Average Error: 13.7 → 0.4
Time: 21.2s
Precision: binary32
\[cosTheta_i > 0.9999 \land cosTheta_i \leq 1 \land 2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1 \land 2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\]
\[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9538992643356323:\\ \;\;\;\;\sqrt[3]{\cos \left(\left(2 \cdot \pi\right) \cdot u2\right)} \cdot \left(\sqrt{-\log \left(1 - u1\right)} \cdot \left(\sqrt[3]{\cos \left(\left(2 \cdot \pi\right) \cdot u2\right)} \cdot \sqrt[3]{\cos \left(\left(2 \cdot \pi\right) \cdot u2\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{\left(u1 \cdot u1\right) \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right) + \left(u1 + 0.25 \cdot {u1}^{4}\right)}\\ \end{array}\]
\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)
↓
\begin{array}{l}
\mathbf{if}\;1 - u1 \leq 0.9538992643356323:\\
\;\;\;\;\sqrt[3]{\cos \left(\left(2 \cdot \pi\right) \cdot u2\right)} \cdot \left(\sqrt{-\log \left(1 - u1\right)} \cdot \left(\sqrt[3]{\cos \left(\left(2 \cdot \pi\right) \cdot u2\right)} \cdot \sqrt[3]{\cos \left(\left(2 \cdot \pi\right) \cdot u2\right)}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{\left(u1 \cdot u1\right) \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right) + \left(u1 + 0.25 \cdot {u1}^{4}\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
 (if (<= (- 1.0 u1) 0.9538992643356323)
   (*
    (cbrt (cos (* (* 2.0 PI) u2)))
    (*
     (sqrt (- (log (- 1.0 u1))))
     (* (cbrt (cos (* (* 2.0 PI) u2))) (cbrt (cos (* (* 2.0 PI) u2))))))
   (*
    (cos (* 2.0 (* PI u2)))
    (sqrt
     (+
      (* (* u1 u1) (+ 0.5 (* u1 0.3333333333333333)))
      (+ u1 (* 0.25 (pow u1 4.0))))))))
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 tmp;
	if ((1.0f - u1) <= 0.9538992643356323f) {
		tmp = cbrtf(cosf((2.0f * ((float) M_PI)) * u2)) * (sqrtf(-logf(1.0f - u1)) * (cbrtf(cosf((2.0f * ((float) M_PI)) * u2)) * cbrtf(cosf((2.0f * ((float) M_PI)) * u2))));
	} else {
		tmp = cosf(2.0f * (((float) M_PI) * u2)) * sqrtf(((u1 * u1) * (0.5f + (u1 * 0.3333333333333333f))) + (u1 + (0.25f * powf(u1, 4.0f))));
	}
	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) < 0.953899264

    1. Initial program 0.7

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

    3. Applied add-cube-cbrt_binary320.8

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\left(\sqrt[3]{\cos \left(\left(2 \cdot \pi\right) \cdot u2\right)} \cdot \sqrt[3]{\cos \left(\left(2 \cdot \pi\right) \cdot u2\right)}\right) \cdot \sqrt[3]{\cos \left(\left(2 \cdot \pi\right) \cdot u2\right)}\right)}\]

    4. Applied associate-*r*_binary320.8

      \[\leadsto \color{blue}{\left(\sqrt{-\log \left(1 - u1\right)} \cdot \left(\sqrt[3]{\cos \left(\left(2 \cdot \pi\right) \cdot u2\right)} \cdot \sqrt[3]{\cos \left(\left(2 \cdot \pi\right) \cdot u2\right)}\right)\right) \cdot \sqrt[3]{\cos \left(\left(2 \cdot \pi\right) \cdot u2\right)}}\]

    if 0.953899264 < (-.f32 1 u1)

    1. Initial program 15.8

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

    2. Taylor expanded around 0 0.3

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

    3. Simplified0.3

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

    5. Applied pow1_binary320.3

      \[\leadsto \sqrt{-\left(\left(\left(u1 \cdot u1\right) \cdot \left(-0.5 - u1 \cdot 0.3333333333333333\right) - u1\right) + {u1}^{4} \cdot -0.25\right)} \cdot \color{blue}{{\cos \left(\left(2 \cdot \pi\right) \cdot u2\right)}^{1}}\]

    6. Applied pow1_binary320.3

      \[\leadsto \color{blue}{{\left(\sqrt{-\left(\left(\left(u1 \cdot u1\right) \cdot \left(-0.5 - u1 \cdot 0.3333333333333333\right) - u1\right) + {u1}^{4} \cdot -0.25\right)}\right)}^{1}} \cdot {\cos \left(\left(2 \cdot \pi\right) \cdot u2\right)}^{1}\]

    7. Applied pow-prod-down_binary320.3

      \[\leadsto \color{blue}{{\left(\sqrt{-\left(\left(\left(u1 \cdot u1\right) \cdot \left(-0.5 - u1 \cdot 0.3333333333333333\right) - u1\right) + {u1}^{4} \cdot -0.25\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\right)}^{1}}\]

    8. Simplified0.3

      \[\leadsto {\color{blue}{\left(\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \sqrt{\left(u1 \cdot u1\right) \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right) + \left(u1 + 0.25 \cdot {u1}^{4}\right)}\right)}}^{1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9538992643356323:\\ \;\;\;\;\sqrt[3]{\cos \left(\left(2 \cdot \pi\right) \cdot u2\right)} \cdot \left(\sqrt{-\log \left(1 - u1\right)} \cdot \left(\sqrt[3]{\cos \left(\left(2 \cdot \pi\right) \cdot u2\right)} \cdot \sqrt[3]{\cos \left(\left(2 \cdot \pi\right) \cdot u2\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{\left(u1 \cdot u1\right) \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right) + \left(u1 + 0.25 \cdot {u1}^{4}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021174 
(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))))