Average Error: 59.6 → 24.9
Time: 8.9s
Precision: binary64
\[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\]
↓
\[\sqrt[3]{{\left(e^{-x}\right)}^{3}}\]
\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}
↓
\sqrt[3]{{\left(e^{-x}\right)}^{3}}
(FPCore (x) :precision binary64 (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))
↓
(FPCore (x) :precision binary64 (cbrt (pow (exp (- x)) 3.0)))
double code(double x) {
	return fmod(exp(x), sqrt(cos(x))) * exp(-x);
}
↓
double code(double x) {
	return cbrt(pow(exp(-x), 3.0));
}

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-cbrt-cube_binary6459.6

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

  5. Simplified59.6

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

  7. Applied add-cbrt-cube_binary6459.6

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

  8. Simplified59.6

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

  10. Applied add-exp-log_binary6459.6

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

  11. Simplified59.6

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

  12. Taylor expanded around inf 24.9

    \[\leadsto \sqrt[3]{{\left(e^{\color{blue}{-1 \cdot x}}\right)}^{3}}\]

  13. Final simplification24.9

    \[\leadsto \sqrt[3]{{\left(e^{-x}\right)}^{3}}\]

Reproduce

herbie shell --seed 2021173 
(FPCore (x)
  :name "expfmod"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))