Average Error: 13.6 → 0.3
Time: 12.6s
Precision: binary32
Cost: 16352
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
↓
\[\sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(u1 \cdot \left(-u1\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
(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) (log1p (* u1 (- u1))))) (cos (* (* 2.0 PI) u2))))
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) - log1pf((u1 * -u1)))) * cosf(((2.0f * ((float) M_PI)) * u2));
}

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(u1) - log1p(Float32(u1 * Float32(-u1))))) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end

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

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.6

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

  2. Applied egg-rr3.0

    \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 - u1 \cdot u1\right) + -1 \cdot \mathsf{log1p}\left(u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

  3. Simplified0.3

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

    [Start]3.0

    \[ \sqrt{-\left(\log \left(1 - u1 \cdot u1\right) + -1 \cdot \mathsf{log1p}\left(u1\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

    mul-1-neg [=>]3.0

    \[ \sqrt{-\left(\log \left(1 - u1 \cdot u1\right) + \color{blue}{\left(-\mathsf{log1p}\left(u1\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

    sub-neg [<=]3.0

    \[ \sqrt{-\color{blue}{\left(\log \left(1 - u1 \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

    sub-neg [=>]3.0

    \[ \sqrt{-\left(\log \color{blue}{\left(1 + \left(-u1 \cdot u1\right)\right)} - \mathsf{log1p}\left(u1\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

    log1p-def [=>]0.3

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

    distribute-rgt-neg-in [=>]0.3

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

  4. Final simplification0.3

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

Alternatives

Alternative 1
Error1.3
Cost13476
\[\begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t_0 \leq 0.00022000000171829015:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos t_0 \cdot \sqrt{u1 + \left(u1 \cdot u1\right) \cdot \left(u1 \cdot 0.3333333333333333 + 0.5\right)}\\ \end{array} \]
Alternative 2
Error1.8
Cost13348
\[\begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t_0 \leq 0.004749999847263098:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos t_0 \cdot \sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)}\\ \end{array} \]
Alternative 3
Error0.3
Cost13312
\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(1 - \frac{\cos \left(2 \cdot \left(\pi \cdot u2\right)\right) + -1}{2} \cdot -2\right) \]
Alternative 4
Error3.0
Cost13156
\[\begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t_0 \leq 0.009700000286102295:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos t_0 \cdot \sqrt{u1}\\ \end{array} \]
Alternative 5
Error0.3
Cost13056
\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \]
Alternative 6
Error6.4
Cost6496
\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \]
Alternative 7
Error7.6
Cost3680
\[\sqrt{u1 + \left(u1 \cdot u1\right) \cdot \left(u1 \cdot \left(u1 \cdot 0.25 + 0.3333333333333333\right) + 0.5\right)} \]
Alternative 8
Error8.0
Cost3552
\[\sqrt{u1 + \left(u1 \cdot u1\right) \cdot \left(u1 \cdot 0.3333333333333333 + 0.5\right)} \]
Alternative 9
Error8.7
Cost3424
\[\sqrt{u1 + \left(u1 \cdot u1\right) \cdot 0.5} \]
Alternative 10
Error11.2
Cost3232
\[\sqrt{u1} \]

Error

Reproduce

herbie shell --seed 2022364 
(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))))