Average Error: 58.9 → 3.1
Time: 46.4s
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 r3240138 = eps;
        double r3240139 = a;
        double r3240140 = b;
        double r3240141 = r3240139 + r3240140;
        double r3240142 = r3240141 * r3240138;
        double r3240143 = exp(r3240142);
        double r3240144 = 1.0;
        double r3240145 = r3240143 - r3240144;
        double r3240146 = r3240138 * r3240145;
        double r3240147 = r3240139 * r3240138;
        double r3240148 = exp(r3240147);
        double r3240149 = r3240148 - r3240144;
        double r3240150 = r3240140 * r3240138;
        double r3240151 = exp(r3240150);
        double r3240152 = r3240151 - r3240144;
        double r3240153 = r3240149 * r3240152;
        double r3240154 = r3240146 / r3240153;
        return r3240154;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r3240155 = 1.0;
        double r3240156 = b;
        double r3240157 = r3240155 / r3240156;
        double r3240158 = a;
        double r3240159 = r3240155 / r3240158;
        double r3240160 = r3240157 + r3240159;
        return r3240160;
}

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
Target14.1
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. Simplified27.7

    \[\leadsto \color{blue}{\frac{\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot b\right)} \cdot \mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}}\]

  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 2019151 +o rules:numerics
(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))))