Average Error: 29.9 → 9.5
Time: 3.0s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -2.7083089784345925 \cdot 10^{-15}:\\ \;\;\;\;\frac{e^{\left(a \cdot x\right) \cdot 3} - {1}^{3}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -2.7083089784345925 \cdot 10^{-15}:\\
\;\;\;\;\frac{e^{\left(a \cdot x\right) \cdot 3} - {1}^{3}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\

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

\end{array}
double f(double a, double x) {
        double r117749 = a;
        double r117750 = x;
        double r117751 = r117749 * r117750;
        double r117752 = exp(r117751);
        double r117753 = 1.0;
        double r117754 = r117752 - r117753;
        return r117754;
}
↓
double f(double a, double x) {
        double r117755 = a;
        double r117756 = x;
        double r117757 = r117755 * r117756;
        double r117758 = -2.7083089784345925e-15;
        bool r117759 = r117757 <= r117758;
        double r117760 = 3.0;
        double r117761 = r117757 * r117760;
        double r117762 = exp(r117761);
        double r117763 = 1.0;
        double r117764 = pow(r117763, r117760);
        double r117765 = r117762 - r117764;
        double r117766 = exp(r117757);
        double r117767 = r117766 + r117763;
        double r117768 = r117766 * r117767;
        double r117769 = r117763 * r117763;
        double r117770 = r117768 + r117769;
        double r117771 = r117765 / r117770;
        double r117772 = 0.5;
        double r117773 = 2.0;
        double r117774 = pow(r117755, r117773);
        double r117775 = r117772 * r117774;
        double r117776 = r117775 * r117756;
        double r117777 = r117755 + r117776;
        double r117778 = r117756 * r117777;
        double r117779 = 0.16666666666666666;
        double r117780 = pow(r117755, r117760);
        double r117781 = pow(r117756, r117760);
        double r117782 = r117780 * r117781;
        double r117783 = r117779 * r117782;
        double r117784 = r117778 + r117783;
        double r117785 = r117759 ? r117771 : r117784;
        return r117785;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.9
Target0.2
Herbie9.5
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -2.7083089784345925e-15

    1. Initial program 0.8

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

    3. Applied flip3--0.8

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

    4. Simplified0.8

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

    6. Applied pow-exp0.7

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

    if -2.7083089784345925e-15 < (* a x)

    1. Initial program 44.7

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

    2. Taylor expanded around 0 14.0

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

    3. Simplified14.0

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

  4. Final simplification9.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -2.7083089784345925 \cdot 10^{-15}:\\ \;\;\;\;\frac{e^{\left(a \cdot x\right) \cdot 3} - {1}^{3}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 
(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))