Average Error: 60.5 → 3.2
Time: 31.8s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\frac{1}{a} + \frac{1}{b}\]
\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}{a} + \frac{1}{b}
double f(double a, double b, double eps) {
        double r126487 = eps;
        double r126488 = a;
        double r126489 = b;
        double r126490 = r126488 + r126489;
        double r126491 = r126490 * r126487;
        double r126492 = exp(r126491);
        double r126493 = 1.0;
        double r126494 = r126492 - r126493;
        double r126495 = r126487 * r126494;
        double r126496 = r126488 * r126487;
        double r126497 = exp(r126496);
        double r126498 = r126497 - r126493;
        double r126499 = r126489 * r126487;
        double r126500 = exp(r126499);
        double r126501 = r126500 - r126493;
        double r126502 = r126498 * r126501;
        double r126503 = r126495 / r126502;
        return r126503;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r126504 = 1.0;
        double r126505 = a;
        double r126506 = r126504 / r126505;
        double r126507 = b;
        double r126508 = r126504 / r126507;
        double r126509 = r126506 + r126508;
        return r126509;
}

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
Target15.4
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 58.4

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

  3. Simplified58.4

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

  4. Taylor expanded around 0 3.2

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

  5. Final simplification3.2

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

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :pre (and (< -1.0 eps) (< eps 1.0))

  :herbie-target
  (/ (+ a b) (* a b))

  (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))))