Average Error: 29.7 → 0.6
Time: 13.0s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -137.5426545982631:\\ \;\;\;\;{\left(e^{\left(3 \cdot a\right) \cdot x}\right)}^{\frac{1}{3}} - 1\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + \left(\frac{1}{2} \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) + \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6}\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -137.5426545982631:\\
\;\;\;\;{\left(e^{\left(3 \cdot a\right) \cdot x}\right)}^{\frac{1}{3}} - 1\\

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

\end{array}
double f(double a, double x) {
        double r1734019 = a;
        double r1734020 = x;
        double r1734021 = r1734019 * r1734020;
        double r1734022 = exp(r1734021);
        double r1734023 = 1.0;
        double r1734024 = r1734022 - r1734023;
        return r1734024;
}
↓
double f(double a, double x) {
        double r1734025 = a;
        double r1734026 = x;
        double r1734027 = r1734025 * r1734026;
        double r1734028 = -137.5426545982631;
        bool r1734029 = r1734027 <= r1734028;
        double r1734030 = 3.0;
        double r1734031 = r1734030 * r1734025;
        double r1734032 = r1734031 * r1734026;
        double r1734033 = exp(r1734032);
        double r1734034 = 0.3333333333333333;
        double r1734035 = pow(r1734033, r1734034);
        double r1734036 = 1.0;
        double r1734037 = r1734035 - r1734036;
        double r1734038 = 0.5;
        double r1734039 = r1734027 * r1734027;
        double r1734040 = r1734038 * r1734039;
        double r1734041 = 0.16666666666666666;
        double r1734042 = r1734027 * r1734041;
        double r1734043 = r1734039 * r1734042;
        double r1734044 = r1734040 + r1734043;
        double r1734045 = r1734027 + r1734044;
        double r1734046 = r1734029 ? r1734037 : r1734045;
        return r1734046;
}

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

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -137.5426545982631

    1. Initial program 0

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

    3. Applied add-cbrt-cube0

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

    4. Simplified0

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

    6. Applied pow1/30

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

    if -137.5426545982631 < (* a x)

    1. Initial program 44.2

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

    2. Taylor expanded around 0 14.9

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

    3. Simplified0.8

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

    5. Applied associate-+r+0.8

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

    6. Simplified0.8

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -137.5426545982631:\\ \;\;\;\;{\left(e^{\left(3 \cdot a\right) \cdot x}\right)}^{\frac{1}{3}} - 1\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + \left(\frac{1}{2} \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) + \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(FPCore (a x)
  :name "expax (section 3.5)"
  :herbie-expected 14

  :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))