Average Error: 28.8 → 9.2
Time: 3.7s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -2.2100171872579915 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}, \sqrt[3]{e^{a \cdot x}}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -2.2100171872579915 \cdot 10^{-7}:\\
\;\;\;\;\mathsf{fma}\left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}, \sqrt[3]{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 r93541 = a;
        double r93542 = x;
        double r93543 = r93541 * r93542;
        double r93544 = exp(r93543);
        double r93545 = 1.0;
        double r93546 = r93544 - r93545;
        return r93546;
}
↓
double f(double a, double x) {
        double r93547 = a;
        double r93548 = x;
        double r93549 = r93547 * r93548;
        double r93550 = -2.2100171872579915e-07;
        bool r93551 = r93549 <= r93550;
        double r93552 = exp(r93549);
        double r93553 = cbrt(r93552);
        double r93554 = r93553 * r93553;
        double r93555 = 1.0;
        double r93556 = -r93555;
        double r93557 = fma(r93554, r93553, r93556);
        double r93558 = 0.5;
        double r93559 = 2.0;
        double r93560 = pow(r93547, r93559);
        double r93561 = pow(r93548, r93559);
        double r93562 = r93560 * r93561;
        double r93563 = 0.16666666666666666;
        double r93564 = 3.0;
        double r93565 = pow(r93547, r93564);
        double r93566 = pow(r93548, r93564);
        double r93567 = r93565 * r93566;
        double r93568 = fma(r93563, r93567, r93549);
        double r93569 = fma(r93558, r93562, r93568);
        double r93570 = r93551 ? r93557 : r93569;
        return r93570;
}

Error

Bits error versus a

Bits error versus x

Target

Original28.8
Target0.2
Herbie9.2
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -2.2100171872579915e-07

    1. Initial program 0.2

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

    3. Applied add-cube-cbrt0.2

      \[\leadsto \color{blue}{\left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}\right) \cdot \sqrt[3]{e^{a \cdot x}}} - 1\]

    4. Applied fma-neg0.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}, \sqrt[3]{e^{a \cdot x}}, -1\right)}\]

    if -2.2100171872579915e-07 < (* a x)

    1. Initial program 44.4

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

    2. Taylor expanded around 0 14.0

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

    3. Simplified14.0

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

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -2.2100171872579915 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}, \sqrt[3]{e^{a \cdot x}}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 +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))