Average Error: 13.6 → 0.4
Time: 13.8s
Precision: binary32
\[cosTheta_i > 0.9999 \land cosTheta_i \leq 1 \land 2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1 \land 2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\]
\[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
↓
\[\begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9720318913459778:\\ \;\;\;\;\begin{array}{l} t_0 := \sqrt[3]{-\log \left(1 - u1\right)}\\ \sqrt{t_0 \cdot t_0} \cdot \left(\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \sqrt{t_0}\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(u1 - \left(u1 \cdot u1\right) \cdot \left(-0.5 - u1 \cdot 0.3333333333333333\right)\right) - {u1}^{4} \cdot -0.25} \cdot \cos \left(\sqrt{2} \cdot \left(u2 \cdot \left(\pi \cdot \sqrt{2}\right)\right)\right)\\ \end{array} \]
\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 0.9720318913459778:\\
\;\;\;\;\begin{array}{l}
t_0 := \sqrt[3]{-\log \left(1 - u1\right)}\\
\sqrt{t_0 \cdot t_0} \cdot \left(\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \sqrt{t_0}\right)
\end{array}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\left(u1 - \left(u1 \cdot u1\right) \cdot \left(-0.5 - u1 \cdot 0.3333333333333333\right)\right) - {u1}^{4} \cdot -0.25} \cdot \cos \left(\sqrt{2} \cdot \left(u2 \cdot \left(\pi \cdot \sqrt{2}\right)\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
 (if (<= (- 1.0 u1) 0.9720318913459778)
   (let* ((t_0 (cbrt (- (log (- 1.0 u1))))))
     (* (sqrt (* t_0 t_0)) (* (cos (* 2.0 (* u2 PI))) (sqrt t_0))))
   (*
    (sqrt
     (-
      (- u1 (* (* u1 u1) (- -0.5 (* u1 0.3333333333333333))))
      (* (pow u1 4.0) -0.25)))
    (cos (* (sqrt 2.0) (* u2 (* PI (sqrt 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 tmp;
	if ((1.0f - u1) <= 0.9720318913459778f) {
		float t_0_1 = cbrtf(-logf(1.0f - u1));
		tmp = sqrtf(t_0_1 * t_0_1) * (cosf(2.0f * (u2 * ((float) M_PI))) * sqrtf(t_0_1));
	} else {
		tmp = sqrtf((u1 - ((u1 * u1) * (-0.5f - (u1 * 0.3333333333333333f)))) - (powf(u1, 4.0f) * -0.25f)) * cosf(sqrtf(2.0f) * (u2 * (((float) M_PI) * sqrtf(2.0f))));
	}
	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) < 0.972031891

    1. Initial program 0.8

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

    3. Applied add-cube-cbrt_binary321.0

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

    4. Applied sqrt-prod_binary321.0

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

    5. Applied associate-*l*_binary321.0

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

    6. Simplified1.0

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

    if 0.972031891 < (-.f32 1 u1)

    1. Initial program 16.0

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

    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 add-sqr-sqrt_binary320.3

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

    6. Simplified0.3

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

    7. Simplified0.3

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

    9. Applied sqrt-prod_binary320.3

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

    10. Applied associate-*l*_binary320.3

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

    11. Simplified0.3

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

    12. Taylor expanded around 0 0.3

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9720318913459778:\\ \;\;\;\;\sqrt{\sqrt[3]{-\log \left(1 - u1\right)} \cdot \sqrt[3]{-\log \left(1 - u1\right)}} \cdot \left(\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \sqrt{\sqrt[3]{-\log \left(1 - u1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(u1 - \left(u1 \cdot u1\right) \cdot \left(-0.5 - u1 \cdot 0.3333333333333333\right)\right) - {u1}^{4} \cdot -0.25} \cdot \cos \left(\sqrt{2} \cdot \left(u2 \cdot \left(\pi \cdot \sqrt{2}\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021190 
(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))))