Average Error: 13.5 → 0.6
Time: 21.5s
Precision: binary32
\[cosTheta_i > 0.9999 \land cosTheta_i \leq 1 \land 2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1 \land 2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\]
\[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9605987668037415:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sqrt[3]{\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \left(\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\left(u1 - \left(u1 \cdot u1\right) \cdot \left(-0.5 - \sqrt{u1} \cdot \left(\sqrt{u1} \cdot 0.3333333333333333\right)\right)\right) - {u1}^{4} \cdot -0.25}\\ \end{array}\]
\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)
↓
\begin{array}{l}
\mathbf{if}\;1 - u1 \leq 0.9605987668037415:\\
\;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sqrt[3]{\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \left(\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\left(u1 - \left(u1 \cdot u1\right) \cdot \left(-0.5 - \sqrt{u1} \cdot \left(\sqrt{u1} \cdot 0.3333333333333333\right)\right)\right) - {u1}^{4} \cdot -0.25}\\

\end{array}
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 PI) u2))))
↓
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9605987668037415)
   (*
    (sqrt (- (log (- 1.0 u1))))
    (cbrt
     (*
      (sin (* (* 2.0 PI) u2))
      (* (sin (* (* 2.0 PI) u2)) (sin (* (* 2.0 PI) u2))))))
   (*
    (sin (* (* 2.0 PI) u2))
    (sqrt
     (-
      (-
       u1
       (* (* u1 u1) (- -0.5 (* (sqrt u1) (* (sqrt u1) 0.3333333333333333)))))
      (* (pow u1 4.0) -0.25))))))
float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf(1.0f - u1)) * sinf((2.0f * ((float) M_PI)) * u2);
}
↓
float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((1.0f - u1) <= 0.9605987668037415f) {
		tmp = sqrtf(-logf(1.0f - u1)) * cbrtf(sinf((2.0f * ((float) M_PI)) * u2) * (sinf((2.0f * ((float) M_PI)) * u2) * sinf((2.0f * ((float) M_PI)) * u2)));
	} else {
		tmp = sinf((2.0f * ((float) M_PI)) * u2) * sqrtf((u1 - ((u1 * u1) * (-0.5f - (sqrtf(u1) * (sqrtf(u1) * 0.3333333333333333f))))) - (powf(u1, 4.0f) * -0.25f));
	}
	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) < 0.960598767

    1. Initial program 0.9

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

    3. Applied add-cbrt-cube_binary320.9

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

    if 0.960598767 < (-.f32 1 u1)

    1. Initial program 15.6

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

    2. Taylor expanded around 0 0.5

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

    3. Simplified0.5

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

    5. Applied add-sqr-sqrt_binary320.5

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

    6. Applied associate-*l*_binary320.5

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

    7. Simplified0.5

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9605987668037415:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sqrt[3]{\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \left(\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\left(u1 - \left(u1 \cdot u1\right) \cdot \left(-0.5 - \sqrt{u1} \cdot \left(\sqrt{u1} \cdot 0.3333333333333333\right)\right)\right) - {u1}^{4} \cdot -0.25}\\ \end{array}\]

Reproduce

herbie shell --seed 2021173 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_y"
  :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)))) (sin (* (* 2.0 PI) u2))))