Average Error: 13.8 → 0.6
Time: 10.0s
Precision: binary32
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{2 \cdot \pi}\\ \mathbf{if}\;1 - u1 \leq 0.9700000286102295:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\frac{2 \cdot \pi}{\frac{1}{u2}}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.3333333333333333, {u1}^{3}, \mathsf{fma}\left(0.5, u1 \cdot u1, \mathsf{fma}\left(0.25, {u1}^{4}, u1\right)\right)\right)\right)}^{0.5} \cdot \sin \left(\left|{t_0}^{2} \cdot \left|t_0 \cdot u2\right|\right|\right)\\ \end{array} \]
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 PI) u2))))
↓
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cbrt (* 2.0 PI))))
   (if (<= (- 1.0 u1) 0.9700000286102295)
     (* (sqrt (- (log (- 1.0 u1)))) (sin (/ (* 2.0 PI) (/ 1.0 u2))))
     (*
      (pow
       (fma
        0.3333333333333333
        (pow u1 3.0)
        (fma 0.5 (* u1 u1) (fma 0.25 (pow u1 4.0) u1)))
       0.5)
      (sin (fabs (* (pow t_0 2.0) (fabs (* t_0 u2)))))))))
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 t_0 = cbrtf((2.0f * ((float) M_PI)));
	float tmp;
	if ((1.0f - u1) <= 0.9700000286102295f) {
		tmp = sqrtf(-logf((1.0f - u1))) * sinf(((2.0f * ((float) M_PI)) / (1.0f / u2)));
	} else {
		tmp = powf(fmaf(0.3333333333333333f, powf(u1, 3.0f), fmaf(0.5f, (u1 * u1), fmaf(0.25f, powf(u1, 4.0f), u1))), 0.5f) * sinf(fabsf((powf(t_0, 2.0f) * fabsf((t_0 * u2)))));
	}
	return tmp;
}

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end
↓
function code(cosTheta_i, u1, u2)
	t_0 = cbrt(Float32(Float32(2.0) * Float32(pi)))
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.9700000286102295))
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) / Float32(Float32(1.0) / u2))));
	else
		tmp = Float32((fma(Float32(0.3333333333333333), (u1 ^ Float32(3.0)), fma(Float32(0.5), Float32(u1 * u1), fma(Float32(0.25), (u1 ^ Float32(4.0)), u1))) ^ Float32(0.5)) * sin(abs(Float32((t_0 ^ Float32(2.0)) * abs(Float32(t_0 * u2))))));
	end
	return tmp
end

\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)
↓
\begin{array}{l}
t_0 := \sqrt[3]{2 \cdot \pi}\\
\mathbf{if}\;1 - u1 \leq 0.9700000286102295:\\
\;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\frac{2 \cdot \pi}{\frac{1}{u2}}\right)\\

\mathbf{else}:\\
\;\;\;\;{\left(\mathsf{fma}\left(0.3333333333333333, {u1}^{3}, \mathsf{fma}\left(0.5, u1 \cdot u1, \mathsf{fma}\left(0.25, {u1}^{4}, u1\right)\right)\right)\right)}^{0.5} \cdot \sin \left(\left|{t_0}^{2} \cdot \left|t_0 \cdot u2\right|\right|\right)\\


\end{array}

Error

Bits error versus cosTheta_i

Bits error versus u1

Bits error versus u2

Derivation

  1. Split input into 2 regimes

  2. if (-.f32 1 u1) < 0.970000029

    1. Initial program 0.9

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

    2. Applied egg-rr0.9

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

    if 0.970000029 < (-.f32 1 u1)

    1. Initial program 16.2

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

    2. Taylor expanded in u1 around 0 0.5

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

    3. Simplified0.5

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

    4. Applied egg-rr0.5

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

    5. Applied egg-rr0.5

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9700000286102295:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\frac{2 \cdot \pi}{\frac{1}{u2}}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.3333333333333333, {u1}^{3}, \mathsf{fma}\left(0.5, u1 \cdot u1, \mathsf{fma}\left(0.25, {u1}^{4}, u1\right)\right)\right)\right)}^{0.5} \cdot \sin \left(\left|{\left(\sqrt[3]{2 \cdot \pi}\right)}^{2} \cdot \left|\sqrt[3]{2 \cdot \pi} \cdot u2\right|\right|\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022155 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_y"
  :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)))) (sin (* (* 2.0 PI) u2))))