Average Error: 59.6 → 23.9
Time: 13.3s
Precision: binary64
\[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
↓
\[\begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := e^{-x}\\ t_2 := t_0 \cdot t_1\\ \mathbf{if}\;t_2 \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 1:\\ \;\;\;\;\frac{t_0}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}
↓
\begin{array}{l}
t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\
t_1 := e^{-x}\\
t_2 := t_0 \cdot t_1\\
\mathbf{if}\;t_2 \leq 0:\\
\;\;\;\;t_1\\

\mathbf{elif}\;t_2 \leq 1:\\
\;\;\;\;\frac{t_0}{e^{x}}\\

\mathbf{else}:\\
\;\;\;\;t_1\\


\end{array}

(FPCore (x) :precision binary64 (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (fmod (exp x) (sqrt (cos x))))
        (t_1 (exp (- x)))
        (t_2 (* t_0 t_1)))
   (if (<= t_2 0.0) t_1 (if (<= t_2 1.0) (/ t_0 (exp x)) t_1))))
double code(double x) {
	return fmod(exp(x), sqrt(cos(x))) * exp(-x);
}
↓
double code(double x) {
	double t_0 = fmod(exp(x), sqrt(cos(x)));
	double t_1 = exp(-x);
	double t_2 = t_0 * t_1;
	double tmp;
	if (t_2 <= 0.0) {
		tmp = t_1;
	} else if (t_2 <= 1.0) {
		tmp = t_0 / exp(x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 0.0 or 1 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

    1. Initial program 61.8

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

    2. Simplified61.8

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

    3. Applied add-exp-log_binary6461.8

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

    4. Applied div-exp_binary6461.8

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

    5. Taylor expanded in x around inf 24.5

      \[\leadsto e^{\color{blue}{-1 \cdot x}} \]

    if 0.0 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 1

    1. Initial program 12.8

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

    2. Simplified12.7

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

    3. Applied pow1_binary6412.7

      \[\leadsto \color{blue}{{\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)}^{1}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification23.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 0:\\ \;\;\;\;e^{-x}\\ \mathbf{elif}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 1:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \]

Reproduce

herbie shell --seed 2022104 
(FPCore (x)
  :name "expfmod"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))