Average Error: 60.3 → 3.5
Time: 34.7s
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 r85193 = eps;
        double r85194 = a;
        double r85195 = b;
        double r85196 = r85194 + r85195;
        double r85197 = r85196 * r85193;
        double r85198 = exp(r85197);
        double r85199 = 1.0;
        double r85200 = r85198 - r85199;
        double r85201 = r85193 * r85200;
        double r85202 = r85194 * r85193;
        double r85203 = exp(r85202);
        double r85204 = r85203 - r85199;
        double r85205 = r85195 * r85193;
        double r85206 = exp(r85205);
        double r85207 = r85206 - r85199;
        double r85208 = r85204 * r85207;
        double r85209 = r85201 / r85208;
        return r85209;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r85210 = 1.0;
        double r85211 = a;
        double r85212 = r85210 / r85211;
        double r85213 = b;
        double r85214 = r85210 / r85213;
        double r85215 = r85212 + r85214;
        return r85215;
}

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

Original60.3
Target15.0
Herbie3.5
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.3

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

  2. Simplified60.3

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

  3. Taylor expanded around 0 57.9

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

  4. Simplified57.9

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

  5. Taylor expanded around 0 56.8

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

  6. Simplified56.6

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

  7. Taylor expanded around 0 3.5

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

  8. Final simplification3.5

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

Reproduce

herbie shell --seed 2019174 +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))))