Average Error: 60.4 → 3.2
Time: 17.1s
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 r99347 = eps;
        double r99348 = a;
        double r99349 = b;
        double r99350 = r99348 + r99349;
        double r99351 = r99350 * r99347;
        double r99352 = exp(r99351);
        double r99353 = 1.0;
        double r99354 = r99352 - r99353;
        double r99355 = r99347 * r99354;
        double r99356 = r99348 * r99347;
        double r99357 = exp(r99356);
        double r99358 = r99357 - r99353;
        double r99359 = r99349 * r99347;
        double r99360 = exp(r99359);
        double r99361 = r99360 - r99353;
        double r99362 = r99358 * r99361;
        double r99363 = r99355 / r99362;
        return r99363;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r99364 = 1.0;
        double r99365 = b;
        double r99366 = r99364 / r99365;
        double r99367 = a;
        double r99368 = r99364 / r99367;
        double r99369 = r99366 + r99368;
        return r99369;
}

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
Target14.8
Herbie3.2
\[\frac{a + b}{a \cdot b}\]

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 3.2

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

  3. Final simplification3.2

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

Reproduce

herbie shell --seed 2020042 
(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))))