Average Error: 29.6 → 8.0
Time: 16.1s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -173.653260833614780267453170381486415863:\\ \;\;\;\;{\left(e^{a}\right)}^{x} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, x, \left(x \cdot \mathsf{fma}\left(\frac{1}{2}, a \cdot a, \frac{1}{6} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot x\right)\right)\right)\right) \cdot x\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -173.653260833614780267453170381486415863:\\
\;\;\;\;{\left(e^{a}\right)}^{x} - 1\\

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

\end{array}
double f(double a, double x) {
        double r41214 = a;
        double r41215 = x;
        double r41216 = r41214 * r41215;
        double r41217 = exp(r41216);
        double r41218 = 1.0;
        double r41219 = r41217 - r41218;
        return r41219;
}
↓
double f(double a, double x) {
        double r41220 = a;
        double r41221 = x;
        double r41222 = r41220 * r41221;
        double r41223 = -173.65326083361478;
        bool r41224 = r41222 <= r41223;
        double r41225 = exp(r41220);
        double r41226 = pow(r41225, r41221);
        double r41227 = 1.0;
        double r41228 = r41226 - r41227;
        double r41229 = 0.5;
        double r41230 = r41220 * r41220;
        double r41231 = 0.16666666666666666;
        double r41232 = r41230 * r41222;
        double r41233 = r41231 * r41232;
        double r41234 = fma(r41229, r41230, r41233);
        double r41235 = r41221 * r41234;
        double r41236 = r41235 * r41221;
        double r41237 = fma(r41220, r41221, r41236);
        double r41238 = r41224 ? r41228 : r41237;
        return r41238;
}

Error

Bits error versus a

Bits error versus x

Target

Original29.6
Target0.2
Herbie8.0
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

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-log-exp14.3

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

    4. Applied exp-to-pow14.3

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

    if -173.65326083361478 < (* a x)

    1. Initial program 44.1

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

    3. Applied add-cbrt-cube44.2

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

    4. Simplified44.2

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

    5. Taylor expanded around 0 14.4

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

    6. Simplified11.7

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

    8. Applied sqr-pow11.7

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

    9. Applied associate-*l*7.8

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

    10. Simplified7.8

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

    12. Applied unpow37.8

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

    13. Applied associate-*l*4.9

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

  4. Final simplification8.0

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

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(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))