Average Error: 33.6 → 33.7
Time: 37.9s
Precision: 64
\[\left|\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}\right) \bmod a\right)\right|\]
↓
\[\left|\left(\left(\log \left(\sqrt{e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}} \cdot \sqrt{e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}}\right)\right) \bmod a\right)\right|\]
\left|\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}\right) \bmod a\right)\right|
↓
\left|\left(\left(\log \left(\sqrt{e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}} \cdot \sqrt{e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}}\right)\right) \bmod a\right)\right|
double f(double a) {
        double r16453 = a;
        double r16454 = expm1(r16453);
        double r16455 = sin(r16454);
        double r16456 = expm1(r16455);
        double r16457 = atan(r16453);
        double r16458 = atan2(r16456, r16457);
        double r16459 = fmod(r16458, r16453);
        double r16460 = fabs(r16459);
        return r16460;
}
↓
double f(double a) {
        double r16461 = a;
        double r16462 = expm1(r16461);
        double r16463 = sin(r16462);
        double r16464 = expm1(r16463);
        double r16465 = atan(r16461);
        double r16466 = atan2(r16464, r16465);
        double r16467 = exp(r16466);
        double r16468 = sqrt(r16467);
        double r16469 = r16468 * r16468;
        double r16470 = log(r16469);
        double r16471 = fmod(r16470, r16461);
        double r16472 = fabs(r16471);
        return r16472;
}

Error

Bits error versus a

Derivation

  1. Initial program 33.6

    \[\left|\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}\right) \bmod a\right)\right|\]
  2. Using strategy rm

  3. Applied add-log-exp33.7

    \[\leadsto \left|\left(\color{blue}{\left(\log \left(e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)\right)} \bmod a\right)\right|\]
  4. Using strategy rm

  5. Applied add-sqr-sqrt33.7

    \[\leadsto \left|\left(\left(\log \color{blue}{\left(\sqrt{e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}} \cdot \sqrt{e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}}\right)}\right) \bmod a\right)\right|\]

  6. Final simplification33.7

    \[\leadsto \left|\left(\left(\log \left(\sqrt{e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}} \cdot \sqrt{e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}}\right)\right) \bmod a\right)\right|\]

Reproduce

herbie shell --seed 2019198 
(FPCore (a)
  :name "Random Jason Timeout Test 006"
  (fabs (fmod (atan2 (expm1 (sin (expm1 a))) (atan a)) a)))