Average Error: 33.7 → 33.7
Time: 36.9s
Precision: 64
\[\left|\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}\right) \bmod a\right)\right|\]
↓
\[\left|\left(\left(\log \left(e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)\right) \bmod a\right)\right|\]
\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(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 r18258 = a;
        double r18259 = expm1(r18258);
        double r18260 = sin(r18259);
        double r18261 = expm1(r18260);
        double r18262 = atan(r18258);
        double r18263 = atan2(r18261, r18262);
        double r18264 = fmod(r18263, r18258);
        double r18265 = fabs(r18264);
        return r18265;
}
↓
double f(double a) {
        double r18266 = a;
        double r18267 = expm1(r18266);
        double r18268 = sin(r18267);
        double r18269 = expm1(r18268);
        double r18270 = atan(r18266);
        double r18271 = atan2(r18269, r18270);
        double r18272 = exp(r18271);
        double r18273 = log(r18272);
        double r18274 = fmod(r18273, r18266);
        double r18275 = fabs(r18274);
        return r18275;
}

Error

Bits error versus a

Derivation

  1. Initial program 33.7

    \[\left|\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}\right) \bmod a\right)\right|\]
  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. Final simplification33.7

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

Reproduce

herbie shell --seed 2019196 
(FPCore (a)
  :name "Random Jason Timeout Test 006"
  (fabs (fmod (atan2 (expm1 (sin (expm1 a))) (atan a)) a)))