Average Error: 29.6 → 0.4
Time: 4.5s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -1.751604029164171324843557453476705632056 \cdot 10^{-4}:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + \left(\frac{1}{2} \cdot {\left(a \cdot x\right)}^{2} + \frac{1}{6} \cdot {\left(a \cdot x\right)}^{3}\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -1.751604029164171324843557453476705632056 \cdot 10^{-4}:\\
\;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\

\mathbf{else}:\\
\;\;\;\;a \cdot x + \left(\frac{1}{2} \cdot {\left(a \cdot x\right)}^{2} + \frac{1}{6} \cdot {\left(a \cdot x\right)}^{3}\right)\\

\end{array}
double f(double a, double x) {
        double r110701 = a;
        double r110702 = x;
        double r110703 = r110701 * r110702;
        double r110704 = exp(r110703);
        double r110705 = 1.0;
        double r110706 = r110704 - r110705;
        return r110706;
}
↓
double f(double a, double x) {
        double r110707 = a;
        double r110708 = x;
        double r110709 = r110707 * r110708;
        double r110710 = -0.00017516040291641713;
        bool r110711 = r110709 <= r110710;
        double r110712 = exp(r110709);
        double r110713 = 1.0;
        double r110714 = r110712 - r110713;
        double r110715 = exp(r110714);
        double r110716 = log(r110715);
        double r110717 = 0.5;
        double r110718 = 2.0;
        double r110719 = pow(r110709, r110718);
        double r110720 = r110717 * r110719;
        double r110721 = 0.16666666666666666;
        double r110722 = 3.0;
        double r110723 = pow(r110709, r110722);
        double r110724 = r110721 * r110723;
        double r110725 = r110720 + r110724;
        double r110726 = r110709 + r110725;
        double r110727 = r110711 ? r110716 : r110726;
        return r110727;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.6
Target0.2
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) < -0.00017516040291641713

    1. Initial program 0.1

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

    3. Applied add-log-exp0.1

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

    4. Applied add-log-exp0.1

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

    5. Applied diff-log0.1

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

    6. Simplified0.1

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

    if -0.00017516040291641713 < (* a x)

    1. Initial program 44.9

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

    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. Simplified14.8

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

    5. Applied pow-prod-down4.7

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

    7. Applied distribute-lft-in4.7

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

    8. Simplified4.7

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

    9. Simplified0.6

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

    11. Applied associate-+l+0.6

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -1.751604029164171324843557453476705632056 \cdot 10^{-4}:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + \left(\frac{1}{2} \cdot {\left(a \cdot x\right)}^{2} + \frac{1}{6} \cdot {\left(a \cdot x\right)}^{3}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.10000000000000001) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

  (- (exp (* a x)) 1))