Average Error: 60.2 → 3.5
Time: 9.7s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\frac{1}{b} + \frac{1}{a}\]
\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 r117366 = eps;
        double r117367 = a;
        double r117368 = b;
        double r117369 = r117367 + r117368;
        double r117370 = r117369 * r117366;
        double r117371 = exp(r117370);
        double r117372 = 1.0;
        double r117373 = r117371 - r117372;
        double r117374 = r117366 * r117373;
        double r117375 = r117367 * r117366;
        double r117376 = exp(r117375);
        double r117377 = r117376 - r117372;
        double r117378 = r117368 * r117366;
        double r117379 = exp(r117378);
        double r117380 = r117379 - r117372;
        double r117381 = r117377 * r117380;
        double r117382 = r117374 / r117381;
        return r117382;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r117383 = 1.0;
        double r117384 = b;
        double r117385 = r117383 / r117384;
        double r117386 = a;
        double r117387 = r117383 / r117386;
        double r117388 = r117385 + r117387;
        return r117388;
}

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
Target15.1
Herbie3.5
\[\frac{a + b}{a \cdot b}\]

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 3.5

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

  3. Final simplification3.5

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

Reproduce

herbie shell --seed 2020065 
(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))))