Average Error: 39.8 → 0.7
Time: 3.9m
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0019097566603386235:\\ \;\;\;\;\frac{\left(e^{x} + e^{x} \cdot \left(e^{x} \cdot e^{x}\right)\right) + e^{x} \cdot e^{x}}{e^{x} \cdot \left(e^{x} \cdot e^{x}\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{2}\right) + \frac{1}{12} \cdot x\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0019097566603386235:\\
\;\;\;\;\frac{\left(e^{x} + e^{x} \cdot \left(e^{x} \cdot e^{x}\right)\right) + e^{x} \cdot e^{x}}{e^{x} \cdot \left(e^{x} \cdot e^{x}\right) + -1}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{x} + \frac{1}{2}\right) + \frac{1}{12} \cdot x\\

\end{array}
double f(double x) {
        double r21247948 = x;
        double r21247949 = exp(r21247948);
        double r21247950 = 1.0;
        double r21247951 = r21247949 - r21247950;
        double r21247952 = r21247949 / r21247951;
        return r21247952;
}
↓
double f(double x) {
        double r21247953 = x;
        double r21247954 = -0.0019097566603386235;
        bool r21247955 = r21247953 <= r21247954;
        double r21247956 = exp(r21247953);
        double r21247957 = r21247956 * r21247956;
        double r21247958 = r21247956 * r21247957;
        double r21247959 = r21247956 + r21247958;
        double r21247960 = r21247959 + r21247957;
        double r21247961 = -1.0;
        double r21247962 = r21247958 + r21247961;
        double r21247963 = r21247960 / r21247962;
        double r21247964 = 1.0;
        double r21247965 = r21247964 / r21247953;
        double r21247966 = 0.5;
        double r21247967 = r21247965 + r21247966;
        double r21247968 = 0.08333333333333333;
        double r21247969 = r21247968 * r21247953;
        double r21247970 = r21247967 + r21247969;
        double r21247971 = r21247955 ? r21247963 : r21247970;
        return r21247971;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.8
Target39.3
Herbie0.7
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.0019097566603386235

    1. Initial program 0.0

      \[\frac{e^{x}}{e^{x} - 1}\]
    2. Using strategy rm

    3. Applied flip3--0.0

      \[\leadsto \frac{e^{x}}{\color{blue}{\frac{{\left(e^{x}\right)}^{3} - {1}^{3}}{e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)}}}\]

    4. Taylor expanded around -inf 0.0

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

    5. Simplified0.0

      \[\leadsto \color{blue}{\frac{\left(e^{x} \cdot \left(e^{x} \cdot e^{x}\right) + e^{x}\right) + e^{x} \cdot e^{x}}{-1 + e^{x} \cdot \left(e^{x} \cdot e^{x}\right)}}\]

    if -0.0019097566603386235 < x

    1. Initial program 60.1

      \[\frac{e^{x}}{e^{x} - 1}\]

    2. Taylor expanded around 0 1.0

      \[\leadsto \color{blue}{\frac{1}{12} \cdot x + \left(\frac{1}{x} + \frac{1}{2}\right)}\]
    3. Using strategy rm

    4. Applied +-commutative1.0

      \[\leadsto \frac{1}{12} \cdot x + \color{blue}{\left(\frac{1}{2} + \frac{1}{x}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0019097566603386235:\\ \;\;\;\;\frac{\left(e^{x} + e^{x} \cdot \left(e^{x} \cdot e^{x}\right)\right) + e^{x} \cdot e^{x}}{e^{x} \cdot \left(e^{x} \cdot e^{x}\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{2}\right) + \frac{1}{12} \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 
(FPCore (x)
  :name "expq2 (section 3.11)"

  :herbie-target
  (/ 1 (- 1 (exp (- x))))

  (/ (exp x) (- (exp x) 1)))