Average Error: 29.1 → 0.4
Time: 12.1s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.001614619952117295458710044542272044054698:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\frac{1}{2}, \left(a \cdot x\right) \cdot {a}^{\left(\frac{2}{2}\right)}, a\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.001614619952117295458710044542272044054698:\\
\;\;\;\;\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}^{\left(\frac{2}{2}\right)}, a\right)\\

\end{array}
double f(double a, double x) {
        double r129359 = a;
        double r129360 = x;
        double r129361 = r129359 * r129360;
        double r129362 = exp(r129361);
        double r129363 = 1.0;
        double r129364 = r129362 - r129363;
        return r129364;
}
↓
double f(double a, double x) {
        double r129365 = a;
        double r129366 = x;
        double r129367 = r129365 * r129366;
        double r129368 = -0.0016146199521172955;
        bool r129369 = r129367 <= r129368;
        double r129370 = exp(r129367);
        double r129371 = 1.0;
        double r129372 = r129370 - r129371;
        double r129373 = exp(r129372);
        double r129374 = log(r129373);
        double r129375 = 0.5;
        double r129376 = 2.0;
        double r129377 = r129376 / r129376;
        double r129378 = pow(r129365, r129377);
        double r129379 = r129367 * r129378;
        double r129380 = fma(r129375, r129379, r129365);
        double r129381 = r129366 * r129380;
        double r129382 = r129369 ? r129374 : r129381;
        return r129382;
}

Error

Bits error versus a

Bits error versus x

Target

Original29.1
Target0.2
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.0016146199521172955

    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 -0.0016146199521172955 < (* a x)

    1. Initial program 43.6

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

    2. Taylor expanded around 0 13.7

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

    3. Simplified10.6

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

    4. Taylor expanded around 0 7.8

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

    5. Simplified4.5

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

    7. Applied sqr-pow4.5

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

    8. Applied associate-*r*0.7

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

    9. Simplified0.7

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.001614619952117295458710044542272044054698:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\frac{1}{2}, \left(a \cdot x\right) \cdot {a}^{\left(\frac{2}{2}\right)}, a\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 +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))