Average Error: 60.1 → 3.6
Time: 37.8s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\frac{1}{a} + \frac{1}{b}\]
\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}{a} + \frac{1}{b}
double f(double a, double b, double eps) {
        double r5959699 = eps;
        double r5959700 = a;
        double r5959701 = b;
        double r5959702 = r5959700 + r5959701;
        double r5959703 = r5959702 * r5959699;
        double r5959704 = exp(r5959703);
        double r5959705 = 1.0;
        double r5959706 = r5959704 - r5959705;
        double r5959707 = r5959699 * r5959706;
        double r5959708 = r5959700 * r5959699;
        double r5959709 = exp(r5959708);
        double r5959710 = r5959709 - r5959705;
        double r5959711 = r5959701 * r5959699;
        double r5959712 = exp(r5959711);
        double r5959713 = r5959712 - r5959705;
        double r5959714 = r5959710 * r5959713;
        double r5959715 = r5959707 / r5959714;
        return r5959715;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r5959716 = 1.0;
        double r5959717 = a;
        double r5959718 = r5959716 / r5959717;
        double r5959719 = b;
        double r5959720 = r5959716 / r5959719;
        double r5959721 = r5959718 + r5959720;
        return r5959721;
}

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
Target15.2
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

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

  4. Taylor expanded around 0 3.6

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

  5. Final simplification3.6

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(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))))