Average Error: 0.3 → 0.4
Time: 18.3s
Precision: binary32
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
↓
\[\sqrt{\frac{\mathsf{fma}\left(u1, u1, u1\right)}{1 - u1 \cdot u1}} \cdot \cos \left(e^{\log 6.28318530718 + \log u2}\right) \]
\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right)
↓
\sqrt{\frac{\mathsf{fma}\left(u1, u1, u1\right)}{1 - u1 \cdot u1}} \cdot \cos \left(e^{\log 6.28318530718 + \log u2}\right)

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (cos (* 6.28318530718 u2))))
↓
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ (fma u1 u1 u1) (- 1.0 (* u1 u1))))
  (cos (exp (+ (log 6.28318530718) (log 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) {
	return sqrtf(fmaf(u1, u1, u1) / (1.0f - (u1 * u1))) * cosf(expf(logf(6.28318530718f) + logf(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 add-exp-log_binary320.3

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

  3. Applied add-exp-log_binary320.4

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

  4. Applied prod-exp_binary320.3

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

  5. Applied flip--_binary320.4

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

  6. Applied associate-/r/_binary320.4

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

  7. Simplified0.4

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

  8. Applied pow1_binary320.4

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

  9. Applied pow1_binary320.4

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

  10. Applied pow-prod-down_binary320.4

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2021280 
(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))))