Average Error: 59.5 → 36.2
Time: 8.4s
Precision: binary64
\[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\]
↓
\[\frac{\left(\left(e^{x}\right) \bmod \left(\log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right) \cdot 2\right)\right)}{e^{x}}\]
\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}
↓
\frac{\left(\left(e^{x}\right) \bmod \left(\log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right) \cdot 2\right)\right)}{e^{x}}
(FPCore (x) :precision binary64 (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))
↓
(FPCore (x)
 :precision binary64
 (/
  (fmod
   (exp x)
   (+
    (log (cbrt (exp (sqrt (cos x)))))
    (* (log (cbrt (exp (sqrt (cos x))))) 2.0)))
  (exp x)))
double code(double x) {
	return fmod(exp(x), sqrt(cos(x))) * exp(-x);
}
↓
double code(double x) {
	return fmod(exp(x), (log(cbrt(exp(sqrt(cos(x))))) + (log(cbrt(exp(sqrt(cos(x))))) * 2.0))) / exp(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-log-exp_binary6459.5

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

  6. Applied add-cube-cbrt_binary6436.2

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

  7. Applied log-prod_binary6436.2

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

  8. Simplified36.2

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

  9. Final simplification36.2

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

Reproduce

herbie shell --seed 2020219 
(FPCore (x)
  :name "expfmod"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))