Average Error: 28.7 → 9.7
Time: 4.5s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -5.305313852292152365229972038607186846093 \cdot 10^{-6}:\\ \;\;\;\;\frac{e^{\left(a \cdot x\right) \cdot 3} \cdot e^{\left(a \cdot x\right) \cdot 3} - {1}^{3} \cdot {1}^{3}}{\left(e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1\right) \cdot \left(e^{\left(a \cdot x\right) \cdot 3} + {1}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(0.1666666666666665186369300499791279435158 \cdot \left({a}^{3} \cdot {x}^{3}\right) + 1 \cdot \left(a \cdot x\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -5.305313852292152365229972038607186846093 \cdot 10^{-6}:\\
\;\;\;\;\frac{e^{\left(a \cdot x\right) \cdot 3} \cdot e^{\left(a \cdot x\right) \cdot 3} - {1}^{3} \cdot {1}^{3}}{\left(e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1\right) \cdot \left(e^{\left(a \cdot x\right) \cdot 3} + {1}^{3}\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(0.1666666666666665186369300499791279435158 \cdot \left({a}^{3} \cdot {x}^{3}\right) + 1 \cdot \left(a \cdot x\right)\right)\\

\end{array}
double f(double a, double x) {
        double r98790 = a;
        double r98791 = x;
        double r98792 = r98790 * r98791;
        double r98793 = exp(r98792);
        double r98794 = 1.0;
        double r98795 = r98793 - r98794;
        return r98795;
}
↓
double f(double a, double x) {
        double r98796 = a;
        double r98797 = x;
        double r98798 = r98796 * r98797;
        double r98799 = -5.305313852292152e-06;
        bool r98800 = r98798 <= r98799;
        double r98801 = 3.0;
        double r98802 = r98798 * r98801;
        double r98803 = exp(r98802);
        double r98804 = r98803 * r98803;
        double r98805 = 1.0;
        double r98806 = pow(r98805, r98801);
        double r98807 = r98806 * r98806;
        double r98808 = r98804 - r98807;
        double r98809 = exp(r98798);
        double r98810 = r98809 + r98805;
        double r98811 = r98809 * r98810;
        double r98812 = r98805 * r98805;
        double r98813 = r98811 + r98812;
        double r98814 = r98803 + r98806;
        double r98815 = r98813 * r98814;
        double r98816 = r98808 / r98815;
        double r98817 = 0.5;
        double r98818 = 2.0;
        double r98819 = pow(r98796, r98818);
        double r98820 = pow(r98797, r98818);
        double r98821 = r98819 * r98820;
        double r98822 = r98817 * r98821;
        double r98823 = 0.16666666666666652;
        double r98824 = pow(r98796, r98801);
        double r98825 = pow(r98797, r98801);
        double r98826 = r98824 * r98825;
        double r98827 = r98823 * r98826;
        double r98828 = r98805 * r98798;
        double r98829 = r98827 + r98828;
        double r98830 = r98822 + r98829;
        double r98831 = r98800 ? r98816 : r98830;
        return r98831;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original28.7
Target0.2
Herbie9.7
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -5.305313852292152e-06

    1. Initial program 0.1

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

    3. Applied flip3--0.1

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

    4. Simplified0.1

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

    6. Applied pow-exp0.1

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

    8. Applied flip--0.1

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

    9. Applied associate-/l/0.1

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

    if -5.305313852292152e-06 < (* a x)

    1. Initial program 44.2

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

    3. Applied flip3--44.2

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

    4. Simplified44.2

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

    5. Taylor expanded around 0 14.9

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(0.1666666666666665186369300499791279435158 \cdot \left({a}^{3} \cdot {x}^{3}\right) + 1 \cdot \left(a \cdot x\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -5.305313852292152365229972038607186846093 \cdot 10^{-6}:\\ \;\;\;\;\frac{e^{\left(a \cdot x\right) \cdot 3} \cdot e^{\left(a \cdot x\right) \cdot 3} - {1}^{3} \cdot {1}^{3}}{\left(e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1\right) \cdot \left(e^{\left(a \cdot x\right) \cdot 3} + {1}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(0.1666666666666665186369300499791279435158 \cdot \left({a}^{3} \cdot {x}^{3}\right) + 1 \cdot \left(a \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019344 
(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))