Average Error: 28.9 → 9.4
Time: 7.2s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le \frac{-52435226478777}{2361183241434822606848}:\\ \;\;\;\;\frac{{\left(e^{\left(a \cdot x\right) \cdot 3}\right)}^{3} - {\left({1}^{3}\right)}^{3}}{\left(1 \cdot \left(e^{a \cdot x} + 1\right) + {\left(e^{a \cdot x}\right)}^{2}\right) \cdot \left(\left({\left(e^{a \cdot x}\right)}^{6} + {1}^{6}\right) + e^{\left(a \cdot x\right) \cdot 3} \cdot {1}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le \frac{-52435226478777}{2361183241434822606848}:\\
\;\;\;\;\frac{{\left(e^{\left(a \cdot x\right) \cdot 3}\right)}^{3} - {\left({1}^{3}\right)}^{3}}{\left(1 \cdot \left(e^{a \cdot x} + 1\right) + {\left(e^{a \cdot x}\right)}^{2}\right) \cdot \left(\left({\left(e^{a \cdot x}\right)}^{6} + {1}^{6}\right) + e^{\left(a \cdot x\right) \cdot 3} \cdot {1}^{3}\right)}\\

\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 r64008 = a;
        double r64009 = x;
        double r64010 = r64008 * r64009;
        double r64011 = exp(r64010);
        double r64012 = 1.0;
        double r64013 = r64011 - r64012;
        return r64013;
}
↓
double f(double a, double x) {
        double r64014 = a;
        double r64015 = x;
        double r64016 = r64014 * r64015;
        double r64017 = -52435226478777.0;
        double r64018 = 2.3611832414348226e+21;
        double r64019 = r64017 / r64018;
        bool r64020 = r64016 <= r64019;
        double r64021 = 3.0;
        double r64022 = r64016 * r64021;
        double r64023 = exp(r64022);
        double r64024 = pow(r64023, r64021);
        double r64025 = 1.0;
        double r64026 = pow(r64025, r64021);
        double r64027 = pow(r64026, r64021);
        double r64028 = r64024 - r64027;
        double r64029 = exp(r64016);
        double r64030 = r64029 + r64025;
        double r64031 = r64025 * r64030;
        double r64032 = 2.0;
        double r64033 = pow(r64029, r64032);
        double r64034 = r64031 + r64033;
        double r64035 = 6.0;
        double r64036 = pow(r64029, r64035);
        double r64037 = pow(r64025, r64035);
        double r64038 = r64036 + r64037;
        double r64039 = r64023 * r64026;
        double r64040 = r64038 + r64039;
        double r64041 = r64034 * r64040;
        double r64042 = r64028 / r64041;
        double r64043 = 0.5;
        double r64044 = pow(r64014, r64032);
        double r64045 = r64043 * r64044;
        double r64046 = r64045 * r64015;
        double r64047 = r64014 + r64046;
        double r64048 = r64015 * r64047;
        double r64049 = 0.16666666666666666;
        double r64050 = pow(r64014, r64021);
        double r64051 = pow(r64015, r64021);
        double r64052 = r64050 * r64051;
        double r64053 = r64049 * r64052;
        double r64054 = r64048 + r64053;
        double r64055 = r64020 ? r64042 : r64054;
        return r64055;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original28.9
Target0.2
Herbie9.4
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -2.2207182212132606e-08

    1. Initial program 0.3

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

    3. Applied flip3--0.3

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

    4. Simplified0.3

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

    6. Applied pow-exp0.3

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

    8. Applied flip3--0.3

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

    9. Applied associate-/l/0.3

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

    10. Simplified0.3

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

    if -2.2207182212132606e-08 < (* a x)

    1. Initial program 44.1

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

    2. Taylor expanded around 0 14.3

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

    3. Simplified14.3

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

  4. Final simplification9.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le \frac{-52435226478777}{2361183241434822606848}:\\ \;\;\;\;\frac{{\left(e^{\left(a \cdot x\right) \cdot 3}\right)}^{3} - {\left({1}^{3}\right)}^{3}}{\left(1 \cdot \left(e^{a \cdot x} + 1\right) + {\left(e^{a \cdot x}\right)}^{2}\right) \cdot \left(\left({\left(e^{a \cdot x}\right)}^{6} + {1}^{6}\right) + e^{\left(a \cdot x\right) \cdot 3} \cdot {1}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.10000000000000001) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

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