Average Error: 60.6 → 3.2
Time: 33.0s
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 r4488452 = eps;
        double r4488453 = a;
        double r4488454 = b;
        double r4488455 = r4488453 + r4488454;
        double r4488456 = r4488455 * r4488452;
        double r4488457 = exp(r4488456);
        double r4488458 = 1.0;
        double r4488459 = r4488457 - r4488458;
        double r4488460 = r4488452 * r4488459;
        double r4488461 = r4488453 * r4488452;
        double r4488462 = exp(r4488461);
        double r4488463 = r4488462 - r4488458;
        double r4488464 = r4488454 * r4488452;
        double r4488465 = exp(r4488464);
        double r4488466 = r4488465 - r4488458;
        double r4488467 = r4488463 * r4488466;
        double r4488468 = r4488460 / r4488467;
        return r4488468;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r4488469 = 1.0;
        double r4488470 = a;
        double r4488471 = r4488469 / r4488470;
        double r4488472 = b;
        double r4488473 = r4488469 / r4488472;
        double r4488474 = r4488471 + r4488473;
        return r4488474;
}

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

Original60.6
Target14.7
Herbie3.2
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.6

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

  2. Taylor expanded around 0 58.0

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

  3. Simplified58.0

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

  5. Applied log1p-expm1-u57.9

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right), \left(b \cdot \left(b \cdot b\right)\right) \cdot \frac{1}{6}, \mathsf{fma}\left(\frac{1}{2} \cdot \left(b \cdot b\right), \varepsilon \cdot \varepsilon, b \cdot \varepsilon\right)\right)\right)\right)}}\]

  6. Simplified56.6

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{fma}\left(\left(\varepsilon \cdot b\right) \cdot \left(\varepsilon \cdot b\right), \frac{1}{2}, \varepsilon \cdot \left(b + \left(\left(\left(\varepsilon \cdot b\right) \cdot \left(\varepsilon \cdot b\right)\right) \cdot b\right) \cdot \frac{1}{6}\right)\right)\right)}\right)}\]

  7. Taylor expanded around 0 3.2

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

  8. Final simplification3.2

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

Reproduce

herbie shell --seed 2019192 +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))))