Average Error: 29.3 → 0.3
Time: 18.8s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.07174886411196652880040147692852769978344:\\ \;\;\;\;\frac{\log \left(e^{e^{\left(a \cdot x\right) \cdot 3} - \left(1 \cdot 1\right) \cdot 1}\right)}{\mathsf{fma}\left(e^{a \cdot x}, 1 + e^{a \cdot x}, 1 \cdot 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \left(a \cdot x\right) \cdot \left(a \cdot x\right), x \cdot \left(\left(\frac{1}{6} \cdot a\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) + a \cdot x\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.07174886411196652880040147692852769978344:\\
\;\;\;\;\frac{\log \left(e^{e^{\left(a \cdot x\right) \cdot 3} - \left(1 \cdot 1\right) \cdot 1}\right)}{\mathsf{fma}\left(e^{a \cdot x}, 1 + e^{a \cdot x}, 1 \cdot 1\right)}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \left(a \cdot x\right) \cdot \left(a \cdot x\right), x \cdot \left(\left(\frac{1}{6} \cdot a\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) + a \cdot x\right)\\

\end{array}
double f(double a, double x) {
        double r4905710 = a;
        double r4905711 = x;
        double r4905712 = r4905710 * r4905711;
        double r4905713 = exp(r4905712);
        double r4905714 = 1.0;
        double r4905715 = r4905713 - r4905714;
        return r4905715;
}
↓
double f(double a, double x) {
        double r4905716 = a;
        double r4905717 = x;
        double r4905718 = r4905716 * r4905717;
        double r4905719 = -0.07174886411196653;
        bool r4905720 = r4905718 <= r4905719;
        double r4905721 = 3.0;
        double r4905722 = r4905718 * r4905721;
        double r4905723 = exp(r4905722);
        double r4905724 = 1.0;
        double r4905725 = r4905724 * r4905724;
        double r4905726 = r4905725 * r4905724;
        double r4905727 = r4905723 - r4905726;
        double r4905728 = exp(r4905727);
        double r4905729 = log(r4905728);
        double r4905730 = exp(r4905718);
        double r4905731 = r4905724 + r4905730;
        double r4905732 = fma(r4905730, r4905731, r4905725);
        double r4905733 = r4905729 / r4905732;
        double r4905734 = 0.5;
        double r4905735 = r4905718 * r4905718;
        double r4905736 = 0.16666666666666666;
        double r4905737 = r4905736 * r4905716;
        double r4905738 = r4905737 * r4905735;
        double r4905739 = r4905717 * r4905738;
        double r4905740 = r4905739 + r4905718;
        double r4905741 = fma(r4905734, r4905735, r4905740);
        double r4905742 = r4905720 ? r4905733 : r4905741;
        return r4905742;
}

Error

Bits error versus a

Bits error versus x

Target

Original29.3
Target0.2
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.07174886411196653

    1. Initial program 0.0

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

    3. Applied flip3--0.0

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

    4. Simplified0.0

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

    5. Simplified0.0

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

    7. Applied add-log-exp0.0

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

    if -0.07174886411196653 < (* a x)

    1. Initial program 44.2

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

    2. Taylor expanded around 0 13.8

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

    3. Simplified0.5

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

    5. Applied distribute-lft-in0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.07174886411196652880040147692852769978344:\\ \;\;\;\;\frac{\log \left(e^{e^{\left(a \cdot x\right) \cdot 3} - \left(1 \cdot 1\right) \cdot 1}\right)}{\mathsf{fma}\left(e^{a \cdot x}, 1 + e^{a \cdot x}, 1 \cdot 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \left(a \cdot x\right) \cdot \left(a \cdot x\right), x \cdot \left(\left(\frac{1}{6} \cdot a\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) + a \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (a x)
  :name "expax (section 3.5)"
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1.0 (+ (/ (* a x) 2.0) (/ (pow (* a x) 2.0) 6.0)))) (- (exp (* a x)) 1.0))

  (- (exp (* a x)) 1.0))