Average Error: 0.3 → 0.3
Time: 6.8s
Precision: binary32
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
↓
\[\begin{array}{l} t_0 := \frac{u1}{1 - {u1}^{3}}\\ \sqrt{t_0 + t_0 \cdot \mathsf{fma}\left(u1, u1, u1\right)} \cdot \cos \left(6.28318530718 \cdot u2\right) \end{array} \]
\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right)
↓
\begin{array}{l}
t_0 := \frac{u1}{1 - {u1}^{3}}\\
\sqrt{t_0 + t_0 \cdot \mathsf{fma}\left(u1, u1, u1\right)} \cdot \cos \left(6.28318530718 \cdot u2\right)
\end{array}

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (cos (* 6.28318530718 u2))))
↓
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (/ u1 (- 1.0 (pow u1 3.0)))))
   (* (sqrt (+ t_0 (* t_0 (fma u1 u1 u1)))) (cos (* 6.28318530718 u2)))))
float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * cosf((6.28318530718f * u2));
}
↓
float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u1 / (1.0f - powf(u1, 3.0f));
	return sqrtf((t_0 + (t_0 * fmaf(u1, u1, u1)))) * cosf((6.28318530718f * u2));
}

Error

Bits error versus cosTheta_i

Bits error versus u1

Bits error versus u2

Derivation

  1. Initial program 0.3

    \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

  2. Applied egg-rr0.4

    \[\leadsto \sqrt{\color{blue}{\frac{u1}{1 - {u1}^{3}} \cdot \left(1 + \mathsf{fma}\left(u1, u1, u1\right)\right)}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

  3. Applied egg-rr0.3

    \[\leadsto \sqrt{\color{blue}{\frac{u1}{1 - {u1}^{3}} \cdot \mathsf{fma}\left(u1, u1, u1\right) + \frac{u1}{1 - {u1}^{3}}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

  4. Final simplification0.3

    \[\leadsto \sqrt{\frac{u1}{1 - {u1}^{3}} + \frac{u1}{1 - {u1}^{3}} \cdot \mathsf{fma}\left(u1, u1, u1\right)} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

Reproduce

herbie shell --seed 2022127 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_x"
  :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 (/ u1 (- 1.0 u1))) (cos (* 6.28318530718 u2))))