Average Error: 58.6 → 3.4
Time: 43.5s
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 r7984983 = eps;
        double r7984984 = a;
        double r7984985 = b;
        double r7984986 = r7984984 + r7984985;
        double r7984987 = r7984986 * r7984983;
        double r7984988 = exp(r7984987);
        double r7984989 = 1.0;
        double r7984990 = r7984988 - r7984989;
        double r7984991 = r7984983 * r7984990;
        double r7984992 = r7984984 * r7984983;
        double r7984993 = exp(r7984992);
        double r7984994 = r7984993 - r7984989;
        double r7984995 = r7984985 * r7984983;
        double r7984996 = exp(r7984995);
        double r7984997 = r7984996 - r7984989;
        double r7984998 = r7984994 * r7984997;
        double r7984999 = r7984991 / r7984998;
        return r7984999;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r7985000 = 1.0;
        double r7985001 = a;
        double r7985002 = r7985000 / r7985001;
        double r7985003 = b;
        double r7985004 = r7985000 / r7985003;
        double r7985005 = r7985002 + r7985004;
        return r7985005;
}

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

Original58.6
Target14.3
Herbie3.4
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 58.6

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

  2. Simplified34.0

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

  3. Taylor expanded around 0 3.4

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

  4. Final simplification3.4

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

Reproduce

herbie shell --seed 2019124 +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))))