?

Average Error: 13.4 → 0.3
Time: 12.2s
Precision: binary32
Cost: 22912

?

\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
↓
\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) - {\sin \left(\pi \cdot u2\right)}^{2}\right)\right) \]
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))
↓
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (- (log1p (- u1))))
  (+ 0.5 (- (* 0.5 (cos (* PI (+ u2 u2)))) (pow (sin (* PI u2)) 2.0)))))
float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf((1.0f - u1))) * cosf(((2.0f * ((float) M_PI)) * u2));
}
↓
float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * (0.5f + ((0.5f * cosf((((float) M_PI) * (u2 + u2)))) - powf(sinf((((float) M_PI) * u2)), 2.0f)));
}

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end
↓
function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(0.5) + Float32(Float32(Float32(0.5) * cos(Float32(Float32(pi) * Float32(u2 + u2)))) - (sin(Float32(Float32(pi) * u2)) ^ Float32(2.0)))))
end

\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)
↓
\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) - {\sin \left(\pi \cdot u2\right)}^{2}\right)\right)

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 13.4

    \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

  2. Simplified0.3

    \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)} \]
    Proof

    [Start]13.4

    \[ \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

    sub-neg [=>]13.4

    \[ \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

    log1p-def [=>]0.3

    \[ \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

    associate-*l* [=>]0.3

    \[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \color{blue}{\left(2 \cdot \left(\pi \cdot u2\right)\right)} \]

  3. Applied egg-rr0.3

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(0.5 + \left(0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) - {\sin \left(\pi \cdot u2\right)}^{2}\right)\right)} \]

  4. Final simplification0.3

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) - {\sin \left(\pi \cdot u2\right)}^{2}\right)\right) \]

Alternatives

Alternative 1
Error3.4
Cost13060
\[\begin{array}{l} \mathbf{if}\;u2 \leq 0.004999999888241291:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{\mathsf{log1p}\left(u1\right)}\\ \end{array} \]
Alternative 2
Error0.3
Cost13056
\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(\left(\pi \cdot u2\right) \cdot 2\right) \]
Alternative 3
Error6.5
Cost6496
\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \]
Alternative 4
Error7.6
Cost3680
\[\sqrt{u1 - \left(u1 \cdot u1\right) \cdot \left(-0.5 + u1 \cdot \left(u1 \cdot -0.25 + -0.3333333333333333\right)\right)} \]
Alternative 5
Error8.0
Cost3552
\[\sqrt{u1 - \left(u1 \cdot u1\right) \cdot \left(-0.5 + u1 \cdot -0.3333333333333333\right)} \]
Alternative 6
Error8.8
Cost3424
\[\sqrt{u1 + 0.5 \cdot \left(u1 \cdot u1\right)} \]
Alternative 7
Error24.0
Cost3296
\[\sqrt{u1} \cdot 3 \]
Alternative 8
Error11.4
Cost3232
\[\sqrt{u1} \]

Error

Reproduce?

herbie shell --seed 2023027 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_x"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))