Average Error: 60.0 → 3.7
Time: 13.5s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\frac{1}{b} + \frac{1}{a}\]
\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 r461 = eps;
        double r462 = a;
        double r463 = b;
        double r464 = r462 + r463;
        double r465 = r464 * r461;
        double r466 = exp(r465);
        double r467 = 1.0;
        double r468 = r466 - r467;
        double r469 = r461 * r468;
        double r470 = r462 * r461;
        double r471 = exp(r470);
        double r472 = r471 - r467;
        double r473 = r463 * r461;
        double r474 = exp(r473);
        double r475 = r474 - r467;
        double r476 = r472 * r475;
        double r477 = r469 / r476;
        return r477;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r478 = 1.0;
        double r479 = b;
        double r480 = r478 / r479;
        double r481 = a;
        double r482 = r478 / r481;
        double r483 = r480 + r482;
        return r483;
}

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

Original60.0
Target15.4
Herbie3.7
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.0

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

  2. Taylor expanded around 0 3.7

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

  3. Final simplification3.7

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

Reproduce

herbie shell --seed 2020025 
(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))))