Average Error: 58.9 → 3.1
Time: 1.3m
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 r13532011 = eps;
        double r13532012 = a;
        double r13532013 = b;
        double r13532014 = r13532012 + r13532013;
        double r13532015 = r13532014 * r13532011;
        double r13532016 = exp(r13532015);
        double r13532017 = 1.0;
        double r13532018 = r13532016 - r13532017;
        double r13532019 = r13532011 * r13532018;
        double r13532020 = r13532012 * r13532011;
        double r13532021 = exp(r13532020);
        double r13532022 = r13532021 - r13532017;
        double r13532023 = r13532013 * r13532011;
        double r13532024 = exp(r13532023);
        double r13532025 = r13532024 - r13532017;
        double r13532026 = r13532022 * r13532025;
        double r13532027 = r13532019 / r13532026;
        return r13532027;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r13532028 = 1.0;
        double r13532029 = a;
        double r13532030 = r13532028 / r13532029;
        double r13532031 = b;
        double r13532032 = r13532028 / r13532031;
        double r13532033 = r13532030 + r13532032;
        return r13532033;
}

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
Target13.7
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. Simplified34.7

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

  3. Taylor expanded around 0 3.1

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

  4. Final simplification3.1

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

Reproduce

herbie shell --seed 2019107 +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))))