Average Error: 59.5 → 59.5
Time: 14.9s
Precision: binary64
\[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
↓
\[\begin{array}{l} t_0 := \log \left(e^{\sqrt[3]{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}}\right)\\ t_0 \cdot \left(t_0 \cdot t_0\right) \end{array} \]
\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}
↓
\begin{array}{l}
t_0 := \log \left(e^{\sqrt[3]{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}}\right)\\
t_0 \cdot \left(t_0 \cdot t_0\right)
\end{array}

(FPCore (x) :precision binary64 (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (log (exp (cbrt (/ (fmod (exp x) (sqrt (cos x))) (exp x)))))))
   (* t_0 (* t_0 t_0))))
double code(double x) {
	return fmod(exp(x), sqrt(cos(x))) * exp(-x);
}
↓
double code(double x) {
	double t_0 = log(exp(cbrt(fmod(exp(x), sqrt(cos(x))) / exp(x))));
	return t_0 * (t_0 * t_0);
}

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. Applied add-cube-cbrt_binary6459.5

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

  4. Applied add-log-exp_binary6459.5

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

  5. Applied add-log-exp_binary6459.5

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

  6. Applied add-log-exp_binary6459.5

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

  7. Final simplification59.5

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

Reproduce

herbie shell --seed 2022067 
(FPCore (x)
  :name "expfmod"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))