Average Error: 29.6 → 0.5
Time: 17.5s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -173.653260833614780267453170381486415863:\\ \;\;\;\;\sqrt[3]{{\left(e^{a \cdot x} - 1\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + x \cdot \left(\left(\left(a \cdot x\right) \cdot a\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6} + \frac{1}{2}\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -173.653260833614780267453170381486415863:\\
\;\;\;\;\sqrt[3]{{\left(e^{a \cdot x} - 1\right)}^{3}}\\

\mathbf{else}:\\
\;\;\;\;a \cdot x + x \cdot \left(\left(\left(a \cdot x\right) \cdot a\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6} + \frac{1}{2}\right)\right)\\

\end{array}
double f(double a, double x) {
        double r56108 = a;
        double r56109 = x;
        double r56110 = r56108 * r56109;
        double r56111 = exp(r56110);
        double r56112 = 1.0;
        double r56113 = r56111 - r56112;
        return r56113;
}
↓
double f(double a, double x) {
        double r56114 = a;
        double r56115 = x;
        double r56116 = r56114 * r56115;
        double r56117 = -173.65326083361478;
        bool r56118 = r56116 <= r56117;
        double r56119 = exp(r56116);
        double r56120 = 1.0;
        double r56121 = r56119 - r56120;
        double r56122 = 3.0;
        double r56123 = pow(r56121, r56122);
        double r56124 = cbrt(r56123);
        double r56125 = r56116 * r56114;
        double r56126 = 0.16666666666666666;
        double r56127 = r56116 * r56126;
        double r56128 = 0.5;
        double r56129 = r56127 + r56128;
        double r56130 = r56125 * r56129;
        double r56131 = r56115 * r56130;
        double r56132 = r56116 + r56131;
        double r56133 = r56118 ? r56124 : r56132;
        return r56133;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.6
Target0.2
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -173.65326083361478

    1. Initial program 0

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

    3. Applied add-cbrt-cube0

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

    4. Simplified0

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

    if -173.65326083361478 < (* a x)

    1. Initial program 44.1

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

    2. Taylor expanded around 0 14.4

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

    3. Simplified4.9

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

    5. Applied associate-*r*4.9

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

    6. Simplified0.7

      \[\leadsto x \cdot \left(a + \color{blue}{\left(\left(a \cdot x\right) \cdot a\right)} \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6} + \frac{1}{2}\right)\right)\]
    7. Using strategy rm

    8. Applied distribute-lft-in0.7

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

    9. Simplified0.7

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -173.653260833614780267453170381486415863:\\ \;\;\;\;\sqrt[3]{{\left(e^{a \cdot x} - 1\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + x \cdot \left(\left(\left(a \cdot x\right) \cdot a\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6} + \frac{1}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(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))