Average Error: 29.7 → 5.3
Time: 12.2s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.00565259617492738353:\\ \;\;\;\;\frac{e^{2 \cdot \left(x \cdot a\right)} - 1 \cdot 1}{e^{a \cdot x} + 1}\\ \mathbf{elif}\;a \cdot x \le 6.429099761039471 \cdot 10^{-102}:\\ \;\;\;\;\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + x \cdot \left(\frac{1}{2} \cdot {a}^{2} + \left(\frac{1}{6} \cdot {a}^{3}\right) \cdot x\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.00565259617492738353:\\
\;\;\;\;\frac{e^{2 \cdot \left(x \cdot a\right)} - 1 \cdot 1}{e^{a \cdot x} + 1}\\

\mathbf{elif}\;a \cdot x \le 6.429099761039471 \cdot 10^{-102}:\\
\;\;\;\;\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x\\

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

\end{array}
double f(double a, double x) {
        double r101970 = a;
        double r101971 = x;
        double r101972 = r101970 * r101971;
        double r101973 = exp(r101972);
        double r101974 = 1.0;
        double r101975 = r101973 - r101974;
        return r101975;
}
↓
double f(double a, double x) {
        double r101976 = a;
        double r101977 = x;
        double r101978 = r101976 * r101977;
        double r101979 = -0.0056525961749273835;
        bool r101980 = r101978 <= r101979;
        double r101981 = 2.0;
        double r101982 = r101977 * r101976;
        double r101983 = r101981 * r101982;
        double r101984 = exp(r101983);
        double r101985 = 1.0;
        double r101986 = r101985 * r101985;
        double r101987 = r101984 - r101986;
        double r101988 = exp(r101978);
        double r101989 = r101988 + r101985;
        double r101990 = r101987 / r101989;
        double r101991 = 6.429099761039471e-102;
        bool r101992 = r101978 <= r101991;
        double r101993 = 0.5;
        double r101994 = pow(r101976, r101981);
        double r101995 = pow(r101977, r101981);
        double r101996 = r101994 * r101995;
        double r101997 = r101993 * r101996;
        double r101998 = r101997 + r101978;
        double r101999 = r101993 * r101994;
        double r102000 = 0.16666666666666666;
        double r102001 = 3.0;
        double r102002 = pow(r101976, r102001);
        double r102003 = r102000 * r102002;
        double r102004 = r102003 * r101977;
        double r102005 = r101999 + r102004;
        double r102006 = r101977 * r102005;
        double r102007 = r101976 + r102006;
        double r102008 = r101977 * r102007;
        double r102009 = r101992 ? r101998 : r102008;
        double r102010 = r101980 ? r101990 : r102009;
        return r102010;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.7
Target0.2
Herbie5.3
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* a x) < -0.0056525961749273835

    1. Initial program 0.0

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

    3. Applied flip--0.0

      \[\leadsto \color{blue}{\frac{e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1}{e^{a \cdot x} + 1}}\]

    4. Simplified0.0

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

    if -0.0056525961749273835 < (* a x) < 6.429099761039471e-102

    1. Initial program 42.7

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

    2. Taylor expanded around 0 10.6

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

    3. Simplified5.4

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

    4. Taylor expanded around 0 5.6

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

    if 6.429099761039471e-102 < (* a x)

    1. Initial program 53.5

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

    2. Taylor expanded around 0 35.0

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

    3. Simplified20.8

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

  4. Final simplification5.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.00565259617492738353:\\ \;\;\;\;\frac{e^{2 \cdot \left(x \cdot a\right)} - 1 \cdot 1}{e^{a \cdot x} + 1}\\ \mathbf{elif}\;a \cdot x \le 6.429099761039471 \cdot 10^{-102}:\\ \;\;\;\;\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + x \cdot \left(\frac{1}{2} \cdot {a}^{2} + \left(\frac{1}{6} \cdot {a}^{3}\right) \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(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))