expax (section 3.5)

Average Error: 29.2 → 1.1

Time: 19.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -11614523509.551191:\\ \;\;\;\;\left(\sqrt[3]{\sqrt{e^{a \cdot x}} - 1} \cdot \left(\sqrt[3]{\sqrt{e^{a \cdot x}} - 1} \cdot \sqrt[3]{\sqrt{e^{a \cdot x}} - 1}\right)\right) \cdot \left(1 + \sqrt{e^{a \cdot x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \left(\left(x \cdot \frac{1}{48}\right) \cdot \left(a \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) + \left(\frac{1}{2} \cdot \left(a \cdot x\right) + \left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{8}\right)\right)\right)\\ \end{array}\]

C

double f(double a, double x) {
        double r4986185 = a;
        double r4986186 = x;
        double r4986187 = r4986185 * r4986186;
        double r4986188 = exp(r4986187);
        double r4986189 = 1.0;
        double r4986190 = r4986188 - r4986189;
        return r4986190;
}

↓

double f(double a, double x) {
        double r4986191 = a;
        double r4986192 = x;
        double r4986193 = r4986191 * r4986192;
        double r4986194 = -11614523509.551191;
        bool r4986195 = r4986193 <= r4986194;
        double r4986196 = exp(r4986193);
        double r4986197 = sqrt(r4986196);
        double r4986198 = 1.0;
        double r4986199 = r4986197 - r4986198;
        double r4986200 = cbrt(r4986199);
        double r4986201 = r4986200 * r4986200;
        double r4986202 = r4986200 * r4986201;
        double r4986203 = r4986198 + r4986197;
        double r4986204 = r4986202 * r4986203;
        double r4986205 = 0.020833333333333332;
        double r4986206 = r4986192 * r4986205;
        double r4986207 = r4986193 * r4986193;
        double r4986208 = r4986191 * r4986207;
        double r4986209 = r4986206 * r4986208;
        double r4986210 = 0.5;
        double r4986211 = r4986210 * r4986193;
        double r4986212 = 0.125;
        double r4986213 = r4986193 * r4986212;
        double r4986214 = r4986193 * r4986213;
        double r4986215 = r4986211 + r4986214;
        double r4986216 = r4986209 + r4986215;
        double r4986217 = r4986203 * r4986216;
        double r4986218 = r4986195 ? r4986204 : r4986217;
        return r4986218;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.2
Target	0.2
Herbie	1.1

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

Derivation

1. Split input into 2 regimes

2. if (* a x) < -11614523509.551191

   1. Initial program 0

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

   3. Applied *-un-lft-identity 0

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

   4. Applied add-sqr-sqrt 0

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

   5. Applied difference-of-squares 0

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

   7. Applied add-cube-cbrt 0

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

   if -11614523509.551191 < (* a x)

   1. Initial program 43.0

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

   3. Applied *-un-lft-identity 43.0

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

   4. Applied add-sqr-sqrt 43.0

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

   5. Applied difference-of-squares 43.0

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

   6. Taylor expanded around 0 14.5

      \[\leadsto \left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(\frac{1}{2} \cdot \left(a \cdot x\right) + \frac{1}{48} \cdot \left({a}^{3} \cdot {x}^{3}\right)\right)\right)}\]

   7. Simplified 1.6

      \[\leadsto \left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \color{blue}{\left(\left(\frac{1}{48} \cdot x\right) \cdot \left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot a\right) + \left(\left(a \cdot x\right) \cdot \frac{1}{2} + \left(\frac{1}{8} \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -11614523509.551191:\\ \;\;\;\;\left(\sqrt[3]{\sqrt{e^{a \cdot x}} - 1} \cdot \left(\sqrt[3]{\sqrt{e^{a \cdot x}} - 1} \cdot \sqrt[3]{\sqrt{e^{a \cdot x}} - 1}\right)\right) \cdot \left(1 + \sqrt{e^{a \cdot x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \left(\left(x \cdot \frac{1}{48}\right) \cdot \left(a \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) + \left(\frac{1}{2} \cdot \left(a \cdot x\right) + \left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{8}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (a x)
  :name "expax (section 3.5)"
  :herbie-expected 14

  :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))