Average Error: 60.5 → 3.2
Time: 34.7s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\frac{1}{b} + \frac{1}{a}\]
\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}{b} + \frac{1}{a}
double f(double a, double b, double eps) {
        double r74818 = eps;
        double r74819 = a;
        double r74820 = b;
        double r74821 = r74819 + r74820;
        double r74822 = r74821 * r74818;
        double r74823 = exp(r74822);
        double r74824 = 1.0;
        double r74825 = r74823 - r74824;
        double r74826 = r74818 * r74825;
        double r74827 = r74819 * r74818;
        double r74828 = exp(r74827);
        double r74829 = r74828 - r74824;
        double r74830 = r74820 * r74818;
        double r74831 = exp(r74830);
        double r74832 = r74831 - r74824;
        double r74833 = r74829 * r74832;
        double r74834 = r74826 / r74833;
        return r74834;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r74835 = 1.0;
        double r74836 = b;
        double r74837 = r74835 / r74836;
        double r74838 = a;
        double r74839 = r74835 / r74838;
        double r74840 = r74837 + r74839;
        return r74840;
}

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

Original60.5
Target15.1
Herbie3.2
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.5

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

  2. Taylor expanded around 0 3.2

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

  3. Final simplification3.2

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

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :precision binary64
  :pre (and (< -1 eps) (< eps 1))

  :herbie-target
  (/ (+ a b) (* a b))

  (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))))