Average Error: 0.5 → 0.5
Time: 13.7s
Precision: binary32
\[cosTheta_i > 0.9999 \land cosTheta_i \leq 1 \land 2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1 \land 2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\]
\[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
↓
\[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right) \cdot \left(\sqrt[3]{6.28318530718} \cdot u2\right)\right) \]
\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right)
↓
\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right) \cdot \left(\sqrt[3]{6.28318530718} \cdot u2\right)\right)

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))
↓
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ u1 (- 1.0 u1)))
  (sin
   (*
    (* (cbrt 6.28318530718) (cbrt 6.28318530718))
    (* (cbrt 6.28318530718) u2)))))
float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(u1 / (1.0f - u1)) * sinf(6.28318530718f * u2);
}
↓
float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(u1 / (1.0f - u1)) * sinf((cbrtf(6.28318530718f) * cbrtf(6.28318530718f)) * (cbrtf(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 0.5

    \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
  2. Using strategy rm

  3. Applied add-cube-cbrt_binary320.5

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

  4. Applied associate-*l*_binary320.5

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

  5. Simplified0.5

    \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right) \cdot \color{blue}{\left(u2 \cdot \sqrt[3]{6.28318530718}\right)}\right) \]
  6. Using strategy rm

  7. Applied *-un-lft-identity_binary320.5

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

  8. Final simplification0.5

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

Reproduce

herbie shell --seed 2021204 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_y"
  :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))) (sin (* 6.28318530718 u2))))