Average Error: 58.5 → 3.5
Time: 40.6s
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 r2931159 = eps;
        double r2931160 = a;
        double r2931161 = b;
        double r2931162 = r2931160 + r2931161;
        double r2931163 = r2931162 * r2931159;
        double r2931164 = exp(r2931163);
        double r2931165 = 1.0;
        double r2931166 = r2931164 - r2931165;
        double r2931167 = r2931159 * r2931166;
        double r2931168 = r2931160 * r2931159;
        double r2931169 = exp(r2931168);
        double r2931170 = r2931169 - r2931165;
        double r2931171 = r2931161 * r2931159;
        double r2931172 = exp(r2931171);
        double r2931173 = r2931172 - r2931165;
        double r2931174 = r2931170 * r2931173;
        double r2931175 = r2931167 / r2931174;
        return r2931175;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r2931176 = 1.0;
        double r2931177 = a;
        double r2931178 = r2931176 / r2931177;
        double r2931179 = b;
        double r2931180 = r2931176 / r2931179;
        double r2931181 = r2931178 + r2931180;
        return r2931181;
}

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

Original58.5
Target13.9
Herbie3.5
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 58.5

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

  2. Simplified37.7

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

  3. Taylor expanded around 0 3.5

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

  4. Final simplification3.5

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

Reproduce

herbie shell --seed 2019132 +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))))