Average Error: 60.4 → 3.2
Time: 17.3s
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 r105329 = eps;
        double r105330 = a;
        double r105331 = b;
        double r105332 = r105330 + r105331;
        double r105333 = r105332 * r105329;
        double r105334 = exp(r105333);
        double r105335 = 1.0;
        double r105336 = r105334 - r105335;
        double r105337 = r105329 * r105336;
        double r105338 = r105330 * r105329;
        double r105339 = exp(r105338);
        double r105340 = r105339 - r105335;
        double r105341 = r105331 * r105329;
        double r105342 = exp(r105341);
        double r105343 = r105342 - r105335;
        double r105344 = r105340 * r105343;
        double r105345 = r105337 / r105344;
        return r105345;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r105346 = 1.0;
        double r105347 = b;
        double r105348 = r105346 / r105347;
        double r105349 = a;
        double r105350 = r105346 / r105349;
        double r105351 = r105348 + r105350;
        return r105351;
}

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))))