Average Error: 0.3 → 0.3
Time: 15.6s
Precision: binary32
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
↓
\[\sqrt[3]{\left(\left(\frac{u1}{1 - u1 \cdot u1} \cdot \left(u1 + 1\right)\right) \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \left(\left(\cos \left(\sqrt{6.28318530718} \cdot \left(\sqrt{6.28318530718} \cdot u2\right)\right) \cdot \cos \left(6.28318530718 \cdot u2\right)\right) \cdot \cos \left(e^{\log 6.28318530718 + \log u2}\right)\right)} \]
\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right)
↓
\sqrt[3]{\left(\left(\frac{u1}{1 - u1 \cdot u1} \cdot \left(u1 + 1\right)\right) \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \left(\left(\cos \left(\sqrt{6.28318530718} \cdot \left(\sqrt{6.28318530718} \cdot u2\right)\right) \cdot \cos \left(6.28318530718 \cdot u2\right)\right) \cdot \cos \left(e^{\log 6.28318530718 + \log u2}\right)\right)}

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (cos (* 6.28318530718 u2))))
↓
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (cbrt
  (*
   (* (* (/ u1 (- 1.0 (* u1 u1))) (+ u1 1.0)) (sqrt (/ u1 (- 1.0 u1))))
   (*
    (*
     (cos (* (sqrt 6.28318530718) (* (sqrt 6.28318530718) u2)))
     (cos (* 6.28318530718 u2)))
    (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 cbrtf((((u1 / (1.0f - (u1 * u1))) * (u1 + 1.0f)) * sqrtf(u1 / (1.0f - u1))) * ((cosf(sqrtf(6.28318530718f) * (sqrtf(6.28318530718f) * u2)) * cosf(6.28318530718f * u2)) * cosf(expf(logf(6.28318530718f) + logf(u2)))));
}

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 0.3

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

  2. Applied add-cbrt-cube_binary320.3

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

  3. Applied add-cbrt-cube_binary320.3

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

  4. Applied cbrt-unprod_binary320.3

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

  5. Applied add-exp-log_binary320.3

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

  6. Applied add-exp-log_binary320.3

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

  7. Applied prod-exp_binary320.3

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

  8. Applied flip--_binary320.3

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

  9. Applied associate-/r/_binary320.3

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

  10. Applied sqrt-prod_binary320.3

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

  11. Applied flip--_binary320.4

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

  12. Applied associate-/r/_binary320.4

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

  13. Applied sqrt-prod_binary320.4

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

  14. Applied swap-sqr_binary320.4

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

  15. Simplified0.4

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

  16. Simplified0.3

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

  17. Applied add-sqr-sqrt_binary320.3

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

  18. Applied associate-*l*_binary320.3

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

  19. Simplified0.3

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

  20. Final simplification0.3

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

Reproduce

herbie shell --seed 2021291 
(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))))