Average Error: 60.4 → 3.3
Time: 19.6s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\frac{1}{b} + \frac{1}{a}\]
\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 r117152 = eps;
        double r117153 = a;
        double r117154 = b;
        double r117155 = r117153 + r117154;
        double r117156 = r117155 * r117152;
        double r117157 = exp(r117156);
        double r117158 = 1.0;
        double r117159 = r117157 - r117158;
        double r117160 = r117152 * r117159;
        double r117161 = r117153 * r117152;
        double r117162 = exp(r117161);
        double r117163 = r117162 - r117158;
        double r117164 = r117154 * r117152;
        double r117165 = exp(r117164);
        double r117166 = r117165 - r117158;
        double r117167 = r117163 * r117166;
        double r117168 = r117160 / r117167;
        return r117168;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r117169 = 1.0;
        double r117170 = b;
        double r117171 = r117169 / r117170;
        double r117172 = a;
        double r117173 = r117169 / r117172;
        double r117174 = r117171 + r117173;
        return r117174;
}

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

Original60.4
Target14.6
Herbie3.3
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.4

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

  2. Taylor expanded around 0 3.3

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

  3. Final simplification3.3

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

Reproduce

herbie shell --seed 2020043 +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))))