Average Error: 41.2 → 0.5
Time: 10.9s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.9688178485685753349443416482245083898306:\\ \;\;\;\;\frac{e^{x}}{e^{x + x} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.9688178485685753349443416482245083898306:\\
\;\;\;\;\frac{e^{x}}{e^{x + x} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)\\

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

\end{array}
double f(double x) {
        double r68953 = x;
        double r68954 = exp(r68953);
        double r68955 = 1.0;
        double r68956 = r68954 - r68955;
        double r68957 = r68954 / r68956;
        return r68957;
}
↓
double f(double x) {
        double r68958 = x;
        double r68959 = exp(r68958);
        double r68960 = 0.9688178485685753;
        bool r68961 = r68959 <= r68960;
        double r68962 = r68958 + r68958;
        double r68963 = exp(r68962);
        double r68964 = 1.0;
        double r68965 = r68964 * r68964;
        double r68966 = r68963 - r68965;
        double r68967 = r68959 / r68966;
        double r68968 = r68959 + r68964;
        double r68969 = r68967 * r68968;
        double r68970 = 0.08333333333333333;
        double r68971 = 1.0;
        double r68972 = r68971 / r68958;
        double r68973 = fma(r68970, r68958, r68972);
        double r68974 = 0.5;
        double r68975 = r68973 + r68974;
        double r68976 = r68961 ? r68969 : r68975;
        return r68976;
}

Error

Bits error versus x

Target

Original41.2
Target40.9
Herbie0.5
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.9688178485685753

    1. Initial program 0.0

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

    3. Applied flip--0.0

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

    4. Applied associate-/r/0.0

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

    5. Simplified0.0

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

    if 0.9688178485685753 < (exp x)

    1. Initial program 61.9

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

    2. Taylor expanded around 0 0.7

      \[\leadsto \color{blue}{\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)}\]

    3. Simplified0.7

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.9688178485685753349443416482245083898306:\\ \;\;\;\;\frac{e^{x}}{e^{x + x} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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