Average Error: 58.6 → 3.4
Time: 3.8m
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 r16723981 = eps;
        double r16723982 = a;
        double r16723983 = b;
        double r16723984 = r16723982 + r16723983;
        double r16723985 = r16723984 * r16723981;
        double r16723986 = exp(r16723985);
        double r16723987 = 1.0;
        double r16723988 = r16723986 - r16723987;
        double r16723989 = r16723981 * r16723988;
        double r16723990 = r16723982 * r16723981;
        double r16723991 = exp(r16723990);
        double r16723992 = r16723991 - r16723987;
        double r16723993 = r16723983 * r16723981;
        double r16723994 = exp(r16723993);
        double r16723995 = r16723994 - r16723987;
        double r16723996 = r16723992 * r16723995;
        double r16723997 = r16723989 / r16723996;
        return r16723997;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r16723998 = 1.0;
        double r16723999 = a;
        double r16724000 = r16723998 / r16723999;
        double r16724001 = b;
        double r16724002 = r16723998 / r16724001;
        double r16724003 = r16724000 + r16724002;
        return r16724003;
}

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

Original58.6
Target14.0
Herbie3.4
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 58.6

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

  2. Simplified34.0

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

  3. Taylor expanded around 0 3.4

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

  4. Final simplification3.4

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

Reproduce

herbie shell --seed 2019119 +o rules:numerics
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :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))))