Average Error: 28.9 → 3.0
Time: 21.2s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -2.220718221213260556924813660106821089357 \cdot 10^{-8}:\\ \;\;\;\;\left(\sqrt[3]{\frac{e^{2 \cdot \left(x \cdot a\right)} - 1 \cdot 1}{e^{a \cdot x} + 1}} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{a \cdot x}} - \sqrt{1}\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left({x}^{2} \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} \cdot a\right)\right)\right) + x\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -2.220718221213260556924813660106821089357 \cdot 10^{-8}:\\
\;\;\;\;\left(\sqrt[3]{\frac{e^{2 \cdot \left(x \cdot a\right)} - 1 \cdot 1}{e^{a \cdot x} + 1}} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{a \cdot x}} - \sqrt{1}\right)}\\

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

\end{array}
double f(double a, double x) {
        double r87108 = a;
        double r87109 = x;
        double r87110 = r87108 * r87109;
        double r87111 = exp(r87110);
        double r87112 = 1.0;
        double r87113 = r87111 - r87112;
        return r87113;
}
↓
double f(double a, double x) {
        double r87114 = a;
        double r87115 = x;
        double r87116 = r87114 * r87115;
        double r87117 = -2.2207182212132606e-08;
        bool r87118 = r87116 <= r87117;
        double r87119 = 2.0;
        double r87120 = r87115 * r87114;
        double r87121 = r87119 * r87120;
        double r87122 = exp(r87121);
        double r87123 = 1.0;
        double r87124 = r87123 * r87123;
        double r87125 = r87122 - r87124;
        double r87126 = exp(r87116);
        double r87127 = r87126 + r87123;
        double r87128 = r87125 / r87127;
        double r87129 = cbrt(r87128);
        double r87130 = r87126 - r87123;
        double r87131 = cbrt(r87130);
        double r87132 = r87129 * r87131;
        double r87133 = sqrt(r87126);
        double r87134 = sqrt(r87123);
        double r87135 = r87133 + r87134;
        double r87136 = r87133 - r87134;
        double r87137 = r87135 * r87136;
        double r87138 = cbrt(r87137);
        double r87139 = r87132 * r87138;
        double r87140 = pow(r87115, r87119);
        double r87141 = 0.5;
        double r87142 = 0.16666666666666666;
        double r87143 = r87142 * r87114;
        double r87144 = r87115 * r87143;
        double r87145 = r87141 + r87144;
        double r87146 = r87140 * r87145;
        double r87147 = r87114 * r87146;
        double r87148 = r87147 + r87115;
        double r87149 = r87114 * r87148;
        double r87150 = r87118 ? r87139 : r87149;
        return r87150;
}

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
Herbie3.0
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

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 add-cube-cbrt0.3

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

    5. Applied add-sqr-sqrt0.3

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

    6. Applied add-sqr-sqrt0.3

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

    7. Applied difference-of-squares0.3

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

    9. Applied flip--0.3

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

    10. Simplified0.3

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

    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. Simplified4.5

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

  4. Final simplification3.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -2.220718221213260556924813660106821089357 \cdot 10^{-8}:\\ \;\;\;\;\left(\sqrt[3]{\frac{e^{2 \cdot \left(x \cdot a\right)} - 1 \cdot 1}{e^{a \cdot x} + 1}} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{a \cdot x}} - \sqrt{1}\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left({x}^{2} \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} \cdot a\right)\right)\right) + x\right)\\ \end{array}\]

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))