Kahan's exp quotient

Average Error: 40.0 → 0.4

Time: 2.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.649035260432496941170044113533776908298 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{1}{36}, {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, x, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{36}, {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, x, 1\right)\right)\right) \cdot \sqrt[3]{\mathsf{fma}\left(\frac{1}{6}, {x}^{2}, \mathsf{fma}\left(\frac{1}{2}, x, 1\right)\right)}\\ \end{array}\]

C

double f(double x) {
        double r111570 = x;
        double r111571 = exp(r111570);
        double r111572 = 1.0;
        double r111573 = r111571 - r111572;
        double r111574 = r111573 / r111570;
        return r111574;
}

↓

double f(double x) {
        double r111575 = x;
        double r111576 = -0.0001649035260432497;
        bool r111577 = r111575 <= r111576;
        double r111578 = 1.0;
        double r111579 = -r111578;
        double r111580 = r111575 + r111575;
        double r111581 = exp(r111580);
        double r111582 = fma(r111579, r111578, r111581);
        double r111583 = exp(r111575);
        double r111584 = r111583 + r111578;
        double r111585 = r111582 / r111584;
        double r111586 = r111585 / r111575;
        double r111587 = 0.027777777777777776;
        double r111588 = 2.0;
        double r111589 = pow(r111575, r111588);
        double r111590 = 0.16666666666666666;
        double r111591 = 1.0;
        double r111592 = fma(r111590, r111575, r111591);
        double r111593 = fma(r111587, r111589, r111592);
        double r111594 = r111593 * r111593;
        double r111595 = 0.5;
        double r111596 = fma(r111595, r111575, r111591);
        double r111597 = fma(r111590, r111589, r111596);
        double r111598 = cbrt(r111597);
        double r111599 = r111594 * r111598;
        double r111600 = r111577 ? r111586 : r111599;
        return r111600;
}

Error

Bits error versus x

Target

Original	40.0
Target	40.5
Herbie	0.4

\[\begin{array}{l} \mathbf{if}\;x \lt 1 \land x \gt -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if x < -0.0001649035260432497

   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

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

   if -0.0001649035260432497 < x

   1. Initial program 60.1

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

   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

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

   5. Applied add-cube-cbrt 0.6

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

   6. Taylor expanded around 0 0.6

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

   7. Simplified 0.6

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

   8. Taylor expanded around 0 0.5

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

   9. Simplified 0.5

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.649035260432496941170044113533776908298 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{1}{36}, {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, x, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{36}, {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, x, 1\right)\right)\right) \cdot \sqrt[3]{\mathsf{fma}\left(\frac{1}{6}, {x}^{2}, \mathsf{fma}\left(\frac{1}{2}, x, 1\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 +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))