Average Error: 60.1 → 3.6
Time: 35.7s
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 r5023130 = eps;
        double r5023131 = a;
        double r5023132 = b;
        double r5023133 = r5023131 + r5023132;
        double r5023134 = r5023133 * r5023130;
        double r5023135 = exp(r5023134);
        double r5023136 = 1.0;
        double r5023137 = r5023135 - r5023136;
        double r5023138 = r5023130 * r5023137;
        double r5023139 = r5023131 * r5023130;
        double r5023140 = exp(r5023139);
        double r5023141 = r5023140 - r5023136;
        double r5023142 = r5023132 * r5023130;
        double r5023143 = exp(r5023142);
        double r5023144 = r5023143 - r5023136;
        double r5023145 = r5023141 * r5023144;
        double r5023146 = r5023138 / r5023145;
        return r5023146;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r5023147 = 1.0;
        double r5023148 = a;
        double r5023149 = r5023147 / r5023148;
        double r5023150 = b;
        double r5023151 = r5023147 / r5023150;
        double r5023152 = r5023149 + r5023151;
        return r5023152;
}

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
Target15.2
Herbie3.6
\[\frac{a + b}{a \cdot b}\]

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 57.9

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot {\varepsilon}^{2}\right) + \left(\frac{1}{6} \cdot \left({a}^{3} \cdot {\varepsilon}^{3}\right) + a \cdot \varepsilon\right)\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

  3. Simplified57.9

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\mathsf{fma}\left(a \cdot a, \frac{1}{2} \cdot \left(\varepsilon \cdot \varepsilon\right), \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right), \frac{1}{6} \cdot \left(a \cdot \left(a \cdot a\right)\right), \varepsilon \cdot a\right)\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

  4. Taylor expanded around 0 3.6

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

  5. Final simplification3.6

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :pre (and (< -1.0 eps) (< eps 1.0))

  :herbie-target
  (/ (+ a b) (* a b))

  (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))))