Average Error: 60.1 → 3.5
Time: 16.3s
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 r125127 = eps;
        double r125128 = a;
        double r125129 = b;
        double r125130 = r125128 + r125129;
        double r125131 = r125130 * r125127;
        double r125132 = exp(r125131);
        double r125133 = 1.0;
        double r125134 = r125132 - r125133;
        double r125135 = r125127 * r125134;
        double r125136 = r125128 * r125127;
        double r125137 = exp(r125136);
        double r125138 = r125137 - r125133;
        double r125139 = r125129 * r125127;
        double r125140 = exp(r125139);
        double r125141 = r125140 - r125133;
        double r125142 = r125138 * r125141;
        double r125143 = r125135 / r125142;
        return r125143;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r125144 = 1.0;
        double r125145 = b;
        double r125146 = r125144 / r125145;
        double r125147 = a;
        double r125148 = r125144 / r125147;
        double r125149 = r125146 + r125148;
        return r125149;
}

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

Original60.1
Target14.9
Herbie3.5
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.1

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

  2. Taylor expanded around 0 3.5

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

  3. Final simplification3.5

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

Reproduce

herbie shell --seed 2019318 
(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))))