Average Error: 59.7 → 59.7
Time: 13.0s
Precision: binary64
\[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{\sqrt{\cos x}}\\ \frac{\left(\left(e^{x}\right) \bmod \left(t_0 \cdot \left(\sqrt[3]{t_0 \cdot t_0} \cdot {\left(\sqrt[3]{{\cos x}^{0.16666666666666666}}\right)}^{4}\right)\right)\right)}{e^{x}} \end{array} \]
\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}
↓
\begin{array}{l}
t_0 := \sqrt[3]{\sqrt{\cos x}}\\
\frac{\left(\left(e^{x}\right) \bmod \left(t_0 \cdot \left(\sqrt[3]{t_0 \cdot t_0} \cdot {\left(\sqrt[3]{{\cos x}^{0.16666666666666666}}\right)}^{4}\right)\right)\right)}{e^{x}}
\end{array}

(FPCore (x) :precision binary64 (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (sqrt (cos x)))))
   (/
    (fmod
     (exp x)
     (*
      t_0
      (*
       (cbrt (* t_0 t_0))
       (pow (cbrt (pow (cos x) 0.16666666666666666)) 4.0))))
    (exp x))))
double code(double x) {
	return fmod(exp(x), sqrt(cos(x))) * exp(-x);
}
↓
double code(double x) {
	double t_0 = cbrt(sqrt(cos(x)));
	return fmod(exp(x), (t_0 * (cbrt(t_0 * t_0) * pow(cbrt(pow(cos(x), 0.16666666666666666)), 4.0)))) / exp(x);
}

Error

Bits error versus x

Derivation

  1. Initial program 59.7

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

  2. Simplified59.7

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

  3. Applied add-cube-cbrt_binary6459.7

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

  4. Applied add-cube-cbrt_binary6459.7

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

  5. Applied cbrt-prod_binary6459.7

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

  6. Applied associate-*l*_binary6459.7

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

  7. Simplified59.7

    \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\left(\sqrt[3]{\sqrt[3]{\sqrt{\cos x}} \cdot \sqrt[3]{\sqrt{\cos x}}} \cdot \color{blue}{{\left(\sqrt[3]{{\cos x}^{0.16666666666666666}}\right)}^{4}}\right) \cdot \sqrt[3]{\sqrt{\cos x}}\right)\right)}{e^{x}} \]

  8. Final simplification59.7

    \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{\sqrt{\cos x}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{\cos x}} \cdot \sqrt[3]{\sqrt{\cos x}}} \cdot {\left(\sqrt[3]{{\cos x}^{0.16666666666666666}}\right)}^{4}\right)\right)\right)}{e^{x}} \]

Reproduce

herbie shell --seed 2021307 
(FPCore (x)
  :name "expfmod"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))