Average Error: 60.3 → 3.4
Time: 1.1m
Precision: 64
\[-1.0 \lt \varepsilon \land \varepsilon \lt 1.0\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1.0\right)}{\left(e^{a \cdot \varepsilon} - 1.0\right) \cdot \left(e^{b \cdot \varepsilon} - 1.0\right)}\]
↓
\[\frac{1}{a} + \frac{1}{b}\]
\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1.0\right)}{\left(e^{a \cdot \varepsilon} - 1.0\right) \cdot \left(e^{b \cdot \varepsilon} - 1.0\right)}
↓
\frac{1}{a} + \frac{1}{b}
double f(double a, double b, double eps) {
        double r4759459 = eps;
        double r4759460 = a;
        double r4759461 = b;
        double r4759462 = r4759460 + r4759461;
        double r4759463 = r4759462 * r4759459;
        double r4759464 = exp(r4759463);
        double r4759465 = 1.0;
        double r4759466 = r4759464 - r4759465;
        double r4759467 = r4759459 * r4759466;
        double r4759468 = r4759460 * r4759459;
        double r4759469 = exp(r4759468);
        double r4759470 = r4759469 - r4759465;
        double r4759471 = r4759461 * r4759459;
        double r4759472 = exp(r4759471);
        double r4759473 = r4759472 - r4759465;
        double r4759474 = r4759470 * r4759473;
        double r4759475 = r4759467 / r4759474;
        return r4759475;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r4759476 = 1.0;
        double r4759477 = a;
        double r4759478 = r4759476 / r4759477;
        double r4759479 = b;
        double r4759480 = r4759476 / r4759479;
        double r4759481 = r4759478 + r4759480;
        return r4759481;
}

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

Original60.3
Target14.6
Herbie3.4
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.3

    \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1.0\right)}{\left(e^{a \cdot \varepsilon} - 1.0\right) \cdot \left(e^{b \cdot \varepsilon} - 1.0\right)}\]

  2. Taylor expanded around 0 57.7

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1.0\right)}{\left(e^{a \cdot \varepsilon} - 1.0\right) \cdot \color{blue}{\left(\varepsilon \cdot b + \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right)\right)\right)}}\]

  3. Simplified57.7

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1.0\right)}{\left(e^{a \cdot \varepsilon} - 1.0\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, b, \mathsf{fma}\left(\frac{1}{2} \cdot \left(\varepsilon \cdot \varepsilon\right), b \cdot b, \left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \frac{1}{6}\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)\right)\right)}}\]

  4. Taylor expanded around 0 3.4

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

  5. Final simplification3.4

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

Reproduce

herbie shell --seed 2019165 +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))))