Average Error: 58.9 → 3.1
Time: 53.9s
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 r7311068 = eps;
        double r7311069 = a;
        double r7311070 = b;
        double r7311071 = r7311069 + r7311070;
        double r7311072 = r7311071 * r7311068;
        double r7311073 = exp(r7311072);
        double r7311074 = 1.0;
        double r7311075 = r7311073 - r7311074;
        double r7311076 = r7311068 * r7311075;
        double r7311077 = r7311069 * r7311068;
        double r7311078 = exp(r7311077);
        double r7311079 = r7311078 - r7311074;
        double r7311080 = r7311070 * r7311068;
        double r7311081 = exp(r7311080);
        double r7311082 = r7311081 - r7311074;
        double r7311083 = r7311079 * r7311082;
        double r7311084 = r7311076 / r7311083;
        return r7311084;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r7311085 = 1.0;
        double r7311086 = a;
        double r7311087 = r7311085 / r7311086;
        double r7311088 = b;
        double r7311089 = r7311085 / r7311088;
        double r7311090 = r7311087 + r7311089;
        return r7311090;
}

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
Target13.7
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. Simplified55.1

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

  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 2019107 
(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))))