Average Error: 0.3 → 0.3
Time: 10.6s
Precision: binary32
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
↓
\[\begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ \sqrt[3]{\frac{u1 \cdot \sqrt{\frac{u1}{1 - u1}}}{1 - u1} \cdot \left(\left(t_0 \cdot t_0\right) \cdot \cos \left({e}^{\left(\log 6.28318530718 + \log u2\right)}\right)\right)} \end{array} \]
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (cos (* 6.28318530718 u2))))
↓
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* 6.28318530718 u2))))
   (cbrt
    (*
     (/ (* u1 (sqrt (/ u1 (- 1.0 u1)))) (- 1.0 u1))
     (* (* t_0 t_0) (cos (pow E (+ (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) {
	float t_0 = cosf((6.28318530718f * u2));
	return cbrtf((((u1 * sqrtf((u1 / (1.0f - u1)))) / (1.0f - u1)) * ((t_0 * t_0) * cosf(powf(((float) M_E), (logf(6.28318530718f) + logf(u2)))))));
}

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * cos(Float32(Float32(6.28318530718) * u2)))
end
↓
function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(6.28318530718) * u2))
	return cbrt(Float32(Float32(Float32(u1 * sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))) / Float32(Float32(1.0) - u1)) * Float32(Float32(t_0 * t_0) * cos((Float32(exp(1)) ^ Float32(log(Float32(6.28318530718)) + log(u2)))))))
end

\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right)
↓
\begin{array}{l}
t_0 := \cos \left(6.28318530718 \cdot u2\right)\\
\sqrt[3]{\frac{u1 \cdot \sqrt{\frac{u1}{1 - u1}}}{1 - u1} \cdot \left(\left(t_0 \cdot t_0\right) \cdot \cos \left({e}^{\left(\log 6.28318530718 + \log u2\right)}\right)\right)}
\end{array}

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 *-un-lft-identity_binary320.3

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

  9. Applied exp-prod_binary320.3

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

  10. Simplified0.3

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

  11. Applied sqrt-div_binary320.3

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

  12. Applied sqrt-div_binary320.4

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

  13. Applied associate-*l/_binary320.4

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

  14. Applied frac-times_binary320.4

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

  15. Simplified0.3

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

  16. Simplified0.3

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

  17. Final simplification0.3

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

Reproduce

herbie shell --seed 2022138 
(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))))