Average Error: 0.3 → 0.3
Time: 11.5s
Precision: binary32
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
↓
\[\begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ \sqrt[3]{\sqrt{\frac{{u1}^{3}}{{\left(1 - u1\right)}^{3}}} \cdot \left(\left(t_0 \cdot t_0\right) \cdot \cos \left(\sqrt{6.28318530718} \cdot \left(u2 \cdot \sqrt{6.28318530718}\right)\right)\right)} \end{array} \]
\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]{\sqrt{\frac{{u1}^{3}}{{\left(1 - u1\right)}^{3}}} \cdot \left(\left(t_0 \cdot t_0\right) \cdot \cos \left(\sqrt{6.28318530718} \cdot \left(u2 \cdot \sqrt{6.28318530718}\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
    (*
     (sqrt (/ (pow u1 3.0) (pow (- 1.0 u1) 3.0)))
     (*
      (* t_0 t_0)
      (cos (* (sqrt 6.28318530718) (* u2 (sqrt 6.28318530718)))))))))
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(sqrtf(powf(u1, 3.0f) / powf((1.0f - u1), 3.0f)) * ((t_0 * t_0) * cosf(sqrtf(6.28318530718f) * (u2 * sqrtf(6.28318530718f)))));
}

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_binary320.4

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

  5. Applied sqrt-unprod_binary320.3

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

  6. Applied sqrt-unprod_binary320.4

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

  7. Simplified0.3

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

  8. Applied add-sqr-sqrt_binary320.3

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

  9. Applied associate-*l*_binary320.3

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

  10. Simplified0.3

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

  11. Applied cube-div_binary320.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2022039 
(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))))