Average Error: 0.3 → 0.3
Time: 16.2s
Precision: binary32
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
↓
\[\sqrt[3]{{\left(\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{\mathsf{fma}\left(u1, u1, u1\right) + {u1}^{3}}{1 - {u1}^{3}}}\right)}^{3}} \]
\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right)
↓
\sqrt[3]{{\left(\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{\mathsf{fma}\left(u1, u1, u1\right) + {u1}^{3}}{1 - {u1}^{3}}}\right)}^{3}}

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

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 flip3--_binary320.4

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

  3. Applied associate-/r/_binary320.4

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

  4. Simplified0.4

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

  5. Applied add-cbrt-cube_binary320.4

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

  6. Simplified0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2021357 
(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))))