Average Error: 59.5 → 35.6
Time: 15.1s
Precision: binary64
\[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
↓
\[\begin{array}{l} t_0 := \sqrt{\mathsf{log1p}\left(\sqrt{\cos x}\right)}\\ t_1 := \sqrt[3]{\mathsf{expm1}\left(t_0 \cdot t_0\right)}\\ \frac{\left(\left(e^{x}\right) \bmod \left(t_1 \cdot \left(t_1 \cdot t_1\right)\right)\right)}{e^{x}} \end{array} \]
\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}
↓
\begin{array}{l}
t_0 := \sqrt{\mathsf{log1p}\left(\sqrt{\cos x}\right)}\\
t_1 := \sqrt[3]{\mathsf{expm1}\left(t_0 \cdot t_0\right)}\\
\frac{\left(\left(e^{x}\right) \bmod \left(t_1 \cdot \left(t_1 \cdot t_1\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 (sqrt (log1p (sqrt (cos x))))) (t_1 (cbrt (expm1 (* t_0 t_0)))))
   (/ (fmod (exp x) (* t_1 (* t_1 t_1))) (exp x))))
double code(double x) {
	return fmod(exp(x), sqrt(cos(x))) * exp(-x);
}
↓
double code(double x) {
	double t_0 = sqrt(log1p(sqrt(cos(x))));
	double t_1 = cbrt(expm1(t_0 * t_0));
	return fmod(exp(x), (t_1 * (t_1 * t_1))) / exp(x);
}

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 expm1-log1p-u_binary6459.5

    \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\cos x}\right)\right)\right)}\right)}{e^{x}} \]

  4. Applied add-sqr-sqrt_binary6457.9

    \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{expm1}\left(\color{blue}{\sqrt{\mathsf{log1p}\left(\sqrt{\cos x}\right)} \cdot \sqrt{\mathsf{log1p}\left(\sqrt{\cos x}\right)}}\right)\right)\right)}{e^{x}} \]

  5. Applied add-cube-cbrt_binary6435.6

    \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\left(\sqrt[3]{\mathsf{expm1}\left(\sqrt{\mathsf{log1p}\left(\sqrt{\cos x}\right)} \cdot \sqrt{\mathsf{log1p}\left(\sqrt{\cos x}\right)}\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\sqrt{\mathsf{log1p}\left(\sqrt{\cos x}\right)} \cdot \sqrt{\mathsf{log1p}\left(\sqrt{\cos x}\right)}\right)}\right) \cdot \sqrt[3]{\mathsf{expm1}\left(\sqrt{\mathsf{log1p}\left(\sqrt{\cos x}\right)} \cdot \sqrt{\mathsf{log1p}\left(\sqrt{\cos x}\right)}\right)}\right)}\right)}{e^{x}} \]

  6. Final simplification35.6

    \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{\mathsf{expm1}\left(\sqrt{\mathsf{log1p}\left(\sqrt{\cos x}\right)} \cdot \sqrt{\mathsf{log1p}\left(\sqrt{\cos x}\right)}\right)} \cdot \left(\sqrt[3]{\mathsf{expm1}\left(\sqrt{\mathsf{log1p}\left(\sqrt{\cos x}\right)} \cdot \sqrt{\mathsf{log1p}\left(\sqrt{\cos x}\right)}\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\sqrt{\mathsf{log1p}\left(\sqrt{\cos x}\right)} \cdot \sqrt{\mathsf{log1p}\left(\sqrt{\cos x}\right)}\right)}\right)\right)\right)}{e^{x}} \]

Reproduce

herbie shell --seed 2022024 
(FPCore (x)
  :name "expfmod"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))