Average Error: 13.8 → 0.5
Time: 16.0s
Precision: binary32
Cost: 19872
\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]
\[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt{2 \cdot \pi}\\ \sin \left(t_0 \cdot \left(uy \cdot t_0\right)\right) \cdot \sqrt{\left(ux - ux \cdot maxCos\right) \cdot \left(2 + \left(ux \cdot maxCos - ux\right)\right)} \end{array} \]
(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 (sqrt (* 2.0 PI))))
   (*
    (sin (* t_0 (* uy t_0)))
    (sqrt (* (- ux (* ux maxCos)) (+ 2.0 (- (* ux maxCos) 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) {
	float t_0 = sqrtf((2.0f * ((float) M_PI)));
	return sinf((t_0 * (uy * t_0))) * sqrtf(((ux - (ux * maxCos)) * (2.0f + ((ux * maxCos) - 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)
	t_0 = sqrt(Float32(Float32(2.0) * Float32(pi)))
	return Float32(sin(Float32(t_0 * Float32(uy * t_0))) * sqrt(Float32(Float32(ux - Float32(ux * maxCos)) * Float32(Float32(2.0) + Float32(Float32(ux * maxCos) - ux)))))
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)
	t_0 = sqrt((single(2.0) * single(pi)));
	tmp = sin((t_0 * (uy * t_0))) * sqrt(((ux - (ux * maxCos)) * (single(2.0) + ((ux * maxCos) - 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)}
↓
\begin{array}{l}
t_0 := \sqrt{2 \cdot \pi}\\
\sin \left(t_0 \cdot \left(uy \cdot t_0\right)\right) \cdot \sqrt{\left(ux - ux \cdot maxCos\right) \cdot \left(2 + \left(ux \cdot maxCos - ux\right)\right)}
\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.8

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

  2. Simplified13.8

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

  3. Taylor expanded in ux around 0 0.5

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

  4. Simplified0.5

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

  5. Applied egg-rr0.9

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

  6. Applied egg-rr0.5

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

  7. Final simplification0.5

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

Alternatives

Alternative 1
Error0.6
Cost16544
\[\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt[3]{{\left(\left(ux - ux \cdot maxCos\right) \cdot \mathsf{fma}\left(ux, maxCos, 2 - ux\right)\right)}^{1.5}} \]
Alternative 2
Error0.5
Cost13312
\[\sqrt{\left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 2\right) - ux\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \]
Alternative 3
Error1.3
Cost10244
\[\begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ \mathbf{if}\;uy \cdot 2 \leq 0.0003499999875202775:\\ \;\;\;\;\sqrt{\left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 2\right) - ux\right)} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\sin t_0 \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]
Alternative 4
Error0.5
Cost10176
\[\sqrt{\left(ux - ux \cdot maxCos\right) \cdot \left(2 + \left(ux \cdot maxCos - ux\right)\right)} \cdot \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \]
Alternative 5
Error1.3
Cost10052
\[\begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0003499999875202775:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{\left(1 - maxCos\right) \cdot \left(ux \cdot \left(\left(2 + ux \cdot maxCos\right) - ux\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]
Alternative 6
Error3.2
Cost9988
\[\begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.004999999888241291:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{\left(1 - maxCos\right) \cdot \left(ux \cdot \left(\left(2 + ux \cdot maxCos\right) - ux\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]
Alternative 7
Error5.9
Cost6976
\[2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{\left(1 - maxCos\right) \cdot \left(ux \cdot \left(\left(2 + ux \cdot maxCos\right) - ux\right)\right)}\right) \]
Alternative 8
Error7.3
Cost6720
\[\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)} \]
Alternative 9
Error11.7
Cost6656
\[\left(uy \cdot \pi\right) \cdot \left(2 \cdot \sqrt{2 \cdot ux}\right) \]
Alternative 10
Error29.7
Cost32
\[0 \]

Error

Reproduce

herbie shell --seed 2022217 
(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)))))))