Average Error: 61.2 → 59.5
Time: 26.6s
Precision: 64
\[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
↓
\[\frac{\mathsf{fma}\left(\pi \cdot \pi, \frac{1}{4}, \left(-1\right) \cdot {\left(\sqrt[3]{\sin^{-1} \left({\left(\sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \left(\sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}\right)}^{6}\right)}{\mathsf{fma}\left(\pi, \frac{1}{2}, \sin^{-1} \left({\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\right)}\]
\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)
↓
\frac{\mathsf{fma}\left(\pi \cdot \pi, \frac{1}{4}, \left(-1\right) \cdot {\left(\sqrt[3]{\sin^{-1} \left({\left(\sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \left(\sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}\right)}^{6}\right)}{\mathsf{fma}\left(\pi, \frac{1}{2}, \sin^{-1} \left({\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\right)}
double f(double a) {
        double r4496 = a;
        double r4497 = cosh(r4496);
        double r4498 = r4496 * r4496;
        double r4499 = fmod(r4497, r4498);
        double r4500 = log1p(r4496);
        double r4501 = pow(r4499, r4500);
        double r4502 = acos(r4501);
        return r4502;
}
↓
double f(double a) {
        double r4503 = atan2(1.0, 0.0);
        double r4504 = r4503 * r4503;
        double r4505 = 0.25;
        double r4506 = 1.0;
        double r4507 = -r4506;
        double r4508 = a;
        double r4509 = cosh(r4508);
        double r4510 = r4508 * r4508;
        double r4511 = fmod(r4509, r4510);
        double r4512 = cbrt(r4511);
        double r4513 = r4512 * r4512;
        double r4514 = r4512 * r4513;
        double r4515 = log1p(r4508);
        double r4516 = pow(r4514, r4515);
        double r4517 = asin(r4516);
        double r4518 = cbrt(r4517);
        double r4519 = 6.0;
        double r4520 = pow(r4518, r4519);
        double r4521 = r4507 * r4520;
        double r4522 = fma(r4504, r4505, r4521);
        double r4523 = 0.5;
        double r4524 = 2.0;
        double r4525 = pow(r4508, r4524);
        double r4526 = fmod(r4509, r4525);
        double r4527 = pow(r4526, r4515);
        double r4528 = asin(r4527);
        double r4529 = fma(r4503, r4523, r4528);
        double r4530 = r4522 / r4529;
        return r4530;
}

Error

Bits error versus a

Derivation

  1. Initial program 61.2

    \[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
  2. Using strategy rm

  3. Applied add-log-exp60.3

    \[\leadsto \cos^{-1} \left({\color{blue}{\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
  4. Using strategy rm

  5. Applied acos-asin60.3

    \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left({\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}\]
  6. Using strategy rm

  7. Applied flip--60.3

    \[\leadsto \color{blue}{\frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \sin^{-1} \left({\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right) \cdot \sin^{-1} \left({\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}{\frac{\pi}{2} + \sin^{-1} \left({\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}}\]

  8. Simplified60.3

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\pi \cdot \pi, \frac{1}{4}, \left(-1\right) \cdot {\left(\sqrt[3]{\sin^{-1} \left({\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}\right)}^{6}\right)}}{\frac{\pi}{2} + \sin^{-1} \left({\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}\]

  9. Simplified60.3

    \[\leadsto \frac{\mathsf{fma}\left(\pi \cdot \pi, \frac{1}{4}, \left(-1\right) \cdot {\left(\sqrt[3]{\sin^{-1} \left({\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}\right)}^{6}\right)}{\color{blue}{\mathsf{fma}\left(\pi, \frac{1}{2}, \sin^{-1} \left({\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\right)}}\]
  10. Using strategy rm

  11. Applied add-cube-cbrt60.3

    \[\leadsto \frac{\mathsf{fma}\left(\pi \cdot \pi, \frac{1}{4}, \left(-1\right) \cdot {\left(\sqrt[3]{\sin^{-1} \left({\left(\log \left(e^{\color{blue}{\left(\sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right) \cdot \sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}\right)}^{6}\right)}{\mathsf{fma}\left(\pi, \frac{1}{2}, \sin^{-1} \left({\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\right)}\]

  12. Applied exp-prod60.3

    \[\leadsto \frac{\mathsf{fma}\left(\pi \cdot \pi, \frac{1}{4}, \left(-1\right) \cdot {\left(\sqrt[3]{\sin^{-1} \left({\left(\log \color{blue}{\left({\left(e^{\sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right)}^{\left(\sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)}\right)}\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}\right)}^{6}\right)}{\mathsf{fma}\left(\pi, \frac{1}{2}, \sin^{-1} \left({\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\right)}\]

  13. Applied log-pow60.2

    \[\leadsto \frac{\mathsf{fma}\left(\pi \cdot \pi, \frac{1}{4}, \left(-1\right) \cdot {\left(\sqrt[3]{\sin^{-1} \left({\color{blue}{\left(\sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \log \left(e^{\sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right)\right)}}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}\right)}^{6}\right)}{\mathsf{fma}\left(\pi, \frac{1}{2}, \sin^{-1} \left({\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\right)}\]

  14. Simplified59.5

    \[\leadsto \frac{\mathsf{fma}\left(\pi \cdot \pi, \frac{1}{4}, \left(-1\right) \cdot {\left(\sqrt[3]{\sin^{-1} \left({\left(\sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \color{blue}{\left(\sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)}\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}\right)}^{6}\right)}{\mathsf{fma}\left(\pi, \frac{1}{2}, \sin^{-1} \left({\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\right)}\]

  15. Final simplification59.5

    \[\leadsto \frac{\mathsf{fma}\left(\pi \cdot \pi, \frac{1}{4}, \left(-1\right) \cdot {\left(\sqrt[3]{\sin^{-1} \left({\left(\sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \left(\sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}\right)}^{6}\right)}{\mathsf{fma}\left(\pi, \frac{1}{2}, \sin^{-1} \left({\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\right)}\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (a)
  :name "Random Jason Timeout Test 012"
  :precision binary64
  (acos (pow (fmod (cosh a) (* a a)) (log1p a))))