Average Error: 58.8 → 3.2
Time: 29.0s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\frac{1}{a} + \frac{1}{b}\]
\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 r3630117 = eps;
        double r3630118 = a;
        double r3630119 = b;
        double r3630120 = r3630118 + r3630119;
        double r3630121 = r3630120 * r3630117;
        double r3630122 = exp(r3630121);
        double r3630123 = 1.0;
        double r3630124 = r3630122 - r3630123;
        double r3630125 = r3630117 * r3630124;
        double r3630126 = r3630118 * r3630117;
        double r3630127 = exp(r3630126);
        double r3630128 = r3630127 - r3630123;
        double r3630129 = r3630119 * r3630117;
        double r3630130 = exp(r3630129);
        double r3630131 = r3630130 - r3630123;
        double r3630132 = r3630128 * r3630131;
        double r3630133 = r3630125 / r3630132;
        return r3630133;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r3630134 = 1.0;
        double r3630135 = a;
        double r3630136 = r3630134 / r3630135;
        double r3630137 = b;
        double r3630138 = r3630134 / r3630137;
        double r3630139 = r3630136 + r3630138;
        return r3630139;
}

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

Original58.8
Target13.9
Herbie3.2
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 58.8

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

  2. Simplified27.9

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

  3. Taylor expanded around 0 3.2

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

  4. Final simplification3.2

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

Reproduce

herbie shell --seed 2019158 +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))))