Average Error: 58.9 → 3.1
Time: 34.1s
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 r2190438 = eps;
        double r2190439 = a;
        double r2190440 = b;
        double r2190441 = r2190439 + r2190440;
        double r2190442 = r2190441 * r2190438;
        double r2190443 = exp(r2190442);
        double r2190444 = 1.0;
        double r2190445 = r2190443 - r2190444;
        double r2190446 = r2190438 * r2190445;
        double r2190447 = r2190439 * r2190438;
        double r2190448 = exp(r2190447);
        double r2190449 = r2190448 - r2190444;
        double r2190450 = r2190440 * r2190438;
        double r2190451 = exp(r2190450);
        double r2190452 = r2190451 - r2190444;
        double r2190453 = r2190449 * r2190452;
        double r2190454 = r2190446 / r2190453;
        return r2190454;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r2190455 = 1.0;
        double r2190456 = a;
        double r2190457 = r2190455 / r2190456;
        double r2190458 = b;
        double r2190459 = r2190455 / r2190458;
        double r2190460 = r2190457 + r2190459;
        return r2190460;
}

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

Original58.9
Target14.2
Herbie3.1
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 58.9

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

  2. Taylor expanded around 0 56.1

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

  3. Simplified56.1

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

  4. Taylor expanded around 0 3.1

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

  5. Final simplification3.1

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

Reproduce

herbie shell --seed 2019153 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :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))))