Average Error: 60.2 → 3.4
Time: 10.4s
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 r117069 = eps;
        double r117070 = a;
        double r117071 = b;
        double r117072 = r117070 + r117071;
        double r117073 = r117072 * r117069;
        double r117074 = exp(r117073);
        double r117075 = 1.0;
        double r117076 = r117074 - r117075;
        double r117077 = r117069 * r117076;
        double r117078 = r117070 * r117069;
        double r117079 = exp(r117078);
        double r117080 = r117079 - r117075;
        double r117081 = r117071 * r117069;
        double r117082 = exp(r117081);
        double r117083 = r117082 - r117075;
        double r117084 = r117080 * r117083;
        double r117085 = r117077 / r117084;
        return r117085;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r117086 = 1.0;
        double r117087 = b;
        double r117088 = r117086 / r117087;
        double r117089 = a;
        double r117090 = r117086 / r117089;
        double r117091 = r117088 + r117090;
        return r117091;
}

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

Original60.2
Target14.7
Herbie3.4
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.2

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

  2. Taylor expanded around 0 3.4

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

  3. Final simplification3.4

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

Reproduce

herbie shell --seed 2020060 +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))))