Average Error: 0.5 → 0.5
Time: 14.6s
Precision: binary32
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
↓
\[\sqrt[3]{\sqrt{\frac{{u1}^{3}}{{\left(1 - u1\right)}^{3}}} \cdot {\sin \left(6.28318530718 \cdot u2\right)}^{3}} \]
\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right)
↓
\sqrt[3]{\sqrt{\frac{{u1}^{3}}{{\left(1 - u1\right)}^{3}}} \cdot {\sin \left(6.28318530718 \cdot u2\right)}^{3}}

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))
↓
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (cbrt
  (*
   (sqrt (/ (pow u1 3.0) (pow (- 1.0 u1) 3.0)))
   (pow (sin (* 6.28318530718 u2)) 3.0))))
float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(u1 / (1.0f - u1)) * sinf(6.28318530718f * u2);
}
↓
float code(float cosTheta_i, float u1, float u2) {
	return cbrtf(sqrtf(powf(u1, 3.0f) / powf((1.0f - u1), 3.0f)) * powf(sinf(6.28318530718f * u2), 3.0f));
}

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. Initial program 0.5

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

  2. Applied add-cbrt-cube_binary320.5

    \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\sqrt[3]{\left(\sin \left(6.28318530718 \cdot u2\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right) \cdot \sin \left(6.28318530718 \cdot u2\right)}} \]

  3. Applied add-cbrt-cube_binary320.5

    \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \sqrt{\frac{u1}{1 - u1}}}} \cdot \sqrt[3]{\left(\sin \left(6.28318530718 \cdot u2\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right) \cdot \sin \left(6.28318530718 \cdot u2\right)} \]

  4. Applied cbrt-unprod_binary320.6

    \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \left(\left(\sin \left(6.28318530718 \cdot u2\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right)}} \]

  5. Taylor expanded in u2 around inf 0.5

    \[\leadsto \sqrt[3]{\color{blue}{\sqrt{\frac{{u1}^{3}}{{\left(1 - u1\right)}^{3}}} \cdot {\sin \left(6.28318530718 \cdot u2\right)}^{3}}} \]

  6. Final simplification0.5

    \[\leadsto \sqrt[3]{\sqrt{\frac{{u1}^{3}}{{\left(1 - u1\right)}^{3}}} \cdot {\sin \left(6.28318530718 \cdot u2\right)}^{3}} \]

Reproduce

herbie shell --seed 2021332 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz 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 (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))