Average Error: 58.8 → 3.2
Time: 31.7s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\frac{1}{a} + \frac{1}{b}\]
\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 r779974 = eps;
        double r779975 = a;
        double r779976 = b;
        double r779977 = r779975 + r779976;
        double r779978 = r779977 * r779974;
        double r779979 = exp(r779978);
        double r779980 = 1.0;
        double r779981 = r779979 - r779980;
        double r779982 = r779974 * r779981;
        double r779983 = r779975 * r779974;
        double r779984 = exp(r779983);
        double r779985 = r779984 - r779980;
        double r779986 = r779976 * r779974;
        double r779987 = exp(r779986);
        double r779988 = r779987 - r779980;
        double r779989 = r779985 * r779988;
        double r779990 = r779982 / r779989;
        return r779990;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r779991 = 1.0;
        double r779992 = a;
        double r779993 = r779991 / r779992;
        double r779994 = b;
        double r779995 = r779991 / r779994;
        double r779996 = r779993 + r779995;
        return r779996;
}

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

Original58.8
Target14.1
Herbie3.2
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 58.8

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

  2. Simplified39.5

    \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right) \cdot \varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot b\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot a\right)}}\]

  3. Taylor expanded around 0 3.2

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

  4. Final simplification3.2

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

Reproduce

herbie shell --seed 2019154 +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))))