Average Error: 60.3 → 3.4
Time: 11.6s
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 r117668 = eps;
        double r117669 = a;
        double r117670 = b;
        double r117671 = r117669 + r117670;
        double r117672 = r117671 * r117668;
        double r117673 = exp(r117672);
        double r117674 = 1.0;
        double r117675 = r117673 - r117674;
        double r117676 = r117668 * r117675;
        double r117677 = r117669 * r117668;
        double r117678 = exp(r117677);
        double r117679 = r117678 - r117674;
        double r117680 = r117670 * r117668;
        double r117681 = exp(r117680);
        double r117682 = r117681 - r117674;
        double r117683 = r117679 * r117682;
        double r117684 = r117676 / r117683;
        return r117684;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r117685 = 1.0;
        double r117686 = b;
        double r117687 = r117685 / r117686;
        double r117688 = a;
        double r117689 = r117685 / r117688;
        double r117690 = r117687 + r117689;
        return r117690;
}

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

Original60.3
Target14.8
Herbie3.4
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.3

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

  2. Taylor expanded around 0 57.8

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

  3. Simplified57.8

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

  4. Taylor expanded around 0 3.4

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

  5. Final simplification3.4

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

Reproduce

herbie shell --seed 2020062 +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))))