Average Error: 60.2 → 3.4
Time: 1.5m
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 r5034770 = eps;
        double r5034771 = a;
        double r5034772 = b;
        double r5034773 = r5034771 + r5034772;
        double r5034774 = r5034773 * r5034770;
        double r5034775 = exp(r5034774);
        double r5034776 = 1.0;
        double r5034777 = r5034775 - r5034776;
        double r5034778 = r5034770 * r5034777;
        double r5034779 = r5034771 * r5034770;
        double r5034780 = exp(r5034779);
        double r5034781 = r5034780 - r5034776;
        double r5034782 = r5034772 * r5034770;
        double r5034783 = exp(r5034782);
        double r5034784 = r5034783 - r5034776;
        double r5034785 = r5034781 * r5034784;
        double r5034786 = r5034778 / r5034785;
        return r5034786;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r5034787 = 1.0;
        double r5034788 = a;
        double r5034789 = r5034787 / r5034788;
        double r5034790 = b;
        double r5034791 = r5034787 / r5034790;
        double r5034792 = r5034789 + r5034791;
        return r5034792;
}

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))))