Average Error: 13.7 → 0.3
Time: 5.7s
Precision: binary32
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
↓
\[\begin{array}{l} t_0 := \sin \left(\pi \cdot u2\right)\\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(u2 \cdot 2\right)\right) + \mathsf{fma}\left(-t_0, t_0, {\sin \log \left(1 + \mathsf{expm1}\left(\pi \cdot u2\right)\right)}^{2}\right)\right) \end{array} \]
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))
↓
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* PI u2))))
   (*
    (sqrt (- (log1p (- u1))))
    (+
     (cos (* PI (* u2 2.0)))
     (fma (- t_0) t_0 (pow (sin (log (+ 1.0 (expm1 (* 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) {
	float t_0 = sinf((((float) M_PI) * u2));
	return sqrtf(-log1pf(-u1)) * (cosf((((float) M_PI) * (u2 * 2.0f))) + fmaf(-t_0, t_0, powf(sinf(logf((1.0f + expm1f((((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)
	t_0 = sin(Float32(Float32(pi) * u2))
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(cos(Float32(Float32(pi) * Float32(u2 * Float32(2.0)))) + fma(Float32(-t_0), t_0, (sin(log(Float32(Float32(1.0) + expm1(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)
↓
\begin{array}{l}
t_0 := \sin \left(\pi \cdot u2\right)\\
\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(u2 \cdot 2\right)\right) + \mathsf{fma}\left(-t_0, t_0, {\sin \log \left(1 + \mathsf{expm1}\left(\pi \cdot u2\right)\right)}^{2}\right)\right)
\end{array}

Error

Bits error versus cosTheta_i

Bits error versus u1

Bits error versus u2

Derivation

  1. Initial program 13.7

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

  2. Simplified0.3

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

  3. Applied egg-rr0.3

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

  4. Applied egg-rr0.3

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(u2 \cdot 2\right)\right) + \mathsf{fma}\left(-\sin \left(\pi \cdot u2\right), \sin \left(\pi \cdot u2\right), {\sin \color{blue}{\log \left(1 + \mathsf{expm1}\left(\pi \cdot u2\right)\right)}}^{2}\right)\right) \]

  5. Final simplification0.3

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(u2 \cdot 2\right)\right) + \mathsf{fma}\left(-\sin \left(\pi \cdot u2\right), \sin \left(\pi \cdot u2\right), {\sin \log \left(1 + \mathsf{expm1}\left(\pi \cdot u2\right)\right)}^{2}\right)\right) \]

Reproduce

herbie shell --seed 2022159 
(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))))