UniformSampleCone, y

?

Percentage Accurate: 58.2% → 98.3%
Time: 16.2s
Precision: binary32
Cost: 19968

?

\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]
\[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
↓
\[\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, 1 + \mathsf{expm1}\left(\mathsf{log1p}\left(1 - \left(maxCos + maxCos\right)\right)\right), \left(1 - maxCos\right) \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot ux\right)\right)\right)} \]
(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
   (fma
    ux
    (+ 1.0 (expm1 (log1p (- 1.0 (+ maxCos maxCos)))))
    (* (- 1.0 maxCos) (* (+ maxCos -1.0) (* ux ux)))))))
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(fmaf(ux, (1.0f + expm1f(log1pf((1.0f - (maxCos + maxCos))))), ((1.0f - maxCos) * ((maxCos + -1.0f) * (ux * ux)))));
}

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(fma(ux, Float32(Float32(1.0) + expm1(log1p(Float32(Float32(1.0) - Float32(maxCos + maxCos))))), Float32(Float32(Float32(1.0) - maxCos) * Float32(Float32(maxCos + Float32(-1.0)) * Float32(ux * ux))))))
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{\mathsf{fma}\left(ux, 1 + \mathsf{expm1}\left(\mathsf{log1p}\left(1 - \left(maxCos + maxCos\right)\right)\right), \left(1 - maxCos\right) \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot ux\right)\right)\right)}

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Herbie found 15 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Bogosity?

Bogosity

Derivation?

  1. Initial program 58.5%

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

  2. Simplified58.4%

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

    [Start]58.5%

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

    associate-*l* [=>]58.5%

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

    sub-neg [=>]58.5%

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

    +-commutative [=>]58.5%

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

    distribute-rgt-neg-in [=>]58.5%

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

    fma-def [=>]58.5%

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

    +-commutative [=>]58.5%

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

    associate-+r- [=>]58.5%

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

    fma-def [=>]58.5%

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

    neg-sub0 [=>]58.5%

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

    +-commutative [=>]58.5%

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

    associate-+r- [=>]58.4%

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

    associate--r- [=>]58.4%

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

    neg-sub0 [<=]58.4%

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

    +-commutative [=>]58.4%

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

    sub-neg [<=]58.4%

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

    fma-def [=>]58.4%

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

  3. Taylor expanded in ux around 0 98.4%

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

  4. Simplified98.5%

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

    [Start]98.4%

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

    +-commutative [=>]98.4%

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

    fma-def [=>]98.4%

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

    associate--l+ [=>]98.5%

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

    mul-1-neg [=>]98.5%

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

    sub-neg [=>]98.5%

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

    metadata-eval [=>]98.5%

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

    distribute-neg-in [=>]98.5%

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

    metadata-eval [=>]98.5%

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

    +-commutative [<=]98.5%

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

    sub-neg [<=]98.5%

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

    sub-neg [=>]98.5%

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

    metadata-eval [=>]98.5%

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

    *-commutative [=>]98.5%

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

    unpow2 [=>]98.5%

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

  5. Applied egg-rr98.5%

    \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \left(maxCos + maxCos\right)\right)\right)}, \left(maxCos + -1\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(1 - maxCos\right)\right)\right)} \]
    Step-by-step derivation

    [Start]98.5%

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

    expm1-log1p-u [=>]98.4%

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

    associate--l- [=>]98.5%

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

  6. Final simplification98.5%

    \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, 1 + \mathsf{expm1}\left(\mathsf{log1p}\left(1 - \left(maxCos + maxCos\right)\right)\right), \left(1 - maxCos\right) \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot ux\right)\right)\right)} \]

Alternatives

Alternative 1
Accuracy98.3%
Cost19968
\[\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, 1 + \mathsf{expm1}\left(\mathsf{log1p}\left(1 - \left(maxCos + maxCos\right)\right)\right), \left(1 - maxCos\right) \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot ux\right)\right)\right)} \]
Alternative 2
Accuracy98.3%
Cost13568
\[\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, 1 + \left(1 + maxCos \cdot -2\right), \left(1 - maxCos\right) \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot ux\right)\right)\right)} \]
Alternative 3
Accuracy98.3%
Cost13536
\[\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right) + \left(1 - maxCos\right) \cdot \left(\left(maxCos + -1\right) \cdot {ux}^{2}\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \]
Alternative 4
Accuracy98.3%
Cost10496
\[\sin \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right) - \left(ux \cdot \left(ux \cdot \left(maxCos \cdot maxCos\right)\right) + \left(maxCos \cdot -2\right) \cdot \left(ux \cdot ux - ux\right)\right)} \]
Alternative 5
Accuracy92.4%
Cost9984
\[\sin \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux} \]
Alternative 6
Accuracy82.2%
Cost9924
\[\begin{array}{l} \mathbf{if}\;ux \leq 0.0013800000306218863:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(ux - \left(1 + ux \cdot maxCos\right)\right) \cdot \left(1 + \left(ux \cdot maxCos - ux\right)\right)}\right)\\ \end{array} \]
Alternative 7
Accuracy92.4%
Cost9920
\[\sin \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)} \]
Alternative 8
Accuracy76.5%
Cost8260
\[\begin{array}{l} t_0 := ux - \left(1 + ux \cdot maxCos\right)\\ t_1 := t_0 \cdot \left(-1 + \left(ux - ux \cdot maxCos\right)\right)\\ \mathbf{if}\;ux \leq 9.000000136438757 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{2 \cdot ux + -2 \cdot \left(ux \cdot maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{\frac{1 + \left(t_0 \cdot \left(1 + \left(ux \cdot maxCos - ux\right)\right)\right) \cdot t_1}{1 + t_1}}\right)\\ \end{array} \]
Alternative 9
Accuracy76.5%
Cost7172
\[\begin{array}{l} t_0 := ux - \left(1 + ux \cdot maxCos\right)\\ \mathbf{if}\;ux \leq 0.00022000000171829015:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{2 \cdot ux + -2 \cdot \left(ux \cdot maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 - t_0 \cdot t_0}\right)\\ \end{array} \]
Alternative 10
Accuracy76.5%
Cost7172
\[\begin{array}{l} \mathbf{if}\;ux \leq 0.00022000000171829015:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{2 \cdot ux + -2 \cdot \left(ux \cdot maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(ux - \left(1 + ux \cdot maxCos\right)\right) \cdot \left(1 + \left(ux \cdot maxCos - ux\right)\right)}\right)\\ \end{array} \]
Alternative 11
Accuracy75.3%
Cost6916
\[\begin{array}{l} \mathbf{if}\;ux \leq 0.00022000000171829015:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(ux + -1\right) \cdot \left(1 - ux\right)}\right)\\ \end{array} \]
Alternative 12
Accuracy75.3%
Cost6916
\[\begin{array}{l} \mathbf{if}\;ux \leq 0.00022000000171829015:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{2 \cdot ux + -2 \cdot \left(ux \cdot maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(ux + -1\right) \cdot \left(1 - ux\right)}\right)\\ \end{array} \]
Alternative 13
Accuracy65.6%
Cost6784
\[2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right) \]
Alternative 14
Accuracy63.1%
Cost6656
\[2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux + ux}\right) \]
Alternative 15
Accuracy7.1%
Cost6592
\[2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{0}\right) \]

Reproduce?

herbie shell --seed 2023165 
(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)))))))