Average Error: 0.3 → 0.4
Time: 12.4s
Precision: binary32
\[cosTheta_i > 0.9999 \land cosTheta_i \leq 1 \land 2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1 \land 2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\]
\[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right)\]
↓
\[\sqrt{\frac{u1}{1 - {u1}^{3}} \cdot \left(1 + \left(u1 + u1 \cdot u1\right)\right)} \cdot \cos \left(6.28318530718 \cdot u2\right)\]
\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right)
↓
\sqrt{\frac{u1}{1 - {u1}^{3}} \cdot \left(1 + \left(u1 + u1 \cdot 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 (* (/ u1 (- 1.0 (pow u1 3.0))) (+ 1.0 (+ u1 (* u1 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((u1 / (1.0f - powf(u1, 3.0f))) * (1.0f + (u1 + (u1 * 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 flip3--_binary320.4

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

  4. Applied associate-/r/_binary320.4

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

  5. Simplified0.4

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

  7. Applied *-un-lft-identity_binary320.4

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

  8. Applied distribute-lft-out_binary320.4

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

  9. Applied associate-*r*_binary320.4

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2021174 
(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))))