Average Error: 29.5 → 0.4
Time: 13.0s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -6.158968166252154678263867892962224459552 \cdot 10^{-6}:\\ \;\;\;\;\sqrt[3]{{\left(e^{a \cdot x} - 1\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\frac{1}{2}, \left(a \cdot x\right) \cdot {a}^{\left(\frac{2}{2}\right)}, a\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -6.158968166252154678263867892962224459552 \cdot 10^{-6}:\\
\;\;\;\;\sqrt[3]{{\left(e^{a \cdot x} - 1\right)}^{3}}\\

\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 r113660 = a;
        double r113661 = x;
        double r113662 = r113660 * r113661;
        double r113663 = exp(r113662);
        double r113664 = 1.0;
        double r113665 = r113663 - r113664;
        return r113665;
}
↓
double f(double a, double x) {
        double r113666 = a;
        double r113667 = x;
        double r113668 = r113666 * r113667;
        double r113669 = -6.158968166252155e-06;
        bool r113670 = r113668 <= r113669;
        double r113671 = exp(r113668);
        double r113672 = 1.0;
        double r113673 = r113671 - r113672;
        double r113674 = 3.0;
        double r113675 = pow(r113673, r113674);
        double r113676 = cbrt(r113675);
        double r113677 = 0.5;
        double r113678 = 2.0;
        double r113679 = r113678 / r113678;
        double r113680 = pow(r113666, r113679);
        double r113681 = r113668 * r113680;
        double r113682 = fma(r113677, r113681, r113666);
        double r113683 = r113667 * r113682;
        double r113684 = r113670 ? r113676 : r113683;
        return r113684;
}

Error

Bits error versus a

Bits error versus x

Target

Original29.5
Target0.1
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) < -6.158968166252155e-06

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube0.1

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

    4. Simplified0.1

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

    if -6.158968166252155e-06 < (* a x)

    1. Initial program 44.7

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

    2. Taylor expanded around 0 14.8

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

    3. Simplified11.5

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

    4. Taylor expanded around 0 8.4

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

    5. Simplified4.8

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

    7. Applied sqr-pow4.8

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

    8. Applied associate-*r*0.6

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

    9. Simplified0.6

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -6.158968166252154678263867892962224459552 \cdot 10^{-6}:\\ \;\;\;\;\sqrt[3]{{\left(e^{a \cdot x} - 1\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\frac{1}{2}, \left(a \cdot x\right) \cdot {a}^{\left(\frac{2}{2}\right)}, a\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +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))