Average Error: 0.3 → 0.3
Time: 9.9s
Precision: binary32
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
↓
\[\cos \left(\sqrt[3]{248.05021344244753 \cdot \left(u2 \cdot \left(u2 \cdot u2\right)\right)}\right) \cdot \sqrt{\frac{\mathsf{fma}\left(u1, u1, u1\right)}{1 - u1 \cdot u1}} \]
\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right)
↓
\cos \left(\sqrt[3]{248.05021344244753 \cdot \left(u2 \cdot \left(u2 \cdot u2\right)\right)}\right) \cdot \sqrt{\frac{\mathsf{fma}\left(u1, u1, u1\right)}{1 - u1 \cdot u1}}

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (cos (* 6.28318530718 u2))))
↓
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (cos (cbrt (* 248.05021344244753 (* u2 (* u2 u2)))))
  (sqrt (/ (fma u1 u1 u1) (- 1.0 (* u1 u1))))))
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 cosf(cbrtf(248.05021344244753f * (u2 * (u2 * u2)))) * sqrtf(fmaf(u1, u1, u1) / (1.0f - (u1 * u1)));
}

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 flip--_binary320.4

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

  3. Applied associate-/r/_binary320.4

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

  4. Simplified0.4

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

  5. Taylor expanded in u2 around inf 0.4

    \[\leadsto \color{blue}{\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{\left(1 + u1\right) \cdot u1}{1 - {u1}^{2}}}} \]

  6. Simplified0.4

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

  7. Applied add-cbrt-cube_binary320.4

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

  8. Applied add-cbrt-cube_binary320.4

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

  9. Applied cbrt-unprod_binary320.4

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

  10. Final simplification0.3

    \[\leadsto \cos \left(\sqrt[3]{248.05021344244753 \cdot \left(u2 \cdot \left(u2 \cdot u2\right)\right)}\right) \cdot \sqrt{\frac{\mathsf{fma}\left(u1, u1, u1\right)}{1 - u1 \cdot u1}} \]

Reproduce

herbie shell --seed 2021329 
(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))))