Average Error: 29.5 → 0.3
Time: 40.4s
Precision: 64
\[e^{a \cdot x} - 1.0\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.0009957798500582489:\\ \;\;\;\;\left(\sqrt[3]{\log \left(e^{e^{a \cdot x} - 1.0}\right)} \cdot \sqrt[3]{\log \left(e^{e^{a \cdot x} - 1.0}\right)}\right) \cdot \sqrt[3]{\log \left(\sqrt{e^{e^{a \cdot x} - 1.0}}\right) + \log \left(\sqrt{e^{e^{a \cdot x} - 1.0}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right), \frac{1}{2}, \mathsf{fma}\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right), \frac{1}{6}, a \cdot x\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1.0
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.0009957798500582489:\\
\;\;\;\;\left(\sqrt[3]{\log \left(e^{e^{a \cdot x} - 1.0}\right)} \cdot \sqrt[3]{\log \left(e^{e^{a \cdot x} - 1.0}\right)}\right) \cdot \sqrt[3]{\log \left(\sqrt{e^{e^{a \cdot x} - 1.0}}\right) + \log \left(\sqrt{e^{e^{a \cdot x} - 1.0}}\right)}\\

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

\end{array}
double f(double a, double x) {
        double r5355256 = a;
        double r5355257 = x;
        double r5355258 = r5355256 * r5355257;
        double r5355259 = exp(r5355258);
        double r5355260 = 1.0;
        double r5355261 = r5355259 - r5355260;
        return r5355261;
}
↓
double f(double a, double x) {
        double r5355262 = a;
        double r5355263 = x;
        double r5355264 = r5355262 * r5355263;
        double r5355265 = -0.0009957798500582489;
        bool r5355266 = r5355264 <= r5355265;
        double r5355267 = exp(r5355264);
        double r5355268 = 1.0;
        double r5355269 = r5355267 - r5355268;
        double r5355270 = exp(r5355269);
        double r5355271 = log(r5355270);
        double r5355272 = cbrt(r5355271);
        double r5355273 = r5355272 * r5355272;
        double r5355274 = sqrt(r5355270);
        double r5355275 = log(r5355274);
        double r5355276 = r5355275 + r5355275;
        double r5355277 = cbrt(r5355276);
        double r5355278 = r5355273 * r5355277;
        double r5355279 = r5355264 * r5355264;
        double r5355280 = 0.5;
        double r5355281 = r5355279 * r5355264;
        double r5355282 = 0.16666666666666666;
        double r5355283 = fma(r5355281, r5355282, r5355264);
        double r5355284 = fma(r5355279, r5355280, r5355283);
        double r5355285 = r5355266 ? r5355278 : r5355284;
        return r5355285;
}

Error

Bits error versus a

Bits error versus x

Target

Original29.5
Target0.2
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1.0 + \left(\frac{a \cdot x}{2.0} + \frac{{\left(a \cdot x\right)}^{2.0}}{6.0}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1.0\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.0009957798500582489

    1. Initial program 0.0

      \[e^{a \cdot x} - 1.0\]
    2. Using strategy rm

    3. Applied add-log-exp0.0

      \[\leadsto e^{a \cdot x} - \color{blue}{\log \left(e^{1.0}\right)}\]

    4. Applied add-log-exp0.0

      \[\leadsto \color{blue}{\log \left(e^{e^{a \cdot x}}\right)} - \log \left(e^{1.0}\right)\]

    5. Applied diff-log0.0

      \[\leadsto \color{blue}{\log \left(\frac{e^{e^{a \cdot x}}}{e^{1.0}}\right)}\]

    6. Simplified0.0

      \[\leadsto \log \color{blue}{\left(e^{e^{a \cdot x} - 1.0}\right)}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt0.0

      \[\leadsto \color{blue}{\left(\sqrt[3]{\log \left(e^{e^{a \cdot x} - 1.0}\right)} \cdot \sqrt[3]{\log \left(e^{e^{a \cdot x} - 1.0}\right)}\right) \cdot \sqrt[3]{\log \left(e^{e^{a \cdot x} - 1.0}\right)}}\]
    9. Using strategy rm

    10. Applied add-sqr-sqrt0.0

      \[\leadsto \left(\sqrt[3]{\log \left(e^{e^{a \cdot x} - 1.0}\right)} \cdot \sqrt[3]{\log \left(e^{e^{a \cdot x} - 1.0}\right)}\right) \cdot \sqrt[3]{\log \color{blue}{\left(\sqrt{e^{e^{a \cdot x} - 1.0}} \cdot \sqrt{e^{e^{a \cdot x} - 1.0}}\right)}}\]

    11. Applied log-prod0.0

      \[\leadsto \left(\sqrt[3]{\log \left(e^{e^{a \cdot x} - 1.0}\right)} \cdot \sqrt[3]{\log \left(e^{e^{a \cdot x} - 1.0}\right)}\right) \cdot \sqrt[3]{\color{blue}{\log \left(\sqrt{e^{e^{a \cdot x} - 1.0}}\right) + \log \left(\sqrt{e^{e^{a \cdot x} - 1.0}}\right)}}\]

    if -0.0009957798500582489 < (* a x)

    1. Initial program 44.8

      \[e^{a \cdot x} - 1.0\]

    2. Taylor expanded around 0 14.5

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

    3. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.0009957798500582489:\\ \;\;\;\;\left(\sqrt[3]{\log \left(e^{e^{a \cdot x} - 1.0}\right)} \cdot \sqrt[3]{\log \left(e^{e^{a \cdot x} - 1.0}\right)}\right) \cdot \sqrt[3]{\log \left(\sqrt{e^{e^{a \cdot x} - 1.0}}\right) + \log \left(\sqrt{e^{e^{a \cdot x} - 1.0}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right), \frac{1}{2}, \mathsf{fma}\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right), \frac{1}{6}, a \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(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))