Kahan's exp quotient

Average Error: 39.8 → 0.6

Time: 25.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{e^{x} - 1}{x} \le 0.0:\\ \;\;\;\;\sqrt[3]{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), 1\right) \cdot \left(\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), 1\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x}}{x} - \frac{1}{x}\\ \end{array}\]

C

double f(double x) {
        double r3305609 = x;
        double r3305610 = exp(r3305609);
        double r3305611 = 1.0;
        double r3305612 = r3305610 - r3305611;
        double r3305613 = r3305612 / r3305609;
        return r3305613;
}

↓

double f(double x) {
        double r3305614 = x;
        double r3305615 = exp(r3305614);
        double r3305616 = 1.0;
        double r3305617 = r3305615 - r3305616;
        double r3305618 = r3305617 / r3305614;
        double r3305619 = 0.0;
        bool r3305620 = r3305618 <= r3305619;
        double r3305621 = 0.16666666666666666;
        double r3305622 = 0.5;
        double r3305623 = fma(r3305621, r3305614, r3305622);
        double r3305624 = 1.0;
        double r3305625 = fma(r3305614, r3305623, r3305624);
        double r3305626 = r3305625 * r3305625;
        double r3305627 = r3305625 * r3305626;
        double r3305628 = cbrt(r3305627);
        double r3305629 = r3305615 / r3305614;
        double r3305630 = r3305616 / r3305614;
        double r3305631 = r3305629 - r3305630;
        double r3305632 = r3305620 ? r3305628 : r3305631;
        return r3305632;
}

Error

Bits error versus x

Target

Original	39.8
Target	40.0
Herbie	0.6

\[\begin{array}{l} \mathbf{if}\;x \lt 1 \land x \gt -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (/ (- (exp x) 1.0) x) < 0.0

   1. Initial program 62.0

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

   2. Taylor expanded around 0 0

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)}\]

   3. Simplified 0

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), 1\right)}\]
   4. Using strategy rm

   5. Applied add-cbrt-cube 0

      \[\leadsto \color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), 1\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), 1\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), 1\right)}}\]

   if 0.0 < (/ (- (exp x) 1.0) x)

   1. Initial program 2.7

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

   3. Applied div-sub 1.7

      \[\leadsto \color{blue}{\frac{e^{x}}{x} - \frac{1}{x}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x} - 1}{x} \le 0.0:\\ \;\;\;\;\sqrt[3]{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), 1\right) \cdot \left(\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), 1\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x}}{x} - \frac{1}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x)
  :name "Kahan's exp quotient"

  :herbie-target
  (if (and (< x 1.0) (> x -1.0)) (/ (- (exp x) 1.0) (log (exp x))) (/ (- (exp x) 1.0) x))

  (/ (- (exp x) 1.0) x))