Average Error: 60.2 → 3.4
Time: 1.3m
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 r6590793 = eps;
        double r6590794 = a;
        double r6590795 = b;
        double r6590796 = r6590794 + r6590795;
        double r6590797 = r6590796 * r6590793;
        double r6590798 = exp(r6590797);
        double r6590799 = 1.0;
        double r6590800 = r6590798 - r6590799;
        double r6590801 = r6590793 * r6590800;
        double r6590802 = r6590794 * r6590793;
        double r6590803 = exp(r6590802);
        double r6590804 = r6590803 - r6590799;
        double r6590805 = r6590795 * r6590793;
        double r6590806 = exp(r6590805);
        double r6590807 = r6590806 - r6590799;
        double r6590808 = r6590804 * r6590807;
        double r6590809 = r6590801 / r6590808;
        return r6590809;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r6590810 = 1.0;
        double r6590811 = a;
        double r6590812 = r6590810 / r6590811;
        double r6590813 = b;
        double r6590814 = r6590810 / r6590813;
        double r6590815 = r6590812 + r6590814;
        return r6590815;
}

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

Original60.2
Target15.6
Herbie3.4
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.2

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

  2. Taylor expanded around 0 57.6

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

  3. Simplified57.6

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

  4. Taylor expanded around 0 56.6

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

  5. Simplified56.2

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

  6. Taylor expanded around 0 3.4

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

  7. Final simplification3.4

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

Reproduce

herbie shell --seed 2019200 +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))))