Average Error: 29.6 → 0.6
Time: 18.3s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.01718427484698143575814199834894679952413:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\frac{1}{2}, \left(a \cdot x\right) \cdot a, a\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.01718427484698143575814199834894679952413:\\
\;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\

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

\end{array}
double f(double a, double x) {
        double r77486 = a;
        double r77487 = x;
        double r77488 = r77486 * r77487;
        double r77489 = exp(r77488);
        double r77490 = 1.0;
        double r77491 = r77489 - r77490;
        return r77491;
}
↓
double f(double a, double x) {
        double r77492 = a;
        double r77493 = x;
        double r77494 = r77492 * r77493;
        double r77495 = -0.017184274846981436;
        bool r77496 = r77494 <= r77495;
        double r77497 = exp(r77494);
        double r77498 = 1.0;
        double r77499 = r77497 - r77498;
        double r77500 = exp(r77499);
        double r77501 = log(r77500);
        double r77502 = 0.5;
        double r77503 = r77494 * r77492;
        double r77504 = fma(r77502, r77503, r77492);
        double r77505 = r77493 * r77504;
        double r77506 = r77496 ? r77501 : r77505;
        return r77506;
}

Error

Bits error versus a

Bits error versus x

Target

Original29.6
Target0.2
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.017184274846981436

    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 -0.017184274846981436 < (* a x)

    1. Initial program 44.7

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

    2. Taylor expanded around 0 15.0

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

    3. Simplified11.9

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

    4. Taylor expanded around 0 9.0

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

    5. Simplified4.9

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

    7. Applied unpow24.9

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

    8. Applied associate-*r*0.9

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

    9. Simplified0.9

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.01718427484698143575814199834894679952413:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\frac{1}{2}, \left(a \cdot x\right) \cdot a, a\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 +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))