Average Error: 58.6 → 3.4
Time: 55.7s
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 r6614240 = eps;
        double r6614241 = a;
        double r6614242 = b;
        double r6614243 = r6614241 + r6614242;
        double r6614244 = r6614243 * r6614240;
        double r6614245 = exp(r6614244);
        double r6614246 = 1.0;
        double r6614247 = r6614245 - r6614246;
        double r6614248 = r6614240 * r6614247;
        double r6614249 = r6614241 * r6614240;
        double r6614250 = exp(r6614249);
        double r6614251 = r6614250 - r6614246;
        double r6614252 = r6614242 * r6614240;
        double r6614253 = exp(r6614252);
        double r6614254 = r6614253 - r6614246;
        double r6614255 = r6614251 * r6614254;
        double r6614256 = r6614248 / r6614255;
        return r6614256;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r6614257 = 1.0;
        double r6614258 = a;
        double r6614259 = r6614257 / r6614258;
        double r6614260 = b;
        double r6614261 = r6614257 / r6614260;
        double r6614262 = r6614259 + r6614261;
        return r6614262;
}

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

Original58.6
Target14.0
Herbie3.4
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 58.6

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

  2. Taylor expanded around 0 56.4

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

  3. Simplified55.3

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

  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 2019119 
(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))))