Average Error: 60.3 → 3.4
Time: 11.1s
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 r93563 = eps;
        double r93564 = a;
        double r93565 = b;
        double r93566 = r93564 + r93565;
        double r93567 = r93566 * r93563;
        double r93568 = exp(r93567);
        double r93569 = 1.0;
        double r93570 = r93568 - r93569;
        double r93571 = r93563 * r93570;
        double r93572 = r93564 * r93563;
        double r93573 = exp(r93572);
        double r93574 = r93573 - r93569;
        double r93575 = r93565 * r93563;
        double r93576 = exp(r93575);
        double r93577 = r93576 - r93569;
        double r93578 = r93574 * r93577;
        double r93579 = r93571 / r93578;
        return r93579;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r93580 = 1.0;
        double r93581 = b;
        double r93582 = r93580 / r93581;
        double r93583 = a;
        double r93584 = r93580 / r93583;
        double r93585 = r93582 + r93584;
        return r93585;
}

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

Original60.3
Target15.0
Herbie3.4
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.3

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

  2. Taylor expanded around 0 57.9

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right) + \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \varepsilon \cdot b\right)\right)}}\]

  3. Simplified57.9

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {\varepsilon}^{3} \cdot {b}^{3}, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, \varepsilon \cdot b\right)\right)}}\]

  4. Taylor expanded around 0 3.4

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

  5. Final simplification3.4

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

Reproduce

herbie shell --seed 2020027 +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))))