Average Error: 60.1 → 3.6
Time: 27.3s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\frac{1}{b} + \frac{1}{a}\]
\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)}
↓
\frac{1}{b} + \frac{1}{a}
double f(double a, double b, double eps) {
        double r102624 = eps;
        double r102625 = a;
        double r102626 = b;
        double r102627 = r102625 + r102626;
        double r102628 = r102627 * r102624;
        double r102629 = exp(r102628);
        double r102630 = 1.0;
        double r102631 = r102629 - r102630;
        double r102632 = r102624 * r102631;
        double r102633 = r102625 * r102624;
        double r102634 = exp(r102633);
        double r102635 = r102634 - r102630;
        double r102636 = r102626 * r102624;
        double r102637 = exp(r102636);
        double r102638 = r102637 - r102630;
        double r102639 = r102635 * r102638;
        double r102640 = r102632 / r102639;
        return r102640;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r102641 = 1.0;
        double r102642 = b;
        double r102643 = r102641 / r102642;
        double r102644 = a;
        double r102645 = r102641 / r102644;
        double r102646 = r102643 + r102645;
        return r102646;
}

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

Original60.1
Target15.3
Herbie3.6
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.1

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

  2. Taylor expanded around 0 57.9

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

  3. Simplified57.9

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

  4. Taylor expanded around 0 3.6

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

  5. Simplified3.6

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

  6. Final simplification3.6

    \[\leadsto \frac{1}{b} + \frac{1}{a}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :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))))