Average Error: 60.1 → 3.6
Time: 12.4s
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 r102478 = eps;
        double r102479 = a;
        double r102480 = b;
        double r102481 = r102479 + r102480;
        double r102482 = r102481 * r102478;
        double r102483 = exp(r102482);
        double r102484 = 1.0;
        double r102485 = r102483 - r102484;
        double r102486 = r102478 * r102485;
        double r102487 = r102479 * r102478;
        double r102488 = exp(r102487);
        double r102489 = r102488 - r102484;
        double r102490 = r102480 * r102478;
        double r102491 = exp(r102490);
        double r102492 = r102491 - r102484;
        double r102493 = r102489 * r102492;
        double r102494 = r102486 / r102493;
        return r102494;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r102495 = 1.0;
        double r102496 = b;
        double r102497 = r102495 / r102496;
        double r102498 = a;
        double r102499 = r102495 / r102498;
        double r102500 = r102497 + r102499;
        return r102500;
}

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

Original60.1
Target14.8
Herbie3.6
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.1

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

  2. Taylor expanded around 0 58.0

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

  3. Simplified58.0

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

  4. Taylor expanded around 0 3.6

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

  5. Final simplification3.6

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

Reproduce

herbie shell --seed 2020033 +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))))