Average Error: 60.3 → 3.5
Time: 34.8s
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 r4971913 = eps;
        double r4971914 = a;
        double r4971915 = b;
        double r4971916 = r4971914 + r4971915;
        double r4971917 = r4971916 * r4971913;
        double r4971918 = exp(r4971917);
        double r4971919 = 1.0;
        double r4971920 = r4971918 - r4971919;
        double r4971921 = r4971913 * r4971920;
        double r4971922 = r4971914 * r4971913;
        double r4971923 = exp(r4971922);
        double r4971924 = r4971923 - r4971919;
        double r4971925 = r4971915 * r4971913;
        double r4971926 = exp(r4971925);
        double r4971927 = r4971926 - r4971919;
        double r4971928 = r4971924 * r4971927;
        double r4971929 = r4971921 / r4971928;
        return r4971929;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r4971930 = 1.0;
        double r4971931 = a;
        double r4971932 = r4971930 / r4971931;
        double r4971933 = b;
        double r4971934 = r4971930 / r4971933;
        double r4971935 = r4971932 + r4971934;
        return r4971935;
}

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

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

  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 +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))))