Average Error: 13.7 → 0.4
Time: 15.8s
Precision: binary32
\[cosTheta_i > 0.9999 \land cosTheta_i \leq 1 \land 2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1 \land 2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\]
\[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
↓
\[\begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;1 - u1 \leq 0.9740946292877197:\\ \;\;\;\;\begin{array}{l} t_1 := \sqrt{t_0}\\ \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(t_1 \cdot t_1\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 + \left(0.25 \cdot {u1}^{4} + \left(u1 \cdot u1\right) \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)\right)} \cdot \cos t_0\\ \end{array} \]
\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)
↓
\begin{array}{l}
t_0 := \left(2 \cdot \pi\right) \cdot u2\\
\mathbf{if}\;1 - u1 \leq 0.9740946292877197:\\
\;\;\;\;\begin{array}{l}
t_1 := \sqrt{t_0}\\
\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(t_1 \cdot t_1\right)
\end{array}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{u1 + \left(0.25 \cdot {u1}^{4} + \left(u1 \cdot u1\right) \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)\right)} \cdot \cos t_0\\


\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 (* (* 2.0 PI) u2)))
   (if (<= (- 1.0 u1) 0.9740946292877197)
     (let* ((t_1 (sqrt t_0)))
       (* (sqrt (- (log (- 1.0 u1)))) (cos (* t_1 t_1))))
     (*
      (sqrt
       (+
        u1
        (+
         (* 0.25 (pow u1 4.0))
         (* (* u1 u1) (+ 0.5 (* u1 0.3333333333333333))))))
      (cos t_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 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if ((1.0f - u1) <= 0.9740946292877197f) {
		float t_1_1 = sqrtf(t_0);
		tmp = sqrtf(-logf(1.0f - u1)) * cosf(t_1_1 * t_1_1);
	} else {
		tmp = sqrtf(u1 + ((0.25f * powf(u1, 4.0f)) + ((u1 * u1) * (0.5f + (u1 * 0.3333333333333333f))))) * cosf(t_0);
	}
	return tmp;
}

Error

Bits error versus cosTheta_i

Bits error versus u1

Bits error versus u2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (-.f32 1 u1) < 0.974094629

    1. Initial program 0.9

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

    3. Applied add-sqr-sqrt_binary320.9

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

    if 0.974094629 < (-.f32 1 u1)

    1. Initial program 16.2

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

    2. Taylor expanded in u1 around 0 0.3

      \[\leadsto \sqrt{-\color{blue}{\left(-\left(0.3333333333333333 \cdot {u1}^{3} + \left(0.5 \cdot {u1}^{2} + \left(0.25 \cdot {u1}^{4} + u1\right)\right)\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

    3. Simplified0.3

      \[\leadsto \color{blue}{\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1 + \left(0.25 \cdot {u1}^{4} + \left(u1 \cdot u1\right) \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)\right)}} \]
    4. Using strategy rm

    5. Applied *-commutative_binary320.3

      \[\leadsto \color{blue}{\sqrt{u1 + \left(0.25 \cdot {u1}^{4} + \left(u1 \cdot u1\right) \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9740946292877197:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\sqrt{\left(2 \cdot \pi\right) \cdot u2} \cdot \sqrt{\left(2 \cdot \pi\right) \cdot u2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 + \left(0.25 \cdot {u1}^{4} + \left(u1 \cdot u1\right) \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021206 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_x"
  :precision binary32
  :pre (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0) (<= 2.328306437e-10 u1 1.0) (<= 2.328306437e-10 u2 1.0))
  (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))