Average Error: 60.2 → 3.5
Time: 10.7s
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 r77423 = eps;
        double r77424 = a;
        double r77425 = b;
        double r77426 = r77424 + r77425;
        double r77427 = r77426 * r77423;
        double r77428 = exp(r77427);
        double r77429 = 1.0;
        double r77430 = r77428 - r77429;
        double r77431 = r77423 * r77430;
        double r77432 = r77424 * r77423;
        double r77433 = exp(r77432);
        double r77434 = r77433 - r77429;
        double r77435 = r77425 * r77423;
        double r77436 = exp(r77435);
        double r77437 = r77436 - r77429;
        double r77438 = r77434 * r77437;
        double r77439 = r77431 / r77438;
        return r77439;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r77440 = 1.0;
        double r77441 = b;
        double r77442 = r77440 / r77441;
        double r77443 = a;
        double r77444 = r77440 / r77443;
        double r77445 = r77442 + r77444;
        return r77445;
}

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

Original60.2
Target15.0
Herbie3.5
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.2

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

  2. Taylor expanded around 0 58.2

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

  3. Simplified58.2

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

  4. Taylor expanded around 0 3.5

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

  5. Final simplification3.5

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

Reproduce

herbie shell --seed 2019353 +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))))