Average Error: 29.4 → 0.3
Time: 3.5s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -5.8466223410137292:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}\right) \cdot \sqrt[3]{e^{a \cdot x}} - 1\\ \mathbf{elif}\;a \cdot x \le 1.81556124406024112 \cdot 10^{-5}:\\ \;\;\;\;\left(a \cdot x + \frac{1}{2} \cdot {\left(a \cdot x\right)}^{2}\right) + \frac{1}{6} \cdot {\left(a \cdot x\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{e^{a \cdot x} - 1} \cdot \sqrt{e^{a \cdot x} - 1}\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -5.8466223410137292:\\
\;\;\;\;\left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}\right) \cdot \sqrt[3]{e^{a \cdot x}} - 1\\

\mathbf{elif}\;a \cdot x \le 1.81556124406024112 \cdot 10^{-5}:\\
\;\;\;\;\left(a \cdot x + \frac{1}{2} \cdot {\left(a \cdot x\right)}^{2}\right) + \frac{1}{6} \cdot {\left(a \cdot x\right)}^{3}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{e^{a \cdot x} - 1} \cdot \sqrt{e^{a \cdot x} - 1}\\

\end{array}
double f(double a, double x) {
        double r98961 = a;
        double r98962 = x;
        double r98963 = r98961 * r98962;
        double r98964 = exp(r98963);
        double r98965 = 1.0;
        double r98966 = r98964 - r98965;
        return r98966;
}
↓
double f(double a, double x) {
        double r98967 = a;
        double r98968 = x;
        double r98969 = r98967 * r98968;
        double r98970 = -5.846622341013729;
        bool r98971 = r98969 <= r98970;
        double r98972 = exp(r98969);
        double r98973 = cbrt(r98972);
        double r98974 = r98973 * r98973;
        double r98975 = r98974 * r98973;
        double r98976 = 1.0;
        double r98977 = r98975 - r98976;
        double r98978 = 1.815561244060241e-05;
        bool r98979 = r98969 <= r98978;
        double r98980 = 0.5;
        double r98981 = 2.0;
        double r98982 = pow(r98969, r98981);
        double r98983 = r98980 * r98982;
        double r98984 = r98969 + r98983;
        double r98985 = 0.16666666666666666;
        double r98986 = 3.0;
        double r98987 = pow(r98969, r98986);
        double r98988 = r98985 * r98987;
        double r98989 = r98984 + r98988;
        double r98990 = r98972 - r98976;
        double r98991 = sqrt(r98990);
        double r98992 = r98991 * r98991;
        double r98993 = r98979 ? r98989 : r98992;
        double r98994 = r98971 ? r98977 : r98993;
        return r98994;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.4
Target0.2
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* a x) < -5.846622341013729

    1. Initial program 0

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

    3. Applied add-cube-cbrt0

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

    if -5.846622341013729 < (* a x) < 1.815561244060241e-05

    1. Initial program 44.4

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

    2. Taylor expanded around 0 14.1

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

    3. Simplified14.1

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

    5. Applied pow-prod-down4.4

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

    7. Applied distribute-lft-in4.4

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

    8. Simplified4.4

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

    9. Simplified0.3

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

    if 1.815561244060241e-05 < (* a x)

    1. Initial program 5.2

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

    3. Applied add-sqr-sqrt5.2

      \[\leadsto \color{blue}{\sqrt{e^{a \cdot x} - 1} \cdot \sqrt{e^{a \cdot x} - 1}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -5.8466223410137292:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}\right) \cdot \sqrt[3]{e^{a \cdot x}} - 1\\ \mathbf{elif}\;a \cdot x \le 1.81556124406024112 \cdot 10^{-5}:\\ \;\;\;\;\left(a \cdot x + \frac{1}{2} \cdot {\left(a \cdot x\right)}^{2}\right) + \frac{1}{6} \cdot {\left(a \cdot x\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{e^{a \cdot x} - 1} \cdot \sqrt{e^{a \cdot x} - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(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))