Kahan's exp quotient

Average Error: 40.1 → 0.3

Time: 43.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.00010990201414498517:\\ \;\;\;\;\frac{\frac{e^{\left(x + x\right) + x} + -1}{e^{x} \cdot \left(e^{x} + 1\right) + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(e^{x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)}\right)\\ \end{array}\]

C

double f(double x) {
        double r16206006 = x;
        double r16206007 = exp(r16206006);
        double r16206008 = 1.0;
        double r16206009 = r16206007 - r16206008;
        double r16206010 = r16206009 / r16206006;
        return r16206010;
}

↓

double f(double x) {
        double r16206011 = x;
        double r16206012 = -0.00010990201414498517;
        bool r16206013 = r16206011 <= r16206012;
        double r16206014 = r16206011 + r16206011;
        double r16206015 = r16206014 + r16206011;
        double r16206016 = exp(r16206015);
        double r16206017 = -1.0;
        double r16206018 = r16206016 + r16206017;
        double r16206019 = exp(r16206011);
        double r16206020 = 1.0;
        double r16206021 = r16206019 + r16206020;
        double r16206022 = r16206019 * r16206021;
        double r16206023 = r16206022 + r16206020;
        double r16206024 = r16206018 / r16206023;
        double r16206025 = r16206024 / r16206011;
        double r16206026 = 0.5;
        double r16206027 = 0.16666666666666666;
        double r16206028 = r16206027 * r16206011;
        double r16206029 = r16206026 + r16206028;
        double r16206030 = r16206011 * r16206029;
        double r16206031 = exp(r16206030);
        double r16206032 = log(r16206031);
        double r16206033 = r16206020 + r16206032;
        double r16206034 = r16206013 ? r16206025 : r16206033;
        return r16206034;
}

Error

Bits error versus x

Target

Original	40.1
Target	39.3
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 < -0.00010990201414498517

   1. Initial program 0.1

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

   3. Applied flip3-- 0.1

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

   4. Simplified 0.0

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

   5. Simplified 0.0

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

   if -0.00010990201414498517 < 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

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

   5. Applied add-log-exp 0.4

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00010990201414498517:\\ \;\;\;\;\frac{\frac{e^{\left(x + x\right) + x} + -1}{e^{x} \cdot \left(e^{x} + 1\right) + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(e^{x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(FPCore (x)
  :name "Kahan's exp quotient"

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

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