Average Error: 60.4 → 3.3
Time: 35.6s
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 r95433 = eps;
        double r95434 = a;
        double r95435 = b;
        double r95436 = r95434 + r95435;
        double r95437 = r95436 * r95433;
        double r95438 = exp(r95437);
        double r95439 = 1.0;
        double r95440 = r95438 - r95439;
        double r95441 = r95433 * r95440;
        double r95442 = r95434 * r95433;
        double r95443 = exp(r95442);
        double r95444 = r95443 - r95439;
        double r95445 = r95435 * r95433;
        double r95446 = exp(r95445);
        double r95447 = r95446 - r95439;
        double r95448 = r95444 * r95447;
        double r95449 = r95441 / r95448;
        return r95449;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r95450 = 1.0;
        double r95451 = b;
        double r95452 = r95450 / r95451;
        double r95453 = a;
        double r95454 = r95450 / r95453;
        double r95455 = r95452 + r95454;
        return r95455;
}

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

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

Derivation

  1. Initial program 60.4

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

  2. Taylor expanded around 0 58.0

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

  3. Simplified58.0

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

  5. Applied pow-prod-down57.5

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

  6. Taylor expanded around 0 3.3

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

  7. Final simplification3.3

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

Reproduce

herbie shell --seed 2019212 +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))))