Average Error: 29.9 → 9.5
Time: 3.9s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -1.924193346987677885938052739847603511691 \cdot 10^{-7}:\\ \;\;\;\;\log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right) + \log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -1.924193346987677885938052739847603511691 \cdot 10^{-7}:\\
\;\;\;\;\log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right) + \log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\

\end{array}
double f(double a, double x) {
        double r93315 = a;
        double r93316 = x;
        double r93317 = r93315 * r93316;
        double r93318 = exp(r93317);
        double r93319 = 1.0;
        double r93320 = r93318 - r93319;
        return r93320;
}
↓
double f(double a, double x) {
        double r93321 = a;
        double r93322 = x;
        double r93323 = r93321 * r93322;
        double r93324 = -1.924193346987678e-07;
        bool r93325 = r93323 <= r93324;
        double r93326 = exp(r93323);
        double r93327 = 1.0;
        double r93328 = r93326 - r93327;
        double r93329 = exp(r93328);
        double r93330 = sqrt(r93329);
        double r93331 = log(r93330);
        double r93332 = r93331 + r93331;
        double r93333 = 0.5;
        double r93334 = 2.0;
        double r93335 = pow(r93321, r93334);
        double r93336 = pow(r93322, r93334);
        double r93337 = r93335 * r93336;
        double r93338 = 0.16666666666666666;
        double r93339 = 3.0;
        double r93340 = pow(r93321, r93339);
        double r93341 = pow(r93322, r93339);
        double r93342 = r93340 * r93341;
        double r93343 = fma(r93338, r93342, r93323);
        double r93344 = fma(r93333, r93337, r93343);
        double r93345 = r93325 ? r93332 : r93344;
        return r93345;
}

Error

Bits error versus a

Bits error versus x

Target

Original29.9
Target0.2
Herbie9.5
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -1.924193346987678e-07

    1. Initial program 0.2

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

    3. Applied add-log-exp0.2

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

    4. Applied add-log-exp0.2

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

    5. Applied diff-log0.2

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

    6. Simplified0.2

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

    8. Applied add-sqr-sqrt0.2

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

    9. Applied log-prod0.2

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

    if -1.924193346987678e-07 < (* a x)

    1. Initial program 44.7

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

    2. Taylor expanded around 0 14.2

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

    3. Simplified14.2

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

  4. Final simplification9.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -1.924193346987677885938052739847603511691 \cdot 10^{-7}:\\ \;\;\;\;\log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right) + \log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 +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))