Average Error: 60.3 → 3.5
Time: 31.9s
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 r7205903 = eps;
        double r7205904 = a;
        double r7205905 = b;
        double r7205906 = r7205904 + r7205905;
        double r7205907 = r7205906 * r7205903;
        double r7205908 = exp(r7205907);
        double r7205909 = 1.0;
        double r7205910 = r7205908 - r7205909;
        double r7205911 = r7205903 * r7205910;
        double r7205912 = r7205904 * r7205903;
        double r7205913 = exp(r7205912);
        double r7205914 = r7205913 - r7205909;
        double r7205915 = r7205905 * r7205903;
        double r7205916 = exp(r7205915);
        double r7205917 = r7205916 - r7205909;
        double r7205918 = r7205914 * r7205917;
        double r7205919 = r7205911 / r7205918;
        return r7205919;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r7205920 = 1.0;
        double r7205921 = a;
        double r7205922 = r7205920 / r7205921;
        double r7205923 = b;
        double r7205924 = r7205920 / r7205923;
        double r7205925 = r7205922 + r7205924;
        return r7205925;
}

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
Target15.0
Herbie3.5
\[\frac{a + b}{a \cdot b}\]

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

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

  3. Simplified57.8

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

  4. Taylor expanded around 0 3.5

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

  5. Final simplification3.5

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

Reproduce

herbie shell --seed 2019174 
(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))))