Average Error: 29.6 → 0.9
Time: 13.1s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -2.126354662771189659841580832377871956851 \cdot 10^{-4}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -2.126354662771189659841580832377871956851 \cdot 10^{-4}:\\
\;\;\;\;e^{a \cdot x} - 1\\

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

\end{array}
double f(double a, double x) {
        double r93408 = a;
        double r93409 = x;
        double r93410 = r93408 * r93409;
        double r93411 = exp(r93410);
        double r93412 = 1.0;
        double r93413 = r93411 - r93412;
        return r93413;
}
↓
double f(double a, double x) {
        double r93414 = a;
        double r93415 = x;
        double r93416 = r93414 * r93415;
        double r93417 = -0.00021263546627711897;
        bool r93418 = r93416 <= r93417;
        double r93419 = exp(r93416);
        double r93420 = 1.0;
        double r93421 = r93419 - r93420;
        double r93422 = r93418 ? r93421 : r93416;
        return r93422;
}

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
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.00021263546627711897

    1. Initial program 0.1

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

    if -0.00021263546627711897 < (* a x)

    1. Initial program 44.0

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

    2. Taylor expanded around 0 14.3

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

    3. Simplified14.3

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

    4. Taylor expanded around 0 8.3

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

    5. Simplified8.3

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

    6. Taylor expanded around 0 1.3

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -2.126354662771189659841580832377871956851 \cdot 10^{-4}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(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))