expax (section 3.5)

Average Error: 29.1 → 0.4

Time: 20.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.006625433188359259630006992125572651275434:\\ \;\;\;\;\frac{e^{\left(a \cdot x\right) \cdot 3} - \left(1 \cdot 1\right) \cdot 1}{\mathsf{fma}\left(e^{a \cdot x}, e^{a \cdot x} + 1, 1 \cdot 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \left(a \cdot x\right) \cdot \left(a \cdot x\right), x \cdot \left(a + \left(a \cdot \frac{1}{6}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right)\right)\\ \end{array}\]

C

double f(double a, double x) {
        double r6846161 = a;
        double r6846162 = x;
        double r6846163 = r6846161 * r6846162;
        double r6846164 = exp(r6846163);
        double r6846165 = 1.0;
        double r6846166 = r6846164 - r6846165;
        return r6846166;
}

↓

double f(double a, double x) {
        double r6846167 = a;
        double r6846168 = x;
        double r6846169 = r6846167 * r6846168;
        double r6846170 = -0.00662543318835926;
        bool r6846171 = r6846169 <= r6846170;
        double r6846172 = 3.0;
        double r6846173 = r6846169 * r6846172;
        double r6846174 = exp(r6846173);
        double r6846175 = 1.0;
        double r6846176 = r6846175 * r6846175;
        double r6846177 = r6846176 * r6846175;
        double r6846178 = r6846174 - r6846177;
        double r6846179 = exp(r6846169);
        double r6846180 = r6846179 + r6846175;
        double r6846181 = fma(r6846179, r6846180, r6846176);
        double r6846182 = r6846178 / r6846181;
        double r6846183 = 0.5;
        double r6846184 = r6846169 * r6846169;
        double r6846185 = 0.16666666666666666;
        double r6846186 = r6846167 * r6846185;
        double r6846187 = r6846186 * r6846184;
        double r6846188 = r6846167 + r6846187;
        double r6846189 = r6846168 * r6846188;
        double r6846190 = fma(r6846183, r6846184, r6846189);
        double r6846191 = r6846171 ? r6846182 : r6846190;
        return r6846191;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.1
Target	0.3
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if (* a x) < -0.00662543318835926

   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

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

   5. Simplified 0.0

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

   if -0.00662543318835926 < (* a x)

   1. Initial program 43.8

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

   2. Taylor expanded around 0 14.5

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

   3. Simplified 0.6

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.006625433188359259630006992125572651275434:\\ \;\;\;\;\frac{e^{\left(a \cdot x\right) \cdot 3} - \left(1 \cdot 1\right) \cdot 1}{\mathsf{fma}\left(e^{a \cdot x}, e^{a \cdot x} + 1, 1 \cdot 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \left(a \cdot x\right) \cdot \left(a \cdot x\right), x \cdot \left(a + \left(a \cdot \frac{1}{6}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 +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))