Average Error: 59.6 → 59.6
Time: 8.9s
Precision: binary64
\[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\]
↓
\[e^{\log \left(\sqrt[3]{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)} \cdot {\left(\sqrt[3]{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}^{2}\right) - x}\]
\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}
↓
e^{\log \left(\sqrt[3]{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)} \cdot {\left(\sqrt[3]{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}^{2}\right) - x}
double code(double x) {
	return ((double) (((double) fmod(((double) exp(x)), ((double) sqrt(((double) cos(x)))))) * ((double) exp(((double) -(x))))));
}
↓
double code(double x) {
	return ((double) exp(((double) (((double) log(((double) (((double) cbrt(((double) fmod(((double) exp(x)), ((double) sqrt(((double) cos(x)))))))) * ((double) pow(((double) cbrt(((double) fmod(((double) exp(x)), ((double) sqrt(((double) cos(x)))))))), 2.0)))))) - x))));
}

Error

Bits error versus x

Derivation

  1. Initial program 59.6

    \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\]

  2. Simplified59.6

    \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}\]
  3. Using strategy rm

  4. Applied add-exp-log59.6

    \[\leadsto \frac{\color{blue}{e^{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}}{e^{x}}\]

  5. Applied div-exp59.6

    \[\leadsto \color{blue}{e^{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}}\]
  6. Using strategy rm

  7. Applied add-cube-cbrt59.6

    \[\leadsto e^{\log \color{blue}{\left(\left(\sqrt[3]{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)} \cdot \sqrt[3]{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right) \cdot \sqrt[3]{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)} - x}\]

  8. Simplified59.6

    \[\leadsto e^{\log \left(\color{blue}{{\left(\sqrt[3]{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}^{2}} \cdot \sqrt[3]{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right) - x}\]

  9. Final simplification59.6

    \[\leadsto e^{\log \left(\sqrt[3]{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)} \cdot {\left(\sqrt[3]{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}^{2}\right) - x}\]

Reproduce

herbie shell --seed 2020199 
(FPCore (x)
  :name "expfmod"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))