Average Error: 60.4 → 3.2
Time: 32.8s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\frac{1}{b} + \frac{1}{a}\]
\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 r85664 = eps;
        double r85665 = a;
        double r85666 = b;
        double r85667 = r85665 + r85666;
        double r85668 = r85667 * r85664;
        double r85669 = exp(r85668);
        double r85670 = 1.0;
        double r85671 = r85669 - r85670;
        double r85672 = r85664 * r85671;
        double r85673 = r85665 * r85664;
        double r85674 = exp(r85673);
        double r85675 = r85674 - r85670;
        double r85676 = r85666 * r85664;
        double r85677 = exp(r85676);
        double r85678 = r85677 - r85670;
        double r85679 = r85675 * r85678;
        double r85680 = r85672 / r85679;
        return r85680;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r85681 = 1.0;
        double r85682 = b;
        double r85683 = r85681 / r85682;
        double r85684 = a;
        double r85685 = r85681 / r85684;
        double r85686 = r85683 + r85685;
        return r85686;
}

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

Original60.4
Target14.8
Herbie3.2
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.4

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

  2. Taylor expanded around 0 58.1

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

  3. Simplified58.0

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

  5. Applied pow-prod-down57.3

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

  6. Taylor expanded around 0 3.2

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

  7. Final simplification3.2

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

Reproduce

herbie shell --seed 2019325 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :precision binary64
  :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))))