Average Error: 30.4 → 3.2
Time: 20.0s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.01186149833516873981775763269297385704704:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(a + \left(\left(a \cdot a\right) \cdot \left(\left(x \cdot \frac{1}{6}\right) \cdot a + \frac{1}{2}\right)\right) \cdot x\right)\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.01186149833516873981775763269297385704704:\\
\;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(a + \left(\left(a \cdot a\right) \cdot \left(\left(x \cdot \frac{1}{6}\right) \cdot a + \frac{1}{2}\right)\right) \cdot x\right)\right)\right)\\

\end{array}
double f(double a, double x) {
        double r73424 = a;
        double r73425 = x;
        double r73426 = r73424 * r73425;
        double r73427 = exp(r73426);
        double r73428 = 1.0;
        double r73429 = r73427 - r73428;
        return r73429;
}
↓
double f(double a, double x) {
        double r73430 = a;
        double r73431 = x;
        double r73432 = r73430 * r73431;
        double r73433 = -0.01186149833516874;
        bool r73434 = r73432 <= r73433;
        double r73435 = exp(r73432);
        double r73436 = 1.0;
        double r73437 = r73435 - r73436;
        double r73438 = exp(r73437);
        double r73439 = log(r73438);
        double r73440 = r73430 * r73430;
        double r73441 = 0.16666666666666666;
        double r73442 = r73431 * r73441;
        double r73443 = r73442 * r73430;
        double r73444 = 0.5;
        double r73445 = r73443 + r73444;
        double r73446 = r73440 * r73445;
        double r73447 = r73446 * r73431;
        double r73448 = r73430 + r73447;
        double r73449 = r73431 * r73448;
        double r73450 = log1p(r73449);
        double r73451 = expm1(r73450);
        double r73452 = r73434 ? r73439 : r73451;
        return r73452;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.4
Target0.2
Herbie3.2
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.01186149833516874

    1. Initial program 0.0

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

    3. Applied add-log-exp0.0

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

    4. Applied add-log-exp0.0

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

    5. Applied diff-log0.0

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

    6. Simplified0.0

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

    if -0.01186149833516874 < (* a x)

    1. Initial program 45.0

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

    2. Taylor expanded around 0 14.9

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

    3. Simplified11.7

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \mathsf{fma}\left(x, \frac{1}{6} \cdot {a}^{3}, \frac{1}{2} \cdot {a}^{2}\right), a \cdot x\right)}\]
    4. Using strategy rm

    5. Applied expm1-log1p-u11.7

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left({x}^{2}, \mathsf{fma}\left(x, \frac{1}{6} \cdot {a}^{3}, \frac{1}{2} \cdot {a}^{2}\right), a \cdot x\right)\right)\right)}\]

    6. Simplified4.7

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(x \cdot \left(a + \left(\left(a \cdot a\right) \cdot \left(\left(x \cdot \frac{1}{6}\right) \cdot a + \frac{1}{2}\right)\right) \cdot x\right)\right)}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.01186149833516873981775763269297385704704:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(a + \left(\left(a \cdot a\right) \cdot \left(\left(x \cdot \frac{1}{6}\right) \cdot a + \frac{1}{2}\right)\right) \cdot x\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.10000000000000001) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

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