Average Error: 60.4 → 3.3
Time: 26.1s
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 r85116 = eps;
        double r85117 = a;
        double r85118 = b;
        double r85119 = r85117 + r85118;
        double r85120 = r85119 * r85116;
        double r85121 = exp(r85120);
        double r85122 = 1.0;
        double r85123 = r85121 - r85122;
        double r85124 = r85116 * r85123;
        double r85125 = r85117 * r85116;
        double r85126 = exp(r85125);
        double r85127 = r85126 - r85122;
        double r85128 = r85118 * r85116;
        double r85129 = exp(r85128);
        double r85130 = r85129 - r85122;
        double r85131 = r85127 * r85130;
        double r85132 = r85124 / r85131;
        return r85132;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r85133 = 1.0;
        double r85134 = b;
        double r85135 = r85133 / r85134;
        double r85136 = a;
        double r85137 = r85133 / r85136;
        double r85138 = r85135 + r85137;
        return r85138;
}

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
Target14.6
Herbie3.3
\[\frac{a + b}{a \cdot b}\]

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 3.3

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

  3. Final simplification3.3

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

Reproduce

herbie shell --seed 2019208 
(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))))