Average Error: 60.5 → 52.9
Time: 14.7s
Precision: binary64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -6.98059513921423274 \cdot 10^{74} \lor \neg \left(a \le 50590990.4366769269\right):\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(b \cdot \left(\left(\left(\frac{1}{2} \cdot {\varepsilon}^{2}\right) \cdot b + \left(\left(\frac{1}{6} \cdot {\varepsilon}^{3}\right) \cdot b\right) \cdot b\right) + \varepsilon\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\varepsilon \cdot \left(\left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \varepsilon + \left(\frac{1}{6} \cdot {a}^{3}\right) \cdot {\varepsilon}^{2}\right) + a\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \end{array}\]
\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}
↓
\begin{array}{l}
\mathbf{if}\;a \le -6.98059513921423274 \cdot 10^{74} \lor \neg \left(a \le 50590990.4366769269\right):\\
\;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(b \cdot \left(\left(\left(\frac{1}{2} \cdot {\varepsilon}^{2}\right) \cdot b + \left(\left(\frac{1}{6} \cdot {\varepsilon}^{3}\right) \cdot b\right) \cdot b\right) + \varepsilon\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\varepsilon \cdot \left(\left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \varepsilon + \left(\frac{1}{6} \cdot {a}^{3}\right) \cdot {\varepsilon}^{2}\right) + a\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\

\end{array}
double code(double a, double b, double eps) {
	return ((double) (((double) (eps * ((double) (((double) exp(((double) (((double) (a + b)) * eps)))) - 1.0)))) / ((double) (((double) (((double) exp(((double) (a * eps)))) - 1.0)) * ((double) (((double) exp(((double) (b * eps)))) - 1.0))))));
}
↓
double code(double a, double b, double eps) {
	double VAR;
	if (((a <= -6.980595139214233e+74) || !(a <= 50590990.43667693))) {
		VAR = ((double) (((double) (eps * ((double) (((double) exp(((double) (((double) (a + b)) * eps)))) - 1.0)))) / ((double) (((double) (((double) exp(((double) (a * eps)))) - 1.0)) * ((double) (b * ((double) (((double) (((double) (((double) (0.5 * ((double) pow(eps, 2.0)))) * b)) + ((double) (((double) (((double) (0.16666666666666666 * ((double) pow(eps, 3.0)))) * b)) * b)))) + eps))))))));
	} else {
		VAR = ((double) (((double) (eps * ((double) (((double) exp(((double) (((double) (a + b)) * eps)))) - 1.0)))) / ((double) (((double) (eps * ((double) (((double) (((double) (((double) (0.5 * ((double) pow(a, 2.0)))) * eps)) + ((double) (((double) (0.16666666666666666 * ((double) pow(a, 3.0)))) * ((double) pow(eps, 2.0)))))) + a)))) * ((double) (((double) exp(((double) (b * eps)))) - 1.0))))));
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original60.5
Target15.1
Herbie52.9
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if a < -6.98059513921423274e74 or 50590990.4366769269 < a

    1. Initial program 55.2

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

    2. Taylor expanded around 0 50.1

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right) + \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \varepsilon \cdot b\right)\right)}}\]

    3. Simplified47.9

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\left(b \cdot \left(\left(\left(\frac{1}{2} \cdot {\varepsilon}^{2}\right) \cdot b + \left(\frac{1}{6} \cdot {\varepsilon}^{3}\right) \cdot {b}^{2}\right) + \varepsilon\right)\right)}}\]
    4. Using strategy rm

    5. Applied unpow247.9

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(b \cdot \left(\left(\left(\frac{1}{2} \cdot {\varepsilon}^{2}\right) \cdot b + \left(\frac{1}{6} \cdot {\varepsilon}^{3}\right) \cdot \color{blue}{\left(b \cdot b\right)}\right) + \varepsilon\right)\right)}\]

    6. Applied associate-*r*46.6

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(b \cdot \left(\left(\left(\frac{1}{2} \cdot {\varepsilon}^{2}\right) \cdot b + \color{blue}{\left(\left(\frac{1}{6} \cdot {\varepsilon}^{3}\right) \cdot b\right) \cdot b}\right) + \varepsilon\right)\right)}\]

    if -6.98059513921423274e74 < a < 50590990.4366769269

    1. Initial program 63.8

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

    2. Taylor expanded around 0 56.7

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(\frac{1}{6} \cdot \left({a}^{3} \cdot {\varepsilon}^{3}\right) + \left(\frac{1}{2} \cdot \left({a}^{2} \cdot {\varepsilon}^{2}\right) + a \cdot \varepsilon\right)\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

    3. Simplified56.7

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(\varepsilon \cdot \left(\left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \varepsilon + \left(\frac{1}{6} \cdot {a}^{3}\right) \cdot {\varepsilon}^{2}\right) + a\right)\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification52.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -6.98059513921423274 \cdot 10^{74} \lor \neg \left(a \le 50590990.4366769269\right):\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(b \cdot \left(\left(\left(\frac{1}{2} \cdot {\varepsilon}^{2}\right) \cdot b + \left(\left(\frac{1}{6} \cdot {\varepsilon}^{3}\right) \cdot b\right) \cdot b\right) + \varepsilon\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\varepsilon \cdot \left(\left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \varepsilon + \left(\frac{1}{6} \cdot {a}^{3}\right) \cdot {\varepsilon}^{2}\right) + a\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020173 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :precision binary64
  :pre (and (< -1.0 eps) (< eps 1.0))

  :herbie-target
  (/ (+ a b) (* a b))

  (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))))