Average Error: 60.5 → 3.2
Time: 14.9s
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 r214255 = eps;
        double r214256 = a;
        double r214257 = b;
        double r214258 = r214256 + r214257;
        double r214259 = r214258 * r214255;
        double r214260 = exp(r214259);
        double r214261 = 1.0;
        double r214262 = r214260 - r214261;
        double r214263 = r214255 * r214262;
        double r214264 = r214256 * r214255;
        double r214265 = exp(r214264);
        double r214266 = r214265 - r214261;
        double r214267 = r214257 * r214255;
        double r214268 = exp(r214267);
        double r214269 = r214268 - r214261;
        double r214270 = r214266 * r214269;
        double r214271 = r214263 / r214270;
        return r214271;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r214272 = 1.0;
        double r214273 = b;
        double r214274 = r214272 / r214273;
        double r214275 = a;
        double r214276 = r214272 / r214275;
        double r214277 = r214274 + r214276;
        return r214277;
}

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

Original60.5
Target15.1
Herbie3.2
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.5

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

  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 2019322 
(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))))