Average Error: 33.8 → 33.9
Time: 17.1s
Precision: 64
\[\left|\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}\right) \bmod a\right)\right|\]
↓
\[\left|\left(\sqrt[3]{\left(\left(\left(\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}} \cdot {\left(\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)}^{1}\right) \cdot \sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right) \bmod a\right)} \cdot \sqrt[3]{\left(\left(\left(\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}} \cdot {\left(\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)}^{1}\right) \cdot \sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right) \bmod a\right)}\right) \cdot \sqrt[3]{\left(\left(\left(\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}} \cdot {\left(\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)}^{1}\right) \cdot \log \left(e^{\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}}\right)\right) \bmod a\right)}\right|\]
\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(\sqrt[3]{\left(\left(\left(\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}} \cdot {\left(\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)}^{1}\right) \cdot \sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right) \bmod a\right)} \cdot \sqrt[3]{\left(\left(\left(\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}} \cdot {\left(\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)}^{1}\right) \cdot \sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right) \bmod a\right)}\right) \cdot \sqrt[3]{\left(\left(\left(\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}} \cdot {\left(\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)}^{1}\right) \cdot \log \left(e^{\sqrt[3]{\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 r11941 = a;
        double r11942 = expm1(r11941);
        double r11943 = sin(r11942);
        double r11944 = expm1(r11943);
        double r11945 = atan(r11941);
        double r11946 = atan2(r11944, r11945);
        double r11947 = fmod(r11946, r11941);
        double r11948 = fabs(r11947);
        return r11948;
}
↓
double f(double a) {
        double r11949 = a;
        double r11950 = expm1(r11949);
        double r11951 = sin(r11950);
        double r11952 = expm1(r11951);
        double r11953 = atan(r11949);
        double r11954 = atan2(r11952, r11953);
        double r11955 = cbrt(r11954);
        double r11956 = 1.0;
        double r11957 = pow(r11955, r11956);
        double r11958 = r11955 * r11957;
        double r11959 = r11958 * r11955;
        double r11960 = fmod(r11959, r11949);
        double r11961 = cbrt(r11960);
        double r11962 = r11961 * r11961;
        double r11963 = exp(r11955);
        double r11964 = log(r11963);
        double r11965 = r11958 * r11964;
        double r11966 = fmod(r11965, r11949);
        double r11967 = cbrt(r11966);
        double r11968 = r11962 * r11967;
        double r11969 = fabs(r11968);
        return r11969;
}

Error

Bits error versus a

Derivation

  1. Initial program 33.8

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

  3. Applied add-cube-cbrt33.9

    \[\leadsto \left|\left(\color{blue}{\left(\left(\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}} \cdot \sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right) \cdot \sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)} \bmod a\right)\right|\]
  4. Using strategy rm

  5. Applied pow133.9

    \[\leadsto \left|\left(\left(\left(\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}} \cdot \color{blue}{{\left(\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)}^{1}}\right) \cdot \sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right) \bmod a\right)\right|\]
  6. Using strategy rm

  7. Applied add-cube-cbrt33.9

    \[\leadsto \left|\color{blue}{\left(\sqrt[3]{\left(\left(\left(\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}} \cdot {\left(\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)}^{1}\right) \cdot \sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right) \bmod a\right)} \cdot \sqrt[3]{\left(\left(\left(\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}} \cdot {\left(\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)}^{1}\right) \cdot \sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right) \bmod a\right)}\right) \cdot \sqrt[3]{\left(\left(\left(\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}} \cdot {\left(\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)}^{1}\right) \cdot \sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right) \bmod a\right)}}\right|\]
  8. Using strategy rm

  9. Applied add-log-exp33.9

    \[\leadsto \left|\left(\sqrt[3]{\left(\left(\left(\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}} \cdot {\left(\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)}^{1}\right) \cdot \sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right) \bmod a\right)} \cdot \sqrt[3]{\left(\left(\left(\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}} \cdot {\left(\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)}^{1}\right) \cdot \sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right) \bmod a\right)}\right) \cdot \sqrt[3]{\left(\left(\left(\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}} \cdot {\left(\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)}^{1}\right) \cdot \color{blue}{\log \left(e^{\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}}\right)}\right) \bmod a\right)}\right|\]

  10. Final simplification33.9

    \[\leadsto \left|\left(\sqrt[3]{\left(\left(\left(\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}} \cdot {\left(\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)}^{1}\right) \cdot \sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right) \bmod a\right)} \cdot \sqrt[3]{\left(\left(\left(\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}} \cdot {\left(\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)}^{1}\right) \cdot \sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right) \bmod a\right)}\right) \cdot \sqrt[3]{\left(\left(\left(\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}} \cdot {\left(\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)}^{1}\right) \cdot \log \left(e^{\sqrt[3]{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}}\right)\right) \bmod a\right)}\right|\]

Reproduce

herbie shell --seed 2020039 
(FPCore (a)
  :name "Random Jason Timeout Test 006"
  :precision binary64
  (fabs (fmod (atan2 (expm1 (sin (expm1 a))) (atan a)) a)))