Average Error: 60.6 → 3.1
Time: 17.4s
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 r102611 = eps;
        double r102612 = a;
        double r102613 = b;
        double r102614 = r102612 + r102613;
        double r102615 = r102614 * r102611;
        double r102616 = exp(r102615);
        double r102617 = 1.0;
        double r102618 = r102616 - r102617;
        double r102619 = r102611 * r102618;
        double r102620 = r102612 * r102611;
        double r102621 = exp(r102620);
        double r102622 = r102621 - r102617;
        double r102623 = r102613 * r102611;
        double r102624 = exp(r102623);
        double r102625 = r102624 - r102617;
        double r102626 = r102622 * r102625;
        double r102627 = r102619 / r102626;
        return r102627;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r102628 = 1.0;
        double r102629 = b;
        double r102630 = r102628 / r102629;
        double r102631 = a;
        double r102632 = r102628 / r102631;
        double r102633 = r102630 + r102632;
        return r102633;
}

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

Original60.6
Target14.8
Herbie3.1
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.6

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

  2. Taylor expanded around 0 58.4

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

  3. Simplified58.4

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

  5. Applied pow-prod-down57.9

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

  6. Taylor expanded around 0 3.1

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

  7. Final simplification3.1

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

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :precision binary64
  :pre (and (< -1 eps) (< eps 1))

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

  (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))))