expax (section 3.5)

Average Error: 29.6 → 0.3

Time: 24.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.000117228758051702:\\ \;\;\;\;\frac{e^{a \cdot x} - 1}{\sqrt{e^{a \cdot x} + 1}} \cdot \frac{e^{a \cdot x} + 1}{\sqrt{e^{a \cdot x} + 1}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + \left(\left(\left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) \cdot \frac{1}{6} + \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \frac{1}{2}\right)\\ \end{array}\]

C

double f(double a, double x) {
        double r4053881 = a;
        double r4053882 = x;
        double r4053883 = r4053881 * r4053882;
        double r4053884 = exp(r4053883);
        double r4053885 = 1.0;
        double r4053886 = r4053884 - r4053885;
        return r4053886;
}

↓

double f(double a, double x) {
        double r4053887 = a;
        double r4053888 = x;
        double r4053889 = r4053887 * r4053888;
        double r4053890 = -0.000117228758051702;
        bool r4053891 = r4053889 <= r4053890;
        double r4053892 = exp(r4053889);
        double r4053893 = 1.0;
        double r4053894 = r4053892 - r4053893;
        double r4053895 = r4053892 + r4053893;
        double r4053896 = sqrt(r4053895);
        double r4053897 = r4053894 / r4053896;
        double r4053898 = r4053895 / r4053896;
        double r4053899 = r4053897 * r4053898;
        double r4053900 = r4053889 * r4053889;
        double r4053901 = r4053889 * r4053900;
        double r4053902 = 0.16666666666666666;
        double r4053903 = r4053901 * r4053902;
        double r4053904 = 0.5;
        double r4053905 = r4053900 * r4053904;
        double r4053906 = r4053903 + r4053905;
        double r4053907 = r4053889 + r4053906;
        double r4053908 = r4053891 ? r4053899 : r4053907;
        return r4053908;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.6
Target	0.1
Herbie	0.3

\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (* a x) < -0.000117228758051702

   1. Initial program 0.0

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

   3. Applied flip-- 0.0

      \[\leadsto \color{blue}{\frac{e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1}{e^{a \cdot x} + 1}}\]

   4. Simplified 0.0

      \[\leadsto \frac{\color{blue}{e^{a \cdot x} \cdot e^{a \cdot x} + -1}}{e^{a \cdot x} + 1}\]
   5. Using strategy rm

   6. Applied add-sqr-sqrt 0.0

      \[\leadsto \frac{e^{a \cdot x} \cdot e^{a \cdot x} + -1}{\color{blue}{\sqrt{e^{a \cdot x} + 1} \cdot \sqrt{e^{a \cdot x} + 1}}}\]

   7. Applied difference-of-sqr--1 0.0

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

   8. Applied times-frac 0.0

      \[\leadsto \color{blue}{\frac{e^{a \cdot x} + 1}{\sqrt{e^{a \cdot x} + 1}} \cdot \frac{e^{a \cdot x} - 1}{\sqrt{e^{a \cdot x} + 1}}}\]

   if -0.000117228758051702 < (* a x)

   1. Initial program 44.3

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

   2. Taylor expanded around 0 13.7

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

   3. Simplified 0.4

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.000117228758051702:\\ \;\;\;\;\frac{e^{a \cdot x} - 1}{\sqrt{e^{a \cdot x} + 1}} \cdot \frac{e^{a \cdot x} + 1}{\sqrt{e^{a \cdot x} + 1}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + \left(\left(\left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) \cdot \frac{1}{6} + \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 
(FPCore (a x)
  :name "expax (section 3.5)"

  :herbie-target
  (if (< (fabs (* a x)) 1/10) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

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