Average Error: 58.9 → 3.1
Time: 31.5s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\frac{1}{a} + \frac{1}{b}\]
\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 r1802805 = eps;
        double r1802806 = a;
        double r1802807 = b;
        double r1802808 = r1802806 + r1802807;
        double r1802809 = r1802808 * r1802805;
        double r1802810 = exp(r1802809);
        double r1802811 = 1.0;
        double r1802812 = r1802810 - r1802811;
        double r1802813 = r1802805 * r1802812;
        double r1802814 = r1802806 * r1802805;
        double r1802815 = exp(r1802814);
        double r1802816 = r1802815 - r1802811;
        double r1802817 = r1802807 * r1802805;
        double r1802818 = exp(r1802817);
        double r1802819 = r1802818 - r1802811;
        double r1802820 = r1802816 * r1802819;
        double r1802821 = r1802813 / r1802820;
        return r1802821;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r1802822 = 1.0;
        double r1802823 = a;
        double r1802824 = r1802822 / r1802823;
        double r1802825 = b;
        double r1802826 = r1802822 / r1802825;
        double r1802827 = r1802824 + r1802826;
        return r1802827;
}

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))))