Average Error: 59.5 → 4.1
Time: 8.0s
Precision: binary64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.84863775765344881 \cdot 10^{176}:\\ \;\;\;\;\frac{b + a}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]
\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}\;b \le -7.84863775765344881 \cdot 10^{176}:\\
\;\;\;\;\frac{b + a}{b \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{b} + \frac{1}{a}\\

\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 ((b <= -7.848637757653449e+176)) {
		VAR = ((double) (((double) (b + a)) / ((double) (b * a))));
	} else {
		VAR = ((double) (((double) (1.0 / b)) + ((double) (1.0 / a))));
	}
	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

Original59.5
Target14.4
Herbie4.1
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if b < -7.84863775765344881e176

    1. Initial program 46.2

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

    2. Taylor expanded around 0 16.7

      \[\leadsto \color{blue}{\frac{1}{b} + \frac{1}{a}}\]
    3. Using strategy rm

    4. Applied frac-add15.5

      \[\leadsto \color{blue}{\frac{1 \cdot a + b \cdot 1}{b \cdot a}}\]

    5. Simplified15.5

      \[\leadsto \frac{\color{blue}{b + a}}{b \cdot a}\]

    if -7.84863775765344881e176 < b

    1. Initial program 59.9

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

    2. Taylor expanded around 0 3.7

      \[\leadsto \color{blue}{\frac{1}{b} + \frac{1}{a}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification4.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.84863775765344881 \cdot 10^{176}:\\ \;\;\;\;\frac{b + a}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020156 
(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))))