Average Error: 58.5 → 3.4
Time: 43.8s
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 r5476503 = eps;
        double r5476504 = a;
        double r5476505 = b;
        double r5476506 = r5476504 + r5476505;
        double r5476507 = r5476506 * r5476503;
        double r5476508 = exp(r5476507);
        double r5476509 = 1.0;
        double r5476510 = r5476508 - r5476509;
        double r5476511 = r5476503 * r5476510;
        double r5476512 = r5476504 * r5476503;
        double r5476513 = exp(r5476512);
        double r5476514 = r5476513 - r5476509;
        double r5476515 = r5476505 * r5476503;
        double r5476516 = exp(r5476515);
        double r5476517 = r5476516 - r5476509;
        double r5476518 = r5476514 * r5476517;
        double r5476519 = r5476511 / r5476518;
        return r5476519;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r5476520 = 1.0;
        double r5476521 = a;
        double r5476522 = r5476520 / r5476521;
        double r5476523 = b;
        double r5476524 = r5476520 / r5476523;
        double r5476525 = r5476522 + r5476524;
        return r5476525;
}

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

Original58.5
Target14.2
Herbie3.4
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 58.5

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

  2. Taylor expanded around 0 56.3

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

  3. Simplified55.2

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

  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 2019168 
(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))))