Average Error: 29.8 → 0.4
Time: 23.0s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -8.606916830493381 \cdot 10^{-05}:\\ \;\;\;\;\sqrt[3]{\left(\sqrt{e^{a \cdot x}} - 1\right) \cdot \left(\left(\sqrt{e^{a \cdot x}} - 1\right) \cdot \left(\sqrt{e^{a \cdot x}} - 1\right)\right)} \cdot \left(1 + \sqrt{e^{a \cdot x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a \cdot x\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot x\right) + \log \left(e^{\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)\right) \cdot \frac{1}{6}}\right)\right) + a \cdot x\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -8.606916830493381 \cdot 10^{-05}:\\
\;\;\;\;\sqrt[3]{\left(\sqrt{e^{a \cdot x}} - 1\right) \cdot \left(\left(\sqrt{e^{a \cdot x}} - 1\right) \cdot \left(\sqrt{e^{a \cdot x}} - 1\right)\right)} \cdot \left(1 + \sqrt{e^{a \cdot x}}\right)\\

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

\end{array}
double f(double a, double x) {
        double r2318078 = a;
        double r2318079 = x;
        double r2318080 = r2318078 * r2318079;
        double r2318081 = exp(r2318080);
        double r2318082 = 1.0;
        double r2318083 = r2318081 - r2318082;
        return r2318083;
}
↓
double f(double a, double x) {
        double r2318084 = a;
        double r2318085 = x;
        double r2318086 = r2318084 * r2318085;
        double r2318087 = -8.606916830493381e-05;
        bool r2318088 = r2318086 <= r2318087;
        double r2318089 = exp(r2318086);
        double r2318090 = sqrt(r2318089);
        double r2318091 = 1.0;
        double r2318092 = r2318090 - r2318091;
        double r2318093 = r2318092 * r2318092;
        double r2318094 = r2318092 * r2318093;
        double r2318095 = cbrt(r2318094);
        double r2318096 = r2318091 + r2318090;
        double r2318097 = r2318095 * r2318096;
        double r2318098 = 0.5;
        double r2318099 = r2318086 * r2318098;
        double r2318100 = r2318099 * r2318086;
        double r2318101 = r2318086 * r2318086;
        double r2318102 = r2318101 * r2318086;
        double r2318103 = 0.16666666666666666;
        double r2318104 = r2318102 * r2318103;
        double r2318105 = exp(r2318104);
        double r2318106 = log(r2318105);
        double r2318107 = r2318100 + r2318106;
        double r2318108 = r2318107 + r2318086;
        double r2318109 = r2318088 ? r2318097 : r2318108;
        return r2318109;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.8
Target0.2
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -8.606916830493381e-05

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.1

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

    4. Applied difference-of-sqr-10.1

      \[\leadsto \color{blue}{\left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \left(\sqrt{e^{a \cdot x}} - 1\right)}\]
    5. Using strategy rm

    6. Applied add-cbrt-cube0.1

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

    if -8.606916830493381e-05 < (* a x)

    1. Initial program 45.2

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

    2. Taylor expanded around 0 14.5

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

    3. Simplified0.4

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

    5. Applied add-log-exp0.5

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -8.606916830493381 \cdot 10^{-05}:\\ \;\;\;\;\sqrt[3]{\left(\sqrt{e^{a \cdot x}} - 1\right) \cdot \left(\left(\sqrt{e^{a \cdot x}} - 1\right) \cdot \left(\sqrt{e^{a \cdot x}} - 1\right)\right)} \cdot \left(1 + \sqrt{e^{a \cdot x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a \cdot x\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot x\right) + \log \left(e^{\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)\right) \cdot \frac{1}{6}}\right)\right) + a \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 
(FPCore (a x)
  :name "expax (section 3.5)"

  :herbie-target
  (if (< (fabs (* a x)) 1/10) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

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