Average Error: 60.5 → 3.9
Time: 18.1s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -3.962065998026786493085274341077373192581 \cdot 10^{-85} \lor \neg \left(a \le 7.398910124761133051502446479192068549373 \cdot 10^{-126}\right):\\ \;\;\;\;\frac{\frac{a + b}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{a + b}{b}\\ \end{array}\]
\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)}
↓
\begin{array}{l}
\mathbf{if}\;a \le -3.962065998026786493085274341077373192581 \cdot 10^{-85} \lor \neg \left(a \le 7.398910124761133051502446479192068549373 \cdot 10^{-126}\right):\\
\;\;\;\;\frac{\frac{a + b}{a}}{b}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{a} \cdot \frac{a + b}{b}\\

\end{array}
double f(double a, double b, double eps) {
        double r112654 = eps;
        double r112655 = a;
        double r112656 = b;
        double r112657 = r112655 + r112656;
        double r112658 = r112657 * r112654;
        double r112659 = exp(r112658);
        double r112660 = 1.0;
        double r112661 = r112659 - r112660;
        double r112662 = r112654 * r112661;
        double r112663 = r112655 * r112654;
        double r112664 = exp(r112663);
        double r112665 = r112664 - r112660;
        double r112666 = r112656 * r112654;
        double r112667 = exp(r112666);
        double r112668 = r112667 - r112660;
        double r112669 = r112665 * r112668;
        double r112670 = r112662 / r112669;
        return r112670;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r112671 = a;
        double r112672 = -3.9620659980267865e-85;
        bool r112673 = r112671 <= r112672;
        double r112674 = 7.398910124761133e-126;
        bool r112675 = r112671 <= r112674;
        double r112676 = !r112675;
        bool r112677 = r112673 || r112676;
        double r112678 = b;
        double r112679 = r112671 + r112678;
        double r112680 = r112679 / r112671;
        double r112681 = r112680 / r112678;
        double r112682 = 1.0;
        double r112683 = r112682 / r112671;
        double r112684 = r112679 / r112678;
        double r112685 = r112683 * r112684;
        double r112686 = r112677 ? r112681 : r112685;
        return r112686;
}

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

Original60.5
Target15.0
Herbie3.9
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if a < -3.9620659980267865e-85 or 7.398910124761133e-126 < a

    1. Initial program 58.4

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

    2. Taylor expanded around 0 5.1

      \[\leadsto \color{blue}{\frac{1}{b} + \frac{1}{a}}\]
    3. Using strategy rm

    4. Applied frac-add10.2

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

    5. Simplified10.2

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

    6. Simplified10.2

      \[\leadsto \frac{a + b}{\color{blue}{a \cdot b}}\]
    7. Using strategy rm

    8. Applied associate-/r*6.1

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

    if -3.9620659980267865e-85 < a < 7.398910124761133e-126

    1. Initial program 64.0

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

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\frac{1}{b} + \frac{1}{a}}\]
    3. Using strategy rm

    4. Applied frac-add23.5

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

    5. Simplified23.5

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

    6. Simplified23.5

      \[\leadsto \frac{a + b}{\color{blue}{a \cdot b}}\]
    7. Using strategy rm

    8. Applied *-un-lft-identity23.5

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

    9. Applied times-frac0.1

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \frac{a + b}{b}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -3.962065998026786493085274341077373192581 \cdot 10^{-85} \lor \neg \left(a \le 7.398910124761133051502446479192068549373 \cdot 10^{-126}\right):\\ \;\;\;\;\frac{\frac{a + b}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{a + b}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 
(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))))