Average Error: 59.7 → 59.7
Time: 8.5s
Precision: binary64
\[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\]
↓
\[\log \left(e^{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}\right)\]
\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}
↓
\log \left(e^{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}\right)
(FPCore (x) :precision binary64 (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))
↓
(FPCore (x)
 :precision binary64
 (log (exp (/ (fmod (exp x) (sqrt (cos x))) (exp 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) log(((double) exp((((double) fmod(((double) exp(x)), ((double) sqrt(((double) cos(x)))))) / ((double) exp(x)))))));
}

Error

Bits error versus x

Derivation

  1. Initial program Error: 59.7 bits

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

  2. SimplifiedError: 59.7 bits

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

  4. Applied add-log-expError: 59.7 bits

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

  5. Final simplificationError: 59.7 bits

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

Reproduce

herbie shell --seed 2020203 
(FPCore (x)
  :name "expfmod"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))