Average Error: 60.4 → 3.2
Time: 19.7s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\frac{1}{b} + \frac{1}{a}\]
\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 r106179 = eps;
        double r106180 = a;
        double r106181 = b;
        double r106182 = r106180 + r106181;
        double r106183 = r106182 * r106179;
        double r106184 = exp(r106183);
        double r106185 = 1.0;
        double r106186 = r106184 - r106185;
        double r106187 = r106179 * r106186;
        double r106188 = r106180 * r106179;
        double r106189 = exp(r106188);
        double r106190 = r106189 - r106185;
        double r106191 = r106181 * r106179;
        double r106192 = exp(r106191);
        double r106193 = r106192 - r106185;
        double r106194 = r106190 * r106193;
        double r106195 = r106187 / r106194;
        return r106195;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r106196 = 1.0;
        double r106197 = b;
        double r106198 = r106196 / r106197;
        double r106199 = a;
        double r106200 = r106196 / r106199;
        double r106201 = r106198 + r106200;
        return r106201;
}

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 3.2

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

  3. Final simplification3.2

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

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(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))))