Average Error: 60.5 → 3.2
Time: 28.5s
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 r81356 = eps;
        double r81357 = a;
        double r81358 = b;
        double r81359 = r81357 + r81358;
        double r81360 = r81359 * r81356;
        double r81361 = exp(r81360);
        double r81362 = 1.0;
        double r81363 = r81361 - r81362;
        double r81364 = r81356 * r81363;
        double r81365 = r81357 * r81356;
        double r81366 = exp(r81365);
        double r81367 = r81366 - r81362;
        double r81368 = r81358 * r81356;
        double r81369 = exp(r81368);
        double r81370 = r81369 - r81362;
        double r81371 = r81367 * r81370;
        double r81372 = r81364 / r81371;
        return r81372;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r81373 = 1.0;
        double r81374 = b;
        double r81375 = r81373 / r81374;
        double r81376 = a;
        double r81377 = r81373 / r81376;
        double r81378 = r81375 + r81377;
        return r81378;
}

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

Original60.5
Target15.4
Herbie3.2
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.5

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

  2. Taylor expanded around 0 58.4

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

  3. Simplified58.4

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

  4. Taylor expanded around 0 3.2

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

  5. Simplified3.2

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

  6. Final simplification3.2

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

Reproduce

herbie shell --seed 2019195 
(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))))