Average Error: 61.3 → 60.3
Time: 26.9s
Precision: 64
\[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
↓
\[\left(\frac{\sqrt{\pi}}{\sqrt{2}} + \sqrt{\sin^{-1} \left({\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}\right) \cdot \left(\frac{\sqrt{\pi}}{\sqrt{2}} - \sqrt{\sin^{-1} \left({\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}\right)\]
\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)
↓
\left(\frac{\sqrt{\pi}}{\sqrt{2}} + \sqrt{\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) \cdot \left(\frac{\sqrt{\pi}}{\sqrt{2}} - \sqrt{\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)
double f(double a) {
        double r3233 = a;
        double r3234 = cosh(r3233);
        double r3235 = r3233 * r3233;
        double r3236 = fmod(r3234, r3235);
        double r3237 = log1p(r3233);
        double r3238 = pow(r3236, r3237);
        double r3239 = acos(r3238);
        return r3239;
}
↓
double f(double a) {
        double r3240 = atan2(1.0, 0.0);
        double r3241 = sqrt(r3240);
        double r3242 = 2.0;
        double r3243 = sqrt(r3242);
        double r3244 = r3241 / r3243;
        double r3245 = a;
        double r3246 = cosh(r3245);
        double r3247 = r3245 * r3245;
        double r3248 = fmod(r3246, r3247);
        double r3249 = exp(r3248);
        double r3250 = log(r3249);
        double r3251 = log1p(r3245);
        double r3252 = pow(r3250, r3251);
        double r3253 = asin(r3252);
        double r3254 = sqrt(r3253);
        double r3255 = r3244 + r3254;
        double r3256 = r3244 - r3254;
        double r3257 = r3255 * r3256;
        return r3257;
}

Error

Bits error versus a

Derivation

  1. Initial program 61.3

    \[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
  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 add-sqr-sqrt60.3

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

  8. Applied add-sqr-sqrt60.3

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

  9. Applied add-sqr-sqrt60.3

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

  10. Applied times-frac60.3

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

  11. Applied difference-of-squares60.3

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

  12. Final simplification60.3

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

Reproduce

herbie shell --seed 2020002 
(FPCore (a)
  :name "Random Jason Timeout Test 012"
  :precision binary64
  (acos (pow (fmod (cosh a) (* a a)) (log1p a))))