Average Error: 59.1 → 2.9
Time: 45.0s
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 r5119490 = eps;
        double r5119491 = a;
        double r5119492 = b;
        double r5119493 = r5119491 + r5119492;
        double r5119494 = r5119493 * r5119490;
        double r5119495 = exp(r5119494);
        double r5119496 = 1.0;
        double r5119497 = r5119495 - r5119496;
        double r5119498 = r5119490 * r5119497;
        double r5119499 = r5119491 * r5119490;
        double r5119500 = exp(r5119499);
        double r5119501 = r5119500 - r5119496;
        double r5119502 = r5119492 * r5119490;
        double r5119503 = exp(r5119502);
        double r5119504 = r5119503 - r5119496;
        double r5119505 = r5119501 * r5119504;
        double r5119506 = r5119498 / r5119505;
        return r5119506;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r5119507 = 1.0;
        double r5119508 = a;
        double r5119509 = r5119507 / r5119508;
        double r5119510 = b;
        double r5119511 = r5119507 / r5119510;
        double r5119512 = r5119509 + r5119511;
        return r5119512;
}

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

Original59.1
Target13.8
Herbie2.9
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 59.1

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

  2. Taylor expanded around 0 56.1

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

  3. Simplified55.1

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

  4. Taylor expanded around 0 2.9

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

  5. Final simplification2.9

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

Reproduce

herbie shell --seed 2019163 
(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))))