Kahan's exp quotient

Average Error: 40.4 → 0.3

Time: 4.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -4.0406550847229628 \cdot 10^{-4}:\\ \;\;\;\;\frac{e^{x}}{x} - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{3}{32} \cdot {x}^{2} + \left(\frac{1}{12} \cdot x + 1\right)\right) \cdot \frac{\sqrt{{\left(\frac{1}{6} \cdot {x}^{2}\right)}^{3} + {\left(\frac{1}{2} \cdot x + 1\right)}^{3}}}{\sqrt[3]{\left(\frac{1}{2} \cdot x + 1\right) \cdot \left(\left(\frac{1}{2} \cdot x + 1\right) - \frac{1}{6} \cdot {x}^{2}\right) + \frac{1}{36} \cdot \left({x}^{2} \cdot {x}^{2}\right)}}\\ \end{array}\]

C

double f(double x) {
        double r89599 = x;
        double r89600 = exp(r89599);
        double r89601 = 1.0;
        double r89602 = r89600 - r89601;
        double r89603 = r89602 / r89599;
        return r89603;
}

↓

double f(double x) {
        double r89604 = x;
        double r89605 = -0.0004040655084722963;
        bool r89606 = r89604 <= r89605;
        double r89607 = exp(r89604);
        double r89608 = r89607 / r89604;
        double r89609 = 1.0;
        double r89610 = r89609 / r89604;
        double r89611 = r89608 - r89610;
        double r89612 = 0.09375;
        double r89613 = 2.0;
        double r89614 = pow(r89604, r89613);
        double r89615 = r89612 * r89614;
        double r89616 = 0.08333333333333333;
        double r89617 = r89616 * r89604;
        double r89618 = 1.0;
        double r89619 = r89617 + r89618;
        double r89620 = r89615 + r89619;
        double r89621 = 0.16666666666666666;
        double r89622 = r89621 * r89614;
        double r89623 = 3.0;
        double r89624 = pow(r89622, r89623);
        double r89625 = 0.5;
        double r89626 = r89625 * r89604;
        double r89627 = r89626 + r89618;
        double r89628 = pow(r89627, r89623);
        double r89629 = r89624 + r89628;
        double r89630 = sqrt(r89629);
        double r89631 = r89627 - r89622;
        double r89632 = r89627 * r89631;
        double r89633 = 0.027777777777777776;
        double r89634 = r89614 * r89614;
        double r89635 = r89633 * r89634;
        double r89636 = r89632 + r89635;
        double r89637 = cbrt(r89636);
        double r89638 = r89630 / r89637;
        double r89639 = r89620 * r89638;
        double r89640 = r89606 ? r89611 : r89639;
        return r89640;
}

Error

Bits error versus x

Target

Original	40.4
Target	41.0
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if x < -0.0004040655084722963

   1. Initial program 0.0

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

   3. Applied div-sub 0.0

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

   if -0.0004040655084722963 < x

   1. Initial program 60.2

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

   2. Taylor expanded around 0 0.4

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

   4. Applied flip3-+ 0.4

      \[\leadsto \color{blue}{\frac{{\left(\frac{1}{6} \cdot {x}^{2}\right)}^{3} + {\left(\frac{1}{2} \cdot x + 1\right)}^{3}}{\left(\frac{1}{6} \cdot {x}^{2}\right) \cdot \left(\frac{1}{6} \cdot {x}^{2}\right) + \left(\left(\frac{1}{2} \cdot x + 1\right) \cdot \left(\frac{1}{2} \cdot x + 1\right) - \left(\frac{1}{6} \cdot {x}^{2}\right) \cdot \left(\frac{1}{2} \cdot x + 1\right)\right)}}\]

   5. Simplified 0.4

      \[\leadsto \frac{{\left(\frac{1}{6} \cdot {x}^{2}\right)}^{3} + {\left(\frac{1}{2} \cdot x + 1\right)}^{3}}{\color{blue}{\left(\frac{1}{2} \cdot x + 1\right) \cdot \left(\left(\frac{1}{2} \cdot x + 1\right) - \frac{1}{6} \cdot {x}^{2}\right) + \frac{1}{36} \cdot \left({x}^{2} \cdot {x}^{2}\right)}}\]
   6. Using strategy rm

   7. Applied add-cube-cbrt 0.5

      \[\leadsto \frac{{\left(\frac{1}{6} \cdot {x}^{2}\right)}^{3} + {\left(\frac{1}{2} \cdot x + 1\right)}^{3}}{\color{blue}{\left(\sqrt[3]{\left(\frac{1}{2} \cdot x + 1\right) \cdot \left(\left(\frac{1}{2} \cdot x + 1\right) - \frac{1}{6} \cdot {x}^{2}\right) + \frac{1}{36} \cdot \left({x}^{2} \cdot {x}^{2}\right)} \cdot \sqrt[3]{\left(\frac{1}{2} \cdot x + 1\right) \cdot \left(\left(\frac{1}{2} \cdot x + 1\right) - \frac{1}{6} \cdot {x}^{2}\right) + \frac{1}{36} \cdot \left({x}^{2} \cdot {x}^{2}\right)}\right) \cdot \sqrt[3]{\left(\frac{1}{2} \cdot x + 1\right) \cdot \left(\left(\frac{1}{2} \cdot x + 1\right) - \frac{1}{6} \cdot {x}^{2}\right) + \frac{1}{36} \cdot \left({x}^{2} \cdot {x}^{2}\right)}}}\]

   8. Applied add-sqr-sqrt 0.5

      \[\leadsto \frac{\color{blue}{\sqrt{{\left(\frac{1}{6} \cdot {x}^{2}\right)}^{3} + {\left(\frac{1}{2} \cdot x + 1\right)}^{3}} \cdot \sqrt{{\left(\frac{1}{6} \cdot {x}^{2}\right)}^{3} + {\left(\frac{1}{2} \cdot x + 1\right)}^{3}}}}{\left(\sqrt[3]{\left(\frac{1}{2} \cdot x + 1\right) \cdot \left(\left(\frac{1}{2} \cdot x + 1\right) - \frac{1}{6} \cdot {x}^{2}\right) + \frac{1}{36} \cdot \left({x}^{2} \cdot {x}^{2}\right)} \cdot \sqrt[3]{\left(\frac{1}{2} \cdot x + 1\right) \cdot \left(\left(\frac{1}{2} \cdot x + 1\right) - \frac{1}{6} \cdot {x}^{2}\right) + \frac{1}{36} \cdot \left({x}^{2} \cdot {x}^{2}\right)}\right) \cdot \sqrt[3]{\left(\frac{1}{2} \cdot x + 1\right) \cdot \left(\left(\frac{1}{2} \cdot x + 1\right) - \frac{1}{6} \cdot {x}^{2}\right) + \frac{1}{36} \cdot \left({x}^{2} \cdot {x}^{2}\right)}}\]

   9. Applied times-frac 0.5

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\frac{1}{6} \cdot {x}^{2}\right)}^{3} + {\left(\frac{1}{2} \cdot x + 1\right)}^{3}}}{\sqrt[3]{\left(\frac{1}{2} \cdot x + 1\right) \cdot \left(\left(\frac{1}{2} \cdot x + 1\right) - \frac{1}{6} \cdot {x}^{2}\right) + \frac{1}{36} \cdot \left({x}^{2} \cdot {x}^{2}\right)} \cdot \sqrt[3]{\left(\frac{1}{2} \cdot x + 1\right) \cdot \left(\left(\frac{1}{2} \cdot x + 1\right) - \frac{1}{6} \cdot {x}^{2}\right) + \frac{1}{36} \cdot \left({x}^{2} \cdot {x}^{2}\right)}} \cdot \frac{\sqrt{{\left(\frac{1}{6} \cdot {x}^{2}\right)}^{3} + {\left(\frac{1}{2} \cdot x + 1\right)}^{3}}}{\sqrt[3]{\left(\frac{1}{2} \cdot x + 1\right) \cdot \left(\left(\frac{1}{2} \cdot x + 1\right) - \frac{1}{6} \cdot {x}^{2}\right) + \frac{1}{36} \cdot \left({x}^{2} \cdot {x}^{2}\right)}}}\]

   10. Taylor expanded around 0 0.4

       \[\leadsto \color{blue}{\left(\frac{3}{32} \cdot {x}^{2} + \left(\frac{1}{12} \cdot x + 1\right)\right)} \cdot \frac{\sqrt{{\left(\frac{1}{6} \cdot {x}^{2}\right)}^{3} + {\left(\frac{1}{2} \cdot x + 1\right)}^{3}}}{\sqrt[3]{\left(\frac{1}{2} \cdot x + 1\right) \cdot \left(\left(\frac{1}{2} \cdot x + 1\right) - \frac{1}{6} \cdot {x}^{2}\right) + \frac{1}{36} \cdot \left({x}^{2} \cdot {x}^{2}\right)}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.0406550847229628 \cdot 10^{-4}:\\ \;\;\;\;\frac{e^{x}}{x} - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{3}{32} \cdot {x}^{2} + \left(\frac{1}{12} \cdot x + 1\right)\right) \cdot \frac{\sqrt{{\left(\frac{1}{6} \cdot {x}^{2}\right)}^{3} + {\left(\frac{1}{2} \cdot x + 1\right)}^{3}}}{\sqrt[3]{\left(\frac{1}{2} \cdot x + 1\right) \cdot \left(\left(\frac{1}{2} \cdot x + 1\right) - \frac{1}{6} \cdot {x}^{2}\right) + \frac{1}{36} \cdot \left({x}^{2} \cdot {x}^{2}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(FPCore (x)
  :name "Kahan's exp quotient"
  :precision binary64

  :herbie-target
  (if (and (< x 1) (> x -1)) (/ (- (exp x) 1) (log (exp x))) (/ (- (exp x) 1) x))

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