Kahan's exp quotient

Average Error: 39.8 → 0.3

Time: 15.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.00010786855983473736:\\ \;\;\;\;\frac{\frac{-1 + e^{3 \cdot x}}{\sqrt{e^{x}} \cdot \left(\left(e^{x} + 1\right) \cdot \sqrt{e^{x}}\right) + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right)\\ \end{array}\]

C

double f(double x) {
        double r3461132 = x;
        double r3461133 = exp(r3461132);
        double r3461134 = 1.0;
        double r3461135 = r3461133 - r3461134;
        double r3461136 = r3461135 / r3461132;
        return r3461136;
}

↓

double f(double x) {
        double r3461137 = x;
        double r3461138 = -0.00010786855983473736;
        bool r3461139 = r3461137 <= r3461138;
        double r3461140 = -1.0;
        double r3461141 = 3.0;
        double r3461142 = r3461141 * r3461137;
        double r3461143 = exp(r3461142);
        double r3461144 = r3461140 + r3461143;
        double r3461145 = exp(r3461137);
        double r3461146 = sqrt(r3461145);
        double r3461147 = 1.0;
        double r3461148 = r3461145 + r3461147;
        double r3461149 = r3461148 * r3461146;
        double r3461150 = r3461146 * r3461149;
        double r3461151 = r3461150 + r3461147;
        double r3461152 = r3461144 / r3461151;
        double r3461153 = r3461152 / r3461137;
        double r3461154 = 0.5;
        double r3461155 = 0.16666666666666666;
        double r3461156 = r3461137 * r3461155;
        double r3461157 = r3461154 + r3461156;
        double r3461158 = r3461137 * r3461157;
        double r3461159 = r3461147 + r3461158;
        double r3461160 = r3461139 ? r3461153 : r3461159;
        return r3461160;
}

Error

Bits error versus x

Target

Original	39.8
Target	39.0
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 < -0.00010786855983473736

   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 0.1

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

   5. Simplified 0.1

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

   7. Applied add-sqr-sqrt 0.1

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

   8. Applied associate-*r* 0.1

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

   if -0.00010786855983473736 < x

   1. Initial program 60.2

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

   2. Taylor expanded around 0 0.5

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

   3. Simplified 0.5

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00010786855983473736:\\ \;\;\;\;\frac{\frac{-1 + e^{3 \cdot x}}{\sqrt{e^{x}} \cdot \left(\left(e^{x} + 1\right) \cdot \sqrt{e^{x}}\right) + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(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))