Average Error: 58.7 → 3.3
Time: 33.6s
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 r4067961 = eps;
        double r4067962 = a;
        double r4067963 = b;
        double r4067964 = r4067962 + r4067963;
        double r4067965 = r4067964 * r4067961;
        double r4067966 = exp(r4067965);
        double r4067967 = 1.0;
        double r4067968 = r4067966 - r4067967;
        double r4067969 = r4067961 * r4067968;
        double r4067970 = r4067962 * r4067961;
        double r4067971 = exp(r4067970);
        double r4067972 = r4067971 - r4067967;
        double r4067973 = r4067963 * r4067961;
        double r4067974 = exp(r4067973);
        double r4067975 = r4067974 - r4067967;
        double r4067976 = r4067972 * r4067975;
        double r4067977 = r4067969 / r4067976;
        return r4067977;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r4067978 = 1.0;
        double r4067979 = a;
        double r4067980 = r4067978 / r4067979;
        double r4067981 = b;
        double r4067982 = r4067978 / r4067981;
        double r4067983 = r4067980 + r4067982;
        return r4067983;
}

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

Original58.7
Target14.5
Herbie3.3
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 58.7

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

  2. Simplified28.1

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

  3. Taylor expanded around 0 3.3

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

  4. Final simplification3.3

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

Reproduce

herbie shell --seed 2019162 +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))))