Average Error: 59.1 → 3.1
Time: 36.9s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le 9.918390622765604 \cdot 10^{+229}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{elif}\;a \le 2.194671702223418 \cdot 10^{+284}:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{expm1}\left(b \cdot \varepsilon\right)}{\mathsf{expm1}\left(\varepsilon \cdot \left(b + a\right)\right)}}{\frac{\varepsilon}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]
\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 9.918390622765604 \cdot 10^{+229}:\\
\;\;\;\;\frac{1}{b} + \frac{1}{a}\\

\mathbf{elif}\;a \le 2.194671702223418 \cdot 10^{+284}:\\
\;\;\;\;\frac{1}{\frac{\frac{\mathsf{expm1}\left(b \cdot \varepsilon\right)}{\mathsf{expm1}\left(\varepsilon \cdot \left(b + a\right)\right)}}{\frac{\varepsilon}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}}}\\

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

\end{array}
double f(double a, double b, double eps) {
        double r5576980 = eps;
        double r5576981 = a;
        double r5576982 = b;
        double r5576983 = r5576981 + r5576982;
        double r5576984 = r5576983 * r5576980;
        double r5576985 = exp(r5576984);
        double r5576986 = 1.0;
        double r5576987 = r5576985 - r5576986;
        double r5576988 = r5576980 * r5576987;
        double r5576989 = r5576981 * r5576980;
        double r5576990 = exp(r5576989);
        double r5576991 = r5576990 - r5576986;
        double r5576992 = r5576982 * r5576980;
        double r5576993 = exp(r5576992);
        double r5576994 = r5576993 - r5576986;
        double r5576995 = r5576991 * r5576994;
        double r5576996 = r5576988 / r5576995;
        return r5576996;
}
↓
double f(double a, double b, double eps) {
        double r5576997 = a;
        double r5576998 = 9.918390622765604e+229;
        bool r5576999 = r5576997 <= r5576998;
        double r5577000 = 1.0;
        double r5577001 = b;
        double r5577002 = r5577000 / r5577001;
        double r5577003 = r5577000 / r5576997;
        double r5577004 = r5577002 + r5577003;
        double r5577005 = 2.194671702223418e+284;
        bool r5577006 = r5576997 <= r5577005;
        double r5577007 = eps;
        double r5577008 = r5577001 * r5577007;
        double r5577009 = expm1(r5577008);
        double r5577010 = r5577001 + r5576997;
        double r5577011 = r5577007 * r5577010;
        double r5577012 = expm1(r5577011);
        double r5577013 = r5577009 / r5577012;
        double r5577014 = r5576997 * r5577007;
        double r5577015 = expm1(r5577014);
        double r5577016 = r5577007 / r5577015;
        double r5577017 = r5577013 / r5577016;
        double r5577018 = r5577000 / r5577017;
        double r5577019 = r5577006 ? r5577018 : r5577004;
        double r5577020 = r5576999 ? r5577004 : r5577019;
        return r5577020;
}

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

Original59.1
Target13.8
Herbie3.1
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if a < 9.918390622765604e+229 or 2.194671702223418e+284 < a

    1. Initial program 59.4

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

    2. Simplified27.8

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

    3. Taylor expanded around 0 2.6

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

    if 9.918390622765604e+229 < a < 2.194671702223418e+284

    1. Initial program 50.7

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

    2. Simplified17.8

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

    4. Applied clear-num17.8

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\mathsf{expm1}\left(\varepsilon \cdot b\right)}{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}}{\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le 9.918390622765604 \cdot 10^{+229}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{elif}\;a \le 2.194671702223418 \cdot 10^{+284}:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{expm1}\left(b \cdot \varepsilon\right)}{\mathsf{expm1}\left(\varepsilon \cdot \left(b + a\right)\right)}}{\frac{\varepsilon}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +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))))