Average Error: 59.5 → 59.4
Time: 8.7s
Precision: binary64
\[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\]
↓
\[\frac{\left(\left(e^{x}\right) \bmod \left(\left|\sqrt[3]{\cos x}\right| \cdot \sqrt{\sqrt[3]{\cos x}}\right)\right)}{e^{x}}\]
\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}
↓
\frac{\left(\left(e^{x}\right) \bmod \left(\left|\sqrt[3]{\cos x}\right| \cdot \sqrt{\sqrt[3]{\cos x}}\right)\right)}{e^{x}}
(FPCore (x) :precision binary64 (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))
↓
(FPCore (x)
 :precision binary64
 (/ (fmod (exp x) (* (fabs (cbrt (cos x))) (sqrt (cbrt (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) fmod(((double) exp(x)), ((double) (((double) fabs(((double) cbrt(((double) cos(x)))))) * ((double) sqrt(((double) cbrt(((double) cos(x)))))))))) / ((double) exp(x)));
}

Error

Bits error versus x

Derivation

  1. Initial program Error: 59.5 bits

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

  2. SimplifiedError: 59.4 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-cube-cbrtError: 59.4 bits

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

  5. Applied sqrt-prodError: 59.4 bits

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

  6. SimplifiedError: 59.4 bits

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

  7. Final simplificationError: 59.4 bits

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

Reproduce

herbie shell --seed 2020204 
(FPCore (x)
  :name "expfmod"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))