Kahan's exp quotient

Average Error: 39.2 → 0.3

Time: 9.5s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.66490381328880487 \cdot 10^{-4}:\\ \;\;\;\;\frac{\left(\sqrt[3]{{\left(e^{x}\right)}^{4} - {1}^{4}} \cdot \sqrt[3]{{\left(e^{x}\right)}^{4} - {1}^{4}}\right) \cdot \sqrt[3]{{\left(e^{x}\right)}^{4} - {1}^{4}}}{\left(\left(e^{x} + 1\right) \cdot x\right) \cdot \left({\left(e^{x}\right)}^{2} + 1 \cdot 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right)\\ \end{array}\]

C

double f(double x) {
        double r66582 = x;
        double r66583 = exp(r66582);
        double r66584 = 1.0;
        double r66585 = r66583 - r66584;
        double r66586 = r66585 / r66582;
        return r66586;
}

↓

double f(double x) {
        double r66587 = x;
        double r66588 = -0.0001664903813288805;
        bool r66589 = r66587 <= r66588;
        double r66590 = exp(r66587);
        double r66591 = 4.0;
        double r66592 = pow(r66590, r66591);
        double r66593 = 1.0;
        double r66594 = pow(r66593, r66591);
        double r66595 = r66592 - r66594;
        double r66596 = cbrt(r66595);
        double r66597 = r66596 * r66596;
        double r66598 = r66597 * r66596;
        double r66599 = r66590 + r66593;
        double r66600 = r66599 * r66587;
        double r66601 = 2.0;
        double r66602 = pow(r66590, r66601);
        double r66603 = r66593 * r66593;
        double r66604 = r66602 + r66603;
        double r66605 = r66600 * r66604;
        double r66606 = r66598 / r66605;
        double r66607 = 1.0;
        double r66608 = 0.16666666666666666;
        double r66609 = r66587 * r66608;
        double r66610 = 0.5;
        double r66611 = r66609 + r66610;
        double r66612 = r66587 * r66611;
        double r66613 = r66607 + r66612;
        double r66614 = r66589 ? r66606 : r66613;
        return r66614;
}

Error

Bits error versus x

Target

Original	39.2
Target	39.6
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 < -0.0001664903813288805

   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. Applied associate-/l/ 0.0

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

   6. Applied flip-- 0.0

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

   7. Applied associate-/l/ 0.0

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

   8. Simplified 0.0

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

   10. Applied add-cube-cbrt 0.0

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

   11. Simplified 0.0

       \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{{\left(e^{x}\right)}^{4} - {1}^{4}} \cdot \sqrt[3]{{\left(e^{x}\right)}^{4} - {1}^{4}}\right)} \cdot \sqrt[3]{\left(e^{x} \cdot e^{x}\right) \cdot \left(e^{x} \cdot e^{x}\right) - \left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)}}{\left(\left(e^{x} + 1\right) \cdot x\right) \cdot \left({\left(e^{x}\right)}^{2} + 1 \cdot 1\right)}\]

   12. Simplified 0.0

       \[\leadsto \frac{\left(\sqrt[3]{{\left(e^{x}\right)}^{4} - {1}^{4}} \cdot \sqrt[3]{{\left(e^{x}\right)}^{4} - {1}^{4}}\right) \cdot \color{blue}{\sqrt[3]{{\left(e^{x}\right)}^{4} - {1}^{4}}}}{\left(\left(e^{x} + 1\right) \cdot x\right) \cdot \left({\left(e^{x}\right)}^{2} + 1 \cdot 1\right)}\]

   if -0.0001664903813288805 < x

   1. Initial program 60.0

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

   2. Taylor expanded around 0 0.5

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

   3. Simplified 0.5

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.66490381328880487 \cdot 10^{-4}:\\ \;\;\;\;\frac{\left(\sqrt[3]{{\left(e^{x}\right)}^{4} - {1}^{4}} \cdot \sqrt[3]{{\left(e^{x}\right)}^{4} - {1}^{4}}\right) \cdot \sqrt[3]{{\left(e^{x}\right)}^{4} - {1}^{4}}}{\left(\left(e^{x} + 1\right) \cdot x\right) \cdot \left({\left(e^{x}\right)}^{2} + 1 \cdot 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 
(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))