?

Average Error: 13.7 → 0.5
Time: 17.8s
Precision: binary32
Cost: 16640

?

\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]
\[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
↓
\[\begin{array}{l} t_0 := \left(1 - maxCos\right) \cdot ux\\ \sqrt[3]{{\sin \left(\pi \cdot \left(2 \cdot uy\right)\right)}^{3} \cdot {\left(t_0 \cdot \left(2 - t_0\right)\right)}^{1.5}} \end{array} \]
(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
 (let* ((t_0 (* (- 1.0 maxCos) ux)))
   (cbrt (* (pow (sin (* PI (* 2.0 uy))) 3.0) (pow (* t_0 (- 2.0 t_0)) 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) {
	float t_0 = (1.0f - maxCos) * ux;
	return cbrtf((powf(sinf((((float) M_PI) * (2.0f * uy))), 3.0f) * powf((t_0 * (2.0f - t_0)), 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)
	t_0 = Float32(Float32(Float32(1.0) - maxCos) * ux)
	return cbrt(Float32((sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) ^ Float32(3.0)) * (Float32(t_0 * Float32(Float32(2.0) - t_0)) ^ 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)}
↓
\begin{array}{l}
t_0 := \left(1 - maxCos\right) \cdot ux\\
\sqrt[3]{{\sin \left(\pi \cdot \left(2 \cdot uy\right)\right)}^{3} \cdot {\left(t_0 \cdot \left(2 - t_0\right)\right)}^{1.5}}
\end{array}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 13.7

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

  2. Simplified13.7

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

    [Start]13.7

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

    associate-*l* [=>]13.7

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

    cancel-sign-sub-inv [=>]13.7

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

    +-commutative [=>]13.7

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

    *-commutative [=>]13.7

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

    fma-def [=>]13.7

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

    +-commutative [=>]13.7

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

    associate-+r- [=>]13.7

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

    fma-def [=>]13.7

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

    neg-sub0 [=>]13.7

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

    +-commutative [=>]13.7

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

    associate-+r- [=>]13.7

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

    associate--r- [=>]13.7

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

    +-commutative [=>]13.7

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

    sub0-neg [=>]13.7

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

    sub-neg [<=]13.7

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

    fma-def [=>]13.7

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

  3. Taylor expanded in ux around -inf 0.5

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

  4. Simplified0.5

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

    [Start]0.5

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

    +-commutative [=>]0.5

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

    mul-1-neg [=>]0.5

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

    unsub-neg [=>]0.5

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

    *-commutative [=>]0.5

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

    mul-1-neg [=>]0.5

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

    sub-neg [<=]0.5

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

    associate-*l* [=>]0.5

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

    unpow2 [=>]0.5

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

    mul-1-neg [=>]0.5

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

    sub-neg [<=]0.5

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

  5. Applied egg-rr0.5

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

  6. Simplified0.5

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

    [Start]0.5

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

    associate-*r* [=>]0.5

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

    *-commutative [<=]0.5

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

    *-commutative [=>]0.5

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

    unpow2 [=>]0.5

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

    distribute-rgt-out-- [=>]0.5

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

    *-commutative [=>]0.5

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

    *-commutative [=>]0.5

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

  7. Final simplification0.5

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

Alternatives

Alternative 1
Error1.3
Cost23113
\[\begin{array}{l} t_0 := \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\\ t_1 := \left(1 - maxCos\right) \cdot ux\\ \mathbf{if}\;t_0 \leq 1.6000000213622911 \cdot 10^{-9} \lor \neg \left(t_0 \leq 0.0012000000569969416\right):\\ \;\;\;\;\sin \left(2 \cdot \left(\pi \cdot uy\right)\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{t_1 \cdot \left(2 - t_1\right)}\right)\right)\\ \end{array} \]
Alternative 2
Error1.3
Cost23049
\[\begin{array}{l} t_0 := \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\\ t_1 := \left(1 - maxCos\right) \cdot ux\\ \mathbf{if}\;t_0 \leq 1.6000000213622911 \cdot 10^{-9} \lor \neg \left(t_0 \leq 0.0012000000569969416\right):\\ \;\;\;\;t_0 \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{t_1 \cdot \left(2 - t_1\right)}\right)\right)\\ \end{array} \]
Alternative 3
Error0.6
Cost13408
\[\begin{array}{l} t_0 := \left(1 - maxCos\right) \cdot ux\\ \sin \left(uy \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt[3]{{\left(t_0 \cdot \left(2 - t_0\right)\right)}^{1.5}} \end{array} \]
Alternative 4
Error0.5
Cost10176
\[\sin \left(uy \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(\left(1 - maxCos\right) \cdot \left(2 - \left(1 - maxCos\right) \cdot ux\right)\right)} \]
Alternative 5
Error0.5
Cost10176
\[\sqrt{\left(1 - maxCos\right) \cdot \left(ux \cdot \left(2 - \left(1 - maxCos\right) \cdot ux\right)\right)} \cdot \sin \left(2 \cdot \left(\pi \cdot uy\right)\right) \]
Alternative 6
Error3.3
Cost9988
\[\begin{array}{l} t_0 := \left(1 - maxCos\right) \cdot ux\\ \mathbf{if}\;2 \cdot uy \leq 0.014999999664723873:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{t_0 \cdot \left(2 - t_0\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(\pi \cdot uy\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]
Alternative 7
Error5.9
Cost6976
\[\begin{array}{l} t_0 := \left(1 - maxCos\right) \cdot ux\\ 2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{t_0 \cdot \left(2 - t_0\right)}\right)\right) \end{array} \]
Alternative 8
Error7.2
Cost6720
\[\pi \cdot \left(\left(2 \cdot uy\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\right) \]
Alternative 9
Error11.6
Cost6656
\[\pi \cdot \left(\left(2 \cdot uy\right) \cdot \sqrt{ux + ux}\right) \]
Alternative 10
Error25.3
Cost6528
\[\sin \left(2 \cdot \left(\pi \cdot uy\right)\right) \]
Alternative 11
Error25.5
Cost3328
\[\pi \cdot \left(2 \cdot uy\right) \]
Alternative 12
Error29.7
Cost32
\[0 \]

Error

Reproduce?

herbie shell --seed 2023054 
(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)))))))