Average Error: 60.7 → 3.1
Time: 29.0s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\frac{1}{b} + \frac{1}{a}\]
\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 r64196 = eps;
        double r64197 = a;
        double r64198 = b;
        double r64199 = r64197 + r64198;
        double r64200 = r64199 * r64196;
        double r64201 = exp(r64200);
        double r64202 = 1.0;
        double r64203 = r64201 - r64202;
        double r64204 = r64196 * r64203;
        double r64205 = r64197 * r64196;
        double r64206 = exp(r64205);
        double r64207 = r64206 - r64202;
        double r64208 = r64198 * r64196;
        double r64209 = exp(r64208);
        double r64210 = r64209 - r64202;
        double r64211 = r64207 * r64210;
        double r64212 = r64204 / r64211;
        return r64212;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r64213 = 1.0;
        double r64214 = b;
        double r64215 = r64213 / r64214;
        double r64216 = a;
        double r64217 = r64213 / r64216;
        double r64218 = r64215 + r64217;
        return r64218;
}

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

Original60.7
Target14.6
Herbie3.1
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.7

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

  2. Taylor expanded around 0 57.8

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

  3. Taylor expanded around 0 3.1

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

  4. Final simplification3.1

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

Reproduce

herbie shell --seed 2019198 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :pre (and (< -1.0 eps) (< eps 1.0))

  :herbie-target
  (/ (+ a b) (* a b))

  (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))))