Average Error: 0.3 → 0.3
Time: 9.9s
Precision: binary32
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
↓
\[{\left(\frac{u1}{1 - u1}\right)}^{0.5} \cdot \cos \left(e^{\log \left(\sqrt{6.28318530718} \cdot \left(\sqrt{6.28318530718} \cdot u2\right)\right)}\right) \]
\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right)
↓
{\left(\frac{u1}{1 - u1}\right)}^{0.5} \cdot \cos \left(e^{\log \left(\sqrt{6.28318530718} \cdot \left(\sqrt{6.28318530718} \cdot u2\right)\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
 (*
  (pow (/ u1 (- 1.0 u1)) 0.5)
  (cos (exp (log (* (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 powf((u1 / (1.0f - u1)), 0.5f) * cosf(expf(logf((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. Applied add-sqr-sqrt_binary320.3

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

  3. Applied associate-*l*_binary320.3

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

  4. Applied pow1/2_binary320.3

    \[\leadsto \color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{0.5}} \cdot \cos \left(\sqrt{6.28318530718} \cdot \left(\sqrt{6.28318530718} \cdot u2\right)\right) \]

  5. Applied add-exp-log_binary320.3

    \[\leadsto {\left(\frac{u1}{1 - u1}\right)}^{0.5} \cdot \cos \color{blue}{\left(e^{\log \left(\sqrt{6.28318530718} \cdot \left(\sqrt{6.28318530718} \cdot u2\right)\right)}\right)} \]

  6. Final simplification0.3

    \[\leadsto {\left(\frac{u1}{1 - u1}\right)}^{0.5} \cdot \cos \left(e^{\log \left(\sqrt{6.28318530718} \cdot \left(\sqrt{6.28318530718} \cdot u2\right)\right)}\right) \]

Reproduce

herbie shell --seed 2022125 
(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))))