expax (section 3.5)

Average Error: 29.3 → 0.3

Time: 20.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.07174886411196652880040147692852769978344:\\ \;\;\;\;\frac{\log \left(e^{e^{\left(3 \cdot a\right) \cdot x} - 1 \cdot \left(1 \cdot 1\right)}\right)}{\mathsf{fma}\left(e^{a \cdot x}, e^{a \cdot x}, \left(1 + e^{a \cdot x}\right) \cdot 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{6}, \left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right), \mathsf{fma}\left(\frac{1}{2}, \left(a \cdot x\right) \cdot \left(a \cdot x\right), a \cdot x\right)\right)\\ \end{array}\]

C

double f(double a, double x) {
        double r4972328 = a;
        double r4972329 = x;
        double r4972330 = r4972328 * r4972329;
        double r4972331 = exp(r4972330);
        double r4972332 = 1.0;
        double r4972333 = r4972331 - r4972332;
        return r4972333;
}

↓

double f(double a, double x) {
        double r4972334 = a;
        double r4972335 = x;
        double r4972336 = r4972334 * r4972335;
        double r4972337 = -0.07174886411196653;
        bool r4972338 = r4972336 <= r4972337;
        double r4972339 = 3.0;
        double r4972340 = r4972339 * r4972334;
        double r4972341 = r4972340 * r4972335;
        double r4972342 = exp(r4972341);
        double r4972343 = 1.0;
        double r4972344 = r4972343 * r4972343;
        double r4972345 = r4972343 * r4972344;
        double r4972346 = r4972342 - r4972345;
        double r4972347 = exp(r4972346);
        double r4972348 = log(r4972347);
        double r4972349 = exp(r4972336);
        double r4972350 = r4972343 + r4972349;
        double r4972351 = r4972350 * r4972343;
        double r4972352 = fma(r4972349, r4972349, r4972351);
        double r4972353 = r4972348 / r4972352;
        double r4972354 = 0.16666666666666666;
        double r4972355 = r4972336 * r4972336;
        double r4972356 = r4972336 * r4972355;
        double r4972357 = 0.5;
        double r4972358 = fma(r4972357, r4972355, r4972336);
        double r4972359 = fma(r4972354, r4972356, r4972358);
        double r4972360 = r4972338 ? r4972353 : r4972359;
        return r4972360;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.3
Target	0.2
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if (* a x) < -0.07174886411196653

   1. Initial program 0.0

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

   3. Applied flip3-- 0.0

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

   4. Simplified 0.0

      \[\leadsto \frac{\color{blue}{e^{x \cdot \mathsf{fma}\left(2, a, a\right)} - \left(1 \cdot 1\right) \cdot 1}}{e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right)}\]

   5. Simplified 0.0

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

   7. Applied add-log-exp 0.0

      \[\leadsto \frac{e^{x \cdot \mathsf{fma}\left(2, a, a\right)} - \color{blue}{\log \left(e^{\left(1 \cdot 1\right) \cdot 1}\right)}}{\mathsf{fma}\left(e^{x \cdot a}, e^{x \cdot a}, 1 \cdot \left(1 + e^{x \cdot a}\right)\right)}\]

   8. Applied add-log-exp 0.0

      \[\leadsto \frac{\color{blue}{\log \left(e^{e^{x \cdot \mathsf{fma}\left(2, a, a\right)}}\right)} - \log \left(e^{\left(1 \cdot 1\right) \cdot 1}\right)}{\mathsf{fma}\left(e^{x \cdot a}, e^{x \cdot a}, 1 \cdot \left(1 + e^{x \cdot a}\right)\right)}\]

   9. Applied diff-log 0.0

      \[\leadsto \frac{\color{blue}{\log \left(\frac{e^{e^{x \cdot \mathsf{fma}\left(2, a, a\right)}}}{e^{\left(1 \cdot 1\right) \cdot 1}}\right)}}{\mathsf{fma}\left(e^{x \cdot a}, e^{x \cdot a}, 1 \cdot \left(1 + e^{x \cdot a}\right)\right)}\]

   10. Simplified 0.0

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

   if -0.07174886411196653 < (* a x)

   1. Initial program 44.2

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

   2. Taylor expanded around 0 13.8

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

   3. Simplified 0.5

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.07174886411196652880040147692852769978344:\\ \;\;\;\;\frac{\log \left(e^{e^{\left(3 \cdot a\right) \cdot x} - 1 \cdot \left(1 \cdot 1\right)}\right)}{\mathsf{fma}\left(e^{a \cdot x}, e^{a \cdot x}, \left(1 + e^{a \cdot x}\right) \cdot 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{6}, \left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right), \mathsf{fma}\left(\frac{1}{2}, \left(a \cdot x\right) \cdot \left(a \cdot x\right), a \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (a x)
  :name "expax (section 3.5)"
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1.0 (+ (/ (* a x) 2.0) (/ (pow (* a x) 2.0) 6.0)))) (- (exp (* a x)) 1.0))

  (- (exp (* a x)) 1.0))