Average Error: 29.2 → 0.4
Time: 3.7s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -1.9732286240242323 \cdot 10^{-4}:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x + \frac{1}{2} \cdot {\left(a \cdot x\right)}^{2}\right) + \frac{1}{6} \cdot {\left(a \cdot x\right)}^{3}\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -1.9732286240242323 \cdot 10^{-4}:\\
\;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\

\mathbf{else}:\\
\;\;\;\;\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}\\

\end{array}
double f(double a, double x) {
        double r69014 = a;
        double r69015 = x;
        double r69016 = r69014 * r69015;
        double r69017 = exp(r69016);
        double r69018 = 1.0;
        double r69019 = r69017 - r69018;
        return r69019;
}
↓
double f(double a, double x) {
        double r69020 = a;
        double r69021 = x;
        double r69022 = r69020 * r69021;
        double r69023 = -0.00019732286240242323;
        bool r69024 = r69022 <= r69023;
        double r69025 = exp(r69022);
        double r69026 = 1.0;
        double r69027 = r69025 - r69026;
        double r69028 = exp(r69027);
        double r69029 = log(r69028);
        double r69030 = 0.5;
        double r69031 = 2.0;
        double r69032 = pow(r69022, r69031);
        double r69033 = r69030 * r69032;
        double r69034 = r69022 + r69033;
        double r69035 = 0.16666666666666666;
        double r69036 = 3.0;
        double r69037 = pow(r69022, r69036);
        double r69038 = r69035 * r69037;
        double r69039 = r69034 + r69038;
        double r69040 = r69024 ? r69029 : r69039;
        return r69040;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.00019732286240242323

    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 -0.00019732286240242323 < (* a x)

    1. Initial program 44.0

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

    2. Taylor expanded around 0 14.6

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

    3. Simplified14.6

      \[\leadsto \color{blue}{x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)}\]
    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. Simplified0.5

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -1.9732286240242323 \cdot 10^{-4}:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x + \frac{1}{2} \cdot {\left(a \cdot x\right)}^{2}\right) + \frac{1}{6} \cdot {\left(a \cdot x\right)}^{3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(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))