Average Error: 59.6 → 59.6
Time: 10.2s
Precision: binary64
\[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\]
↓
\[\log \left(\sqrt{e^{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}}\right) + e^{\log \log \left(\sqrt{e^{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}}\right)}\]
\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}
↓
\log \left(\sqrt{e^{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}}\right) + e^{\log \log \left(\sqrt{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 (sqrt (exp (/ (fmod (exp x) (sqrt (cos x))) (exp x)))))
  (exp (log (log (sqrt (exp (/ (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 log(sqrt(exp(fmod(exp(x), sqrt(cos(x))) / exp(x)))) + exp(log(log(sqrt(exp(fmod(exp(x), sqrt(cos(x))) / 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. Simplified59.5

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

  4. Applied add-log-exp_binary6459.6

    \[\leadsto \color{blue}{\log \left(e^{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}\right)}\]
  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-exp-log_binary6459.6

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

  10. Final simplification59.6

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

Reproduce

herbie shell --seed 2020231 
(FPCore (x)
  :name "expfmod"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))