Average Error: 0.3 → 0.3
Time: 19.5s
Precision: binary32
\[cosTheta_i > 0.9999 \land cosTheta_i \leq 1 \land 2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1 \land 2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\]
\[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
↓
\[\sqrt{\sqrt[3]{\frac{u1 \cdot \left(u1 \cdot u1\right)}{\left(1 - u1\right) \cdot \left(\left(1 - u1\right) \cdot \left(1 - u1\right)\right)}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right)
↓
\sqrt{\sqrt[3]{\frac{u1 \cdot \left(u1 \cdot u1\right)}{\left(1 - u1\right) \cdot \left(\left(1 - u1\right) \cdot \left(1 - u1\right)\right)}}} \cdot \cos \left(6.28318530718 \cdot 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 (cbrt (/ (* u1 (* u1 u1)) (* (- 1.0 u1) (* (- 1.0 u1) (- 1.0 u1))))))
  (cos (* 6.28318530718 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(cbrtf((u1 * (u1 * u1)) / ((1.0f - u1) * ((1.0f - u1) * (1.0f - u1))))) * cosf(6.28318530718f * 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. Using strategy rm

  3. Applied add-cbrt-cube_binary320.3

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

  4. Applied add-cbrt-cube_binary320.3

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

  5. Applied cbrt-undiv_binary320.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2021204 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_x"
  :precision binary32
  :pre (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0) (<= 2.328306437e-10 u1 1.0) (<= 2.328306437e-10 u2 1.0))
  (* (sqrt (/ u1 (- 1.0 u1))) (cos (* 6.28318530718 u2))))