Average Error: 60.3 → 3.4
Time: 11.6s
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 r113399 = eps;
        double r113400 = a;
        double r113401 = b;
        double r113402 = r113400 + r113401;
        double r113403 = r113402 * r113399;
        double r113404 = exp(r113403);
        double r113405 = 1.0;
        double r113406 = r113404 - r113405;
        double r113407 = r113399 * r113406;
        double r113408 = r113400 * r113399;
        double r113409 = exp(r113408);
        double r113410 = r113409 - r113405;
        double r113411 = r113401 * r113399;
        double r113412 = exp(r113411);
        double r113413 = r113412 - r113405;
        double r113414 = r113410 * r113413;
        double r113415 = r113407 / r113414;
        return r113415;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r113416 = 1.0;
        double r113417 = b;
        double r113418 = r113416 / r113417;
        double r113419 = a;
        double r113420 = r113416 / r113419;
        double r113421 = r113418 + r113420;
        return r113421;
}

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
Target14.7
Herbie3.4
\[\frac{a + b}{a \cdot b}\]

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. Taylor expanded around 0 57.8

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

  3. Simplified57.8

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

  5. Applied pow-prod-down57.3

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

  6. Simplified57.3

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

  7. Taylor expanded around 0 3.4

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

  8. Final simplification3.4

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

Reproduce

herbie shell --seed 2020081 +o rules:numerics
(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))))