Average Error: 59.4 → 59.4
Time: 8.6s
Precision: binary64
\[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\]
↓
\[\sqrt{\left(\left(e^{x}\right) \bmod \log \left(\sqrt[3]{{\left(e^{\sqrt{\cos x}}\right)}^{3}}\right)\right)} \cdot \frac{\sqrt{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}{e^{x}}\]
\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}
↓
\sqrt{\left(\left(e^{x}\right) \bmod \log \left(\sqrt[3]{{\left(e^{\sqrt{\cos x}}\right)}^{3}}\right)\right)} \cdot \frac{\sqrt{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}{e^{x}}
(FPCore (x) :precision binary64 (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))
↓
(FPCore (x)
 :precision binary64
 (*
  (sqrt (fmod (exp x) (log (cbrt (pow (exp (sqrt (cos x))) 3.0)))))
  (/ (sqrt (fmod (exp x) (sqrt (cos x)))) (exp x))))
double code(double x) {
	return fmod(exp(x), sqrt(cos(x))) * exp(-x);
}
↓
double code(double x) {
	return sqrt(fmod(exp(x), log(cbrt(pow(exp(sqrt(cos(x))), 3.0))))) * (sqrt(fmod(exp(x), sqrt(cos(x)))) / exp(x));
}

Error

Bits error versus x

Derivation

  1. Initial program 59.4

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

  2. Simplified59.4

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

  4. Applied *-un-lft-identity_binary64_76059.4

    \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{\color{blue}{1 \cdot e^{x}}}\]

  5. Applied add-sqr-sqrt_binary64_78259.4

    \[\leadsto \frac{\color{blue}{\sqrt{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)} \cdot \sqrt{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}}{1 \cdot e^{x}}\]

  6. Applied times-frac_binary64_76659.4

    \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}{1} \cdot \frac{\sqrt{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}{e^{x}}}\]

  7. Simplified59.4

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

  9. Applied add-log-exp_binary64_79959.4

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

  11. Applied add-cbrt-cube_binary64_79659.4

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

  12. Simplified59.4

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

  13. Final simplification59.4

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

Reproduce

herbie shell --seed 2020356 
(FPCore (x)
  :name "expfmod"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))