Average Error: 59.6 → 59.6
Time: 9.0s
Precision: binary64
\[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\]
↓
\[\log \left(\sqrt{e^{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}}\right) + \frac{\frac{\sqrt[3]{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}{\sqrt[3]{e^{x}}}}{2} \cdot \frac{\sqrt[3]{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)} \cdot \sqrt[3]{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}}\]
\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}
↓
\log \left(\sqrt{e^{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}}\right) + \frac{\frac{\sqrt[3]{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}{\sqrt[3]{e^{x}}}}{2} \cdot \frac{\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)}}{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}}
(FPCore (x) :precision binary64 (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))
↓
(FPCore (x)
 :precision binary64
 (+
  (log (sqrt (exp (/ (fmod (exp x) (sqrt (cos x))) (exp x)))))
  (*
   (/ (/ (cbrt (fmod (exp x) (sqrt (cos x)))) (cbrt (exp x))) 2.0)
   (/
    (*
     (cbrt (fmod (exp x) (sqrt (cos x))))
     (cbrt (fmod (exp x) (sqrt (cos x)))))
    (* (cbrt (exp x)) (cbrt (exp x)))))))
double code(double x) {
	return fmod(exp(x), sqrt(cos(x))) * exp(-x);
}
↓
double code(double x) {
	return log(sqrt(exp(fmod(exp(x), sqrt(cos(x))) / exp(x)))) + (((cbrt(fmod(exp(x), sqrt(cos(x)))) / cbrt(exp(x))) / 2.0) * ((cbrt(fmod(exp(x), sqrt(cos(x)))) * cbrt(fmod(exp(x), sqrt(cos(x))))) / (cbrt(exp(x)) * cbrt(exp(x)))));
}

Error

Bits error versus x

Derivation

  1. Initial program 59.6

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

  2. Simplified59.6

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

  4. Applied add-log-exp_binary6459.6

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

  6. Applied add-sqr-sqrt_binary6459.6

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

  7. Applied log-prod_binary6459.6

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

  9. Applied add-cube-cbrt_binary6459.6

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

  10. Applied add-cube-cbrt_binary6459.6

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

  11. Applied times-frac_binary6459.6

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

  12. Applied exp-prod_binary6459.6

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

  13. Applied sqrt-pow1_binary6459.6

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

  14. Applied log-pow_binary6459.6

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

  15. Simplified59.6

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

  16. Final simplification59.6

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

Reproduce

herbie shell --seed 2020233 
(FPCore (x)
  :name "expfmod"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))