UniformSampleCone, y

?

Percentage Accurate: 57.8% → 98.5%
Time: 23.5s
Precision: binary32
Cost: 26432

?

\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]
\[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
↓
\[\sin \left(\sqrt[3]{{\left(\sqrt{2 \cdot \pi}\right)}^{6} \cdot {uy}^{3}}\right) \cdot \sqrt{\mathsf{fma}\left(ux, \left(1 - maxCos\right) + \left(1 - maxCos\right), \left(1 - maxCos\right) \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot ux\right)\right)\right)} \]
(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 (cbrt (* (pow (sqrt (* 2.0 PI)) 6.0) (pow uy 3.0))))
  (sqrt
   (fma
    ux
    (+ (- 1.0 maxCos) (- 1.0 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(cbrtf((powf(sqrtf((2.0f * ((float) M_PI))), 6.0f) * powf(uy, 3.0f)))) * sqrtf(fmaf(ux, ((1.0f - maxCos) + (1.0f - 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(cbrt(Float32((sqrt(Float32(Float32(2.0) * Float32(pi))) ^ Float32(6.0)) * (uy ^ Float32(3.0))))) * sqrt(fma(ux, Float32(Float32(Float32(1.0) - maxCos) + Float32(Float32(1.0) - 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(\sqrt[3]{{\left(\sqrt{2 \cdot \pi}\right)}^{6} \cdot {uy}^{3}}\right) \cdot \sqrt{\mathsf{fma}\left(ux, \left(1 - maxCos\right) + \left(1 - maxCos\right), \left(1 - maxCos\right) \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot ux\right)\right)\right)}

Local Percentage Accuracy?

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 10 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each of Herbie's proposed alternatives. Up and to the right is better. Each dot represents an alternative program; the red square represents the initial program.

Derivation?

  1. Initial program 60.0%

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

  2. Simplified60.0%

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

    [Start]60.0

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

    associate-*l* [=>]60.0

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

    sub-neg [=>]60.0

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

    +-commutative [=>]60.0

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

    distribute-rgt-neg-in [=>]60.0

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

    fma-def [=>]60.0

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

    +-commutative [=>]60.0

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

    associate-+r- [=>]60.0

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

    fma-def [=>]60.0

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

    neg-sub0 [=>]60.0

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

    +-commutative [=>]60.0

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

    associate-+r- [=>]60.0

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

    associate--r- [=>]60.0

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

    neg-sub0 [<=]60.0

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

    +-commutative [=>]60.0

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

    sub-neg [<=]60.0

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

    fma-def [=>]60.0

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

  3. Taylor expanded in ux around 0 98.1%

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

  4. Simplified98.1%

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

    [Start]98.1

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

    +-commutative [=>]98.1

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

    fma-def [=>]98.1

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

    +-commutative [=>]98.1

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

    mul-1-neg [=>]98.1

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

    sub-neg [=>]98.1

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

    metadata-eval [=>]98.1

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

    distribute-neg-in [=>]98.1

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

    metadata-eval [=>]98.1

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

    +-commutative [<=]98.1

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

    sub-neg [<=]98.1

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

    associate--l+ [=>]98.1

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

    associate-*r* [=>]98.1

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

    *-commutative [=>]98.1

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

    associate-*l* [=>]98.1

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

    sub-neg [=>]98.1

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

    metadata-eval [=>]98.1

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

    +-commutative [=>]98.1

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

    unpow2 [=>]98.1

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

  5. Applied egg-rr98.2%

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

    [Start]98.1

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

    *-commutative [=>]98.1

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

    add-cbrt-cube [=>]98.1

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

    add-cbrt-cube [=>]98.1

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

    cbrt-unprod [=>]98.1

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

    pow3 [=>]98.1

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

    pow3 [=>]98.2

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

  6. Applied egg-rr98.4%

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

    [Start]98.2

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

    add-sqr-sqrt [=>]98.2

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

    unpow-prod-down [=>]98.4

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

  7. Simplified98.4%

    \[\leadsto \sin \left(\sqrt[3]{\color{blue}{{\left(\sqrt{2 \cdot \pi}\right)}^{6}} \cdot {uy}^{3}}\right) \cdot \sqrt{\mathsf{fma}\left(ux, \left(1 - maxCos\right) + \left(1 - maxCos\right), \left(1 - maxCos\right) \cdot \left(\left(-1 + maxCos\right) \cdot \left(ux \cdot ux\right)\right)\right)} \]

  8. Final simplification98.4%

    \[\leadsto \sin \left(\sqrt[3]{{\left(\sqrt{2 \cdot \pi}\right)}^{6} \cdot {uy}^{3}}\right) \cdot \sqrt{\mathsf{fma}\left(ux, \left(1 - maxCos\right) + \left(1 - maxCos\right), \left(1 - maxCos\right) \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot ux\right)\right)\right)} \]

Alternatives

Alternative 1
Accuracy98.3%
Cost23232
\[\sqrt{\mathsf{fma}\left(ux, \left(1 - maxCos\right) + \left(1 - maxCos\right), \left(1 - maxCos\right) \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot ux\right)\right)\right)} \cdot \sin \left(\sqrt[3]{{uy}^{3} \cdot {\left(2 \cdot \pi\right)}^{3}}\right) \]
Alternative 2
Accuracy98.3%
Cost13408
\[\sqrt{ux \cdot \left(\left(2 + maxCos \cdot -2\right) - ux \cdot {\left(maxCos + -1\right)}^{2}\right)} \cdot \sin \left(2 \cdot \left(\pi \cdot uy\right)\right) \]
Alternative 3
Accuracy97.7%
Cost13312
\[\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(\left(2 - ux\right) - maxCos \cdot \mathsf{fma}\left(-2, ux, 2\right)\right)} \]
Alternative 4
Accuracy97.7%
Cost10240
\[\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right) - maxCos \cdot \left(ux \cdot \left(2 + ux \cdot -2\right)\right)} \]
Alternative 5
Accuracy95.0%
Cost10052
\[\begin{array}{l} \mathbf{if}\;maxCos \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\sin \left(2 \cdot \left(\pi \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \]
Alternative 6
Accuracy92.5%
Cost9920
\[\sin \left(2 \cdot \left(\pi \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)} \]
Alternative 7
Accuracy77.4%
Cost6752
\[2 \cdot \left(\pi \cdot \left(uy \cdot \sqrt{ux \cdot \left(\left(--2\right) - ux\right)}\right)\right) \]
Alternative 8
Accuracy77.4%
Cost6720
\[2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(2 - ux\right)}\right)\right) \]
Alternative 9
Accuracy63.4%
Cost6656
\[2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{2 \cdot ux}\right)\right) \]

Reproduce?

herbie shell --seed 2023160 
(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)))))))