Average Error: 30.0 → 0.8
Time: 3.2s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -8.885468282259285557368902561548723584295 \cdot 10^{-9}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot a\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -8.885468282259285557368902561548723584295 \cdot 10^{-9}:\\
\;\;\;\;e^{a \cdot x} - 1\\

\mathbf{else}:\\
\;\;\;\;x \cdot a\\

\end{array}
double f(double a, double x) {
        double r119117 = a;
        double r119118 = x;
        double r119119 = r119117 * r119118;
        double r119120 = exp(r119119);
        double r119121 = 1.0;
        double r119122 = r119120 - r119121;
        return r119122;
}
↓
double f(double a, double x) {
        double r119123 = a;
        double r119124 = x;
        double r119125 = r119123 * r119124;
        double r119126 = -8.885468282259286e-09;
        bool r119127 = r119125 <= r119126;
        double r119128 = exp(r119125);
        double r119129 = 1.0;
        double r119130 = r119128 - r119129;
        double r119131 = r119124 * r119123;
        double r119132 = r119127 ? r119130 : r119131;
        return r119132;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.0
Target0.2
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -8.885468282259286e-09

    1. Initial program 0.2

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

    if -8.885468282259286e-09 < (* a x)

    1. Initial program 44.9

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

    2. Taylor expanded around 0 14.5

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

    3. Simplified14.5

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

    4. Taylor expanded around 0 8.2

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

    5. Simplified4.4

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

    6. Taylor expanded around 0 1.1

      \[\leadsto \color{blue}{a \cdot x}\]

    7. Simplified1.1

      \[\leadsto \color{blue}{x \cdot a}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -8.885468282259285557368902561548723584295 \cdot 10^{-9}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot a\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(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))