Average Error: 59.5 → 59.5
Time: 9.1s
Precision: binary64
\[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\]
↓
\[e^{\left(\sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)} \cdot \sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right) \cdot \sqrt[3]{\log \left(\sqrt[3]{\log \left(e^{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}\right) \cdot 2 + \log \left(\sqrt[3]{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)} - x}\]
\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}
↓
e^{\left(\sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)} \cdot \sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right) \cdot \sqrt[3]{\log \left(\sqrt[3]{\log \left(e^{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}\right) \cdot 2 + \log \left(\sqrt[3]{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)} - x}
(FPCore (x) :precision binary64 (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))
↓
(FPCore (x)
 :precision binary64
 (exp
  (-
   (*
    (*
     (cbrt (log (fmod (exp x) (sqrt (cos x)))))
     (cbrt (log (fmod (exp x) (sqrt (cos x))))))
    (cbrt
     (+
      (* (log (cbrt (log (exp (fmod (exp x) (sqrt (cos x))))))) 2.0)
      (log (cbrt (fmod (exp x) (sqrt (cos x))))))))
   x)))
double code(double x) {
	return fmod(exp(x), sqrt(cos(x))) * exp(-x);
}
↓
double code(double x) {
	return exp(((cbrt(log(fmod(exp(x), sqrt(cos(x))))) * cbrt(log(fmod(exp(x), sqrt(cos(x)))))) * cbrt((log(cbrt(log(exp(fmod(exp(x), sqrt(cos(x))))))) * 2.0) + log(cbrt(fmod(exp(x), sqrt(cos(x))))))) - x);
}

Error

Bits error versus x

Derivation

  1. Initial program 59.5

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

  2. Simplified59.5

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

  4. Applied add-exp-log_binary6459.5

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

  5. Applied div-exp_binary6459.5

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

  7. Applied add-cube-cbrt_binary6459.5

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

  9. Applied add-cube-cbrt_binary6459.5

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

  10. Applied log-prod_binary6459.5

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

  11. Simplified59.5

    \[\leadsto e^{\left(\sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)} \cdot \sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right) \cdot \sqrt[3]{\color{blue}{\log \left(\sqrt[3]{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right) \cdot 2} + \log \left(\sqrt[3]{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)} - x}\]
  12. Using strategy rm

  13. Applied add-log-exp_binary6459.5

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

  14. Final simplification59.5

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

Reproduce

herbie shell --seed 2020232 
(FPCore (x)
  :name "expfmod"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))