Kahan's exp quotient

Average Error: 39.9 → 0.3

Time: 14.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.668284299855764371729727191251413387363 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{e^{x + x} - 1 \cdot 1}{e^{x} + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{12}, \frac{1}{3}\right), 1\right) \cdot \mathsf{fma}\left(x, \frac{1}{6} + x \cdot \frac{1}{36}, 1\right)\\ \end{array}\]

C

double f(double x) {
        double r109517 = x;
        double r109518 = exp(r109517);
        double r109519 = 1.0;
        double r109520 = r109518 - r109519;
        double r109521 = r109520 / r109517;
        return r109521;
}

↓

double f(double x) {
        double r109522 = x;
        double r109523 = -0.00016682842998557644;
        bool r109524 = r109522 <= r109523;
        double r109525 = r109522 + r109522;
        double r109526 = exp(r109525);
        double r109527 = 1.0;
        double r109528 = r109527 * r109527;
        double r109529 = r109526 - r109528;
        double r109530 = exp(r109522);
        double r109531 = r109530 + r109527;
        double r109532 = r109529 / r109531;
        double r109533 = r109532 / r109522;
        double r109534 = 0.08333333333333333;
        double r109535 = 0.3333333333333333;
        double r109536 = fma(r109522, r109534, r109535);
        double r109537 = 1.0;
        double r109538 = fma(r109522, r109536, r109537);
        double r109539 = 0.16666666666666666;
        double r109540 = 0.027777777777777776;
        double r109541 = r109522 * r109540;
        double r109542 = r109539 + r109541;
        double r109543 = fma(r109522, r109542, r109537);
        double r109544 = r109538 * r109543;
        double r109545 = r109524 ? r109533 : r109544;
        return r109545;
}

Error

Bits error versus x

Target

Original	39.9
Target	40.4
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 < -0.00016682842998557644

   1. Initial program 0.0

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

   3. Applied flip-- 0.0

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

   4. Simplified 0.0

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

   if -0.00016682842998557644 < x

   1. Initial program 60.1

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

   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. Simplified 0.4

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

   5. Applied add-cube-cbrt 0.5

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

   6. Simplified 0.5

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

   7. Simplified 0.5

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

   8. Taylor expanded around 0 0.4

      \[\leadsto \color{blue}{\left(\frac{1}{12} \cdot {x}^{2} + \left(\frac{1}{3} \cdot x + 1\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right), 1\right)}\]

   9. Simplified 0.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{12}, \frac{1}{3}\right), 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right), 1\right)}\]

   10. Taylor expanded around 0 0.4

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{12}, \frac{1}{3}\right), 1\right) \cdot \color{blue}{\left(\frac{1}{36} \cdot {x}^{2} + \left(\frac{1}{6} \cdot x + 1\right)\right)}\]

   11. Simplified 0.4

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.668284299855764371729727191251413387363 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{e^{x + x} - 1 \cdot 1}{e^{x} + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{12}, \frac{1}{3}\right), 1\right) \cdot \mathsf{fma}\left(x, \frac{1}{6} + x \cdot \frac{1}{36}, 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(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))