Average Error: 28.9 → 5.4
Time: 3.6s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -3.32894132256475227 \cdot 10^{-11}:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {\left(x \cdot a\right)}^{3}, a \cdot x\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -3.32894132256475227 \cdot 10^{-11}:\\
\;\;\;\;\log \left(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}, {\left(x \cdot a\right)}^{3}, a \cdot x\right)\right)\\

\end{array}
double f(double a, double x) {
        double r119588 = a;
        double r119589 = x;
        double r119590 = r119588 * r119589;
        double r119591 = exp(r119590);
        double r119592 = 1.0;
        double r119593 = r119591 - r119592;
        return r119593;
}
↓
double f(double a, double x) {
        double r119594 = a;
        double r119595 = x;
        double r119596 = r119594 * r119595;
        double r119597 = -3.328941322564752e-11;
        bool r119598 = r119596 <= r119597;
        double r119599 = exp(r119596);
        double r119600 = 1.0;
        double r119601 = r119599 - r119600;
        double r119602 = exp(r119601);
        double r119603 = log(r119602);
        double r119604 = 0.5;
        double r119605 = 2.0;
        double r119606 = pow(r119594, r119605);
        double r119607 = pow(r119595, r119605);
        double r119608 = r119606 * r119607;
        double r119609 = 0.16666666666666666;
        double r119610 = r119595 * r119594;
        double r119611 = 3.0;
        double r119612 = pow(r119610, r119611);
        double r119613 = fma(r119609, r119612, r119596);
        double r119614 = fma(r119604, r119608, r119613);
        double r119615 = r119598 ? r119603 : r119614;
        return r119615;
}

Error

Bits error versus a

Bits error versus x

Target

Original28.9
Target0.2
Herbie5.4
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -3.328941322564752e-11

    1. Initial program 0.6

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

    3. Applied add-log-exp0.6

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

    4. Applied add-log-exp0.6

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

    5. Applied diff-log0.6

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

    6. Simplified0.6

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

    if -3.328941322564752e-11 < (* a x)

    1. Initial program 44.0

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

    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

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

    5. Applied pow-prod-down8.0

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

    6. Simplified8.0

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

  4. Final simplification5.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -3.32894132256475227 \cdot 10^{-11}:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {\left(x \cdot a\right)}^{3}, a \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +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))