Average Error: 0.3 → 0.4
Time: 15.7s
Precision: binary32
\[cosTheta_i > 0.9999 \land cosTheta_i \leq 1 \land 2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1 \land 2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\]
\[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right)\]
↓
\[\sqrt{\frac{u1}{1 - u1 \cdot u1} \cdot \left(u1 + 1\right)} \cdot \cos \left(\sqrt{6.28318530718} \cdot \left(\sqrt{6.28318530718} \cdot u2\right)\right)\]
\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right)
↓
\sqrt{\frac{u1}{1 - u1 \cdot u1} \cdot \left(u1 + 1\right)} \cdot \cos \left(\sqrt{6.28318530718} \cdot \left(\sqrt{6.28318530718} \cdot u2\right)\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 (* u1 u1))) (+ u1 1.0)))
  (cos (* (sqrt 6.28318530718) (* (sqrt 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 - (u1 * u1))) * (u1 + 1.0f)) * cosf(sqrtf(6.28318530718f) * (sqrtf(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-sqr-sqrt_binary320.3

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

  4. Applied associate-*l*_binary320.3

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

  5. Simplified0.3

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

  7. Applied flip--_binary320.4

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

  8. Applied associate-/r/_binary320.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2021175 
(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))))