expax (section 3.5)

Average Error: 29.8 → 9.8

Time: 4.0s

Precision: 64

Math

\[e^{a \cdot x} - 1\]

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -8.293963492047654929060613942291516367562 \cdot 10^{-20}:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\ \end{array}\]

C

double f(double a, double x) {
        double r87579 = a;
        double r87580 = x;
        double r87581 = r87579 * r87580;
        double r87582 = exp(r87581);
        double r87583 = 1.0;
        double r87584 = r87582 - r87583;
        return r87584;
}

↓

double f(double a, double x) {
        double r87585 = a;
        double r87586 = x;
        double r87587 = r87585 * r87586;
        double r87588 = -8.293963492047655e-20;
        bool r87589 = r87587 <= r87588;
        double r87590 = exp(r87587);
        double r87591 = 1.0;
        double r87592 = r87590 - r87591;
        double r87593 = exp(r87592);
        double r87594 = log(r87593);
        double r87595 = 0.5;
        double r87596 = 2.0;
        double r87597 = pow(r87585, r87596);
        double r87598 = pow(r87586, r87596);
        double r87599 = r87597 * r87598;
        double r87600 = 0.16666666666666666;
        double r87601 = 3.0;
        double r87602 = pow(r87585, r87601);
        double r87603 = pow(r87586, r87601);
        double r87604 = r87602 * r87603;
        double r87605 = fma(r87600, r87604, r87587);
        double r87606 = fma(r87595, r87599, r87605);
        double r87607 = r87589 ? r87594 : r87606;
        return r87607;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.8
Target	0.1
Herbie	9.8

\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (* a x) < -8.293963492047655e-20

   1. Initial program 1.9

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

   3. Applied add-log-exp 1.9

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

   4. Applied add-log-exp 1.9

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

   5. Applied diff-log 2.0

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

   6. Simplified 1.9

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

   if -8.293963492047655e-20 < (* a x)

   1. Initial program 45.3

      \[e^{a \cdot x} - 1\]

   2. Taylor expanded around 0 14.2

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(\frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right) + a \cdot x\right)}\]

   3. Simplified 14.2

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

4. Final simplification 9.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -8.293963492047654929060613942291516367562 \cdot 10^{-20}:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

  (- (exp (* a x)) 1))