Average Error: 28.9 → 5.1
Time: 13.1s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -3.32894132256475227 \cdot 10^{-11}:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, a, \left(x \cdot \sqrt{\mathsf{fma}\left(\frac{1}{2}, a \cdot a, \left(\frac{1}{6} \cdot {a}^{3}\right) \cdot x\right)}\right) \cdot \left(x \cdot \sqrt{\mathsf{fma}\left(\frac{1}{2}, a \cdot a, \left(\frac{1}{6} \cdot {a}^{3}\right) \cdot x\right)}\right)\right)\\ \end{array}\]
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(x, a, \left(x \cdot \sqrt{\mathsf{fma}\left(\frac{1}{2}, a \cdot a, \left(\frac{1}{6} \cdot {a}^{3}\right) \cdot x\right)}\right) \cdot \left(x \cdot \sqrt{\mathsf{fma}\left(\frac{1}{2}, a \cdot a, \left(\frac{1}{6} \cdot {a}^{3}\right) \cdot x\right)}\right)\right)\\

\end{array}
double f(double a, double x) {
        double r159749 = a;
        double r159750 = x;
        double r159751 = r159749 * r159750;
        double r159752 = exp(r159751);
        double r159753 = 1.0;
        double r159754 = r159752 - r159753;
        return r159754;
}
↓
double f(double a, double x) {
        double r159755 = a;
        double r159756 = x;
        double r159757 = r159755 * r159756;
        double r159758 = -3.328941322564752e-11;
        bool r159759 = r159757 <= r159758;
        double r159760 = exp(r159757);
        double r159761 = 1.0;
        double r159762 = r159760 - r159761;
        double r159763 = exp(r159762);
        double r159764 = log(r159763);
        double r159765 = 0.5;
        double r159766 = r159755 * r159755;
        double r159767 = 0.16666666666666666;
        double r159768 = 3.0;
        double r159769 = pow(r159755, r159768);
        double r159770 = r159767 * r159769;
        double r159771 = r159770 * r159756;
        double r159772 = fma(r159765, r159766, r159771);
        double r159773 = sqrt(r159772);
        double r159774 = r159756 * r159773;
        double r159775 = r159774 * r159774;
        double r159776 = fma(r159756, r159755, r159775);
        double r159777 = r159759 ? r159764 : r159776;
        return r159777;
}

Error

Bits error versus a

Bits error versus x

Target

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

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. Simplified11.1

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

    5. Applied add-sqr-sqrt11.1

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

    6. Applied unswap-sqr7.5

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

    7. Simplified7.5

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

    8. Simplified7.5

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

  4. Final simplification5.1

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

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))