Kahan's exp quotient

Average Error: 39.8 → 0.3

Time: 11.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.489848388411735063616148089238322427263 \cdot 10^{-4}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\log \left(e^{e^{x} - 1}\right)} \cdot \sqrt[3]{\log \left(e^{e^{x} - 1}\right)}\right) \cdot \sqrt[3]{\log \left(e^{e^{x} - 1}\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{1}{6} \cdot x + \frac{1}{2}\right) \cdot x\\ \end{array}\]

C

double f(double x) {
        double r3857341 = x;
        double r3857342 = exp(r3857341);
        double r3857343 = 1.0;
        double r3857344 = r3857342 - r3857343;
        double r3857345 = r3857344 / r3857341;
        return r3857345;
}

↓

double f(double x) {
        double r3857346 = x;
        double r3857347 = -0.0001489848388411735;
        bool r3857348 = r3857346 <= r3857347;
        double r3857349 = exp(r3857346);
        double r3857350 = 1.0;
        double r3857351 = r3857349 - r3857350;
        double r3857352 = exp(r3857351);
        double r3857353 = log(r3857352);
        double r3857354 = cbrt(r3857353);
        double r3857355 = r3857354 * r3857354;
        double r3857356 = r3857355 * r3857354;
        double r3857357 = r3857356 / r3857346;
        double r3857358 = 1.0;
        double r3857359 = 0.16666666666666666;
        double r3857360 = r3857359 * r3857346;
        double r3857361 = 0.5;
        double r3857362 = r3857360 + r3857361;
        double r3857363 = r3857362 * r3857346;
        double r3857364 = r3857358 + r3857363;
        double r3857365 = r3857348 ? r3857357 : r3857364;
        return r3857365;
}

Error

Bits error versus x

Target

Original	39.8
Target	40.2
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 < -0.0001489848388411735

   1. Initial program 0.1

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

   3. Applied add-log-exp 0.1

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

   4. Applied add-log-exp 0.1

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

   5. Applied diff-log 0.1

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

   6. Simplified 0.1

      \[\leadsto \frac{\log \color{blue}{\left(e^{e^{x} - 1}\right)}}{x}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 0.1

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

   if -0.0001489848388411735 < x

   1. Initial program 60.2

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

   2. Taylor expanded around 0 0.4

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

   3. Simplified 0.4

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.489848388411735063616148089238322427263 \cdot 10^{-4}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\log \left(e^{e^{x} - 1}\right)} \cdot \sqrt[3]{\log \left(e^{e^{x} - 1}\right)}\right) \cdot \sqrt[3]{\log \left(e^{e^{x} - 1}\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{1}{6} \cdot x + \frac{1}{2}\right) \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(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))