot \left(1 - maxCos\right)\right)}^{2}}\right)} - 1\right) \]

    expm1-def [=>]98.2

    \[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \left(ux \cdot \left(1 - maxCos\right)\right) - {\left(ux \cdot \left(1 - maxCos\right)\right)}^{2}}\right)\right)} \]

    expm1-log1p [=>]98.3

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

    unpow2 [=>]98.3

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

    *-commutative [=>]98.3

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

    swap-sqr [=>]98.3

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

    unpow2 [<=]98.3

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

    unpow2 [<=]98.3

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

    *-commutative [<=]98.3

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

    *-commutative [<=]98.3

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

    associate-*r* [=>]98.3

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

    *-commutative [=>]98.3

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

    unpow2 [=>]98.3

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

    unpow2 [=>]98.3

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

    swap-sqr [<=]98.3

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

    unpow2 [<=]98.3

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

    *-commutative [=>]98.3

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

  7. Final simplification98.3%

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

Alternatives

Alternative 1
Accuracy98.3%
Cost10176
\[\sin \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(\left(1 - maxCos\right) \cdot \left(2 - ux \cdot \left(1 - maxCos\right)\right)\right)} \]
Alternative 2
Accuracy95.7%
Cost10052
\[\begin{array}{l} t_0 := ux \cdot \left(1 - maxCos\right)\\ \mathbf{if}\;maxCos \leq 9.000000318337698 \cdot 10^{-6}:\\ \;\;\;\;\sin \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{t_0 \cdot \left(2 - t_0\right)}\right)\\ \end{array} \]
Alternative 3
Accuracy89.4%
Cost9988
\[\begin{array}{l} t_0 := ux \cdot \left(1 - maxCos\right)\\ \mathbf{if}\;uy \cdot 2 \leq 0.004000000189989805:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{t_0 \cdot \left(2 - t_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]
Alternative 4
Accuracy95.7%
Cost9988
\[\begin{array}{l} t_0 := ux \cdot \left(1 - maxCos\right)\\ \mathbf{if}\;maxCos \leq 9.000000318337698 \cdot 10^{-6}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{t_0 \cdot \left(2 - t_0\right)}\right)\\ \end{array} \]
Alternative 5
Accuracy81.3%
Cost6976
\[\begin{array}{l} t_0 := ux \cdot \left(1 - maxCos\right)\\ 2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{t_0 \cdot \left(2 - t_0\right)}\right) \end{array} \]
Alternative 6
Accuracy79.0%
Cost6852
\[\begin{array}{l} \mathbf{if}\;maxCos \leq 6.499999926745659 \cdot 10^{-6}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(2 + maxCos \cdot -2\right)}\right)\right)\\ \end{array} \]
Alternative 7
Accuracy79.0%
Cost6852
\[\begin{array}{l} \mathbf{if}\;maxCos \leq 6.499999926745659 \cdot 10^{-6}:\\ \;\;\;\;2 \cdot \left(\sqrt{2 \cdot ux - ux \cdot ux} \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(2 + maxCos \cdot -2\right)}\right)\right)\\ \end{array} \]
Alternative 8
Accuracy76.9%
Cost6720
\[2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\right) \]
Alternative 9
Accuracy63.1%
Cost6656
\[2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{2 \cdot ux}\right)\right) \]

Error

Reproduce?

herbie shell --seed 2023129 
(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)))))))