Average Error: 13.8 → 0.3
Time: 13.5s
Precision: binary32
Cost: 19456
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
↓
\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \log \left(e^{u2 \cdot \pi}\right)\right) \]
(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)))) (cos (* 2.0 (log (exp (* u2 PI)))))))
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)) * cosf((2.0f * logf(expf((u2 * ((float) M_PI))))));
}

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)))) * cos(Float32(Float32(2.0) * log(exp(Float32(u2 * Float32(pi)))))))
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 \cos \left(2 \cdot \log \left(e^{u2 \cdot \pi}\right)\right)

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.8

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

  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]13.8

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

    sub-neg [=>]13.8

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

    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-rr0.4

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

  4. Taylor expanded in u2 around inf 0.3

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

  5. Final simplification0.3

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

Alternatives

Alternative 1
Error0.3
Cost19456
\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot e^{\log \left(u2 \cdot \pi\right)}\right) \]
Alternative 2
Error2.9
Cost16356
\[\begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \leq 0.9999961853027344:\\ \;\;\;\;\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]
Alternative 3
Error0.3
Cost13056
\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right) \]
Alternative 4
Error1.4
Cost10244
\[\begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9549999833106995:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 + \left(u1 \cdot u1\right) \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)} \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\\ \end{array} \]
Alternative 5
Error6.4
Cost6496
\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \]
Alternative 6
Error7.4
Cost3680
\[\sqrt{u1 + \left(u1 \cdot u1\right) \cdot \left(0.5 - u1 \cdot \left(-0.3333333333333333 + u1 \cdot -0.25\right)\right)} \]
Alternative 7
Error7.8
Cost3552
\[\sqrt{u1 + u1 \cdot \left(u1 \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)\right)} \]
Alternative 8
Error8.6
Cost3424
\[\sqrt{u1 - \left(u1 \cdot u1\right) \cdot -0.5} \]
Alternative 9
Error11.2
Cost3232
\[\sqrt{u1} \]

Error

Reproduce

herbie shell --seed 2023016 
(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))))