Average Error: 29.3 → 0.3
Time: 13.2s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le \frac{-7138742798571315}{36893488147419103232}:\\ \;\;\;\;\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{a \cdot x}} - \sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x + \frac{{\left(a \cdot x\right)}^{2}}{2}\right) + \frac{{\left(a \cdot x\right)}^{3}}{6}\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le \frac{-7138742798571315}{36893488147419103232}:\\
\;\;\;\;\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{a \cdot x}} - \sqrt{1}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(a \cdot x + \frac{{\left(a \cdot x\right)}^{2}}{2}\right) + \frac{{\left(a \cdot x\right)}^{3}}{6}\\

\end{array}
double f(double a, double x) {
        double r92090 = a;
        double r92091 = x;
        double r92092 = r92090 * r92091;
        double r92093 = exp(r92092);
        double r92094 = 1.0;
        double r92095 = r92093 - r92094;
        return r92095;
}
↓
double f(double a, double x) {
        double r92096 = a;
        double r92097 = x;
        double r92098 = r92096 * r92097;
        double r92099 = -7138742798571315.0;
        double r92100 = 3.6893488147419103e+19;
        double r92101 = r92099 / r92100;
        bool r92102 = r92098 <= r92101;
        double r92103 = exp(r92098);
        double r92104 = sqrt(r92103);
        double r92105 = 1.0;
        double r92106 = sqrt(r92105);
        double r92107 = r92104 + r92106;
        double r92108 = r92104 - r92106;
        double r92109 = r92107 * r92108;
        double r92110 = 2.0;
        double r92111 = pow(r92098, r92110);
        double r92112 = r92111 / r92110;
        double r92113 = r92098 + r92112;
        double r92114 = 3.0;
        double r92115 = pow(r92098, r92114);
        double r92116 = 6.0;
        double r92117 = r92115 / r92116;
        double r92118 = r92113 + r92117;
        double r92119 = r92102 ? r92109 : r92118;
        return r92119;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.3
Target0.2
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.00019349601127565674

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

      \[\leadsto e^{a \cdot x} - \color{blue}{\sqrt{1} \cdot \sqrt{1}}\]

    4. Applied add-sqr-sqrt0.1

      \[\leadsto \color{blue}{\sqrt{e^{a \cdot x}} \cdot \sqrt{e^{a \cdot x}}} - \sqrt{1} \cdot \sqrt{1}\]

    5. Applied difference-of-squares0.1

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

    if -0.00019349601127565674 < (* a x)

    1. Initial program 44.2

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

    2. Taylor expanded around 0 13.8

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

    3. Simplified13.8

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

    5. Applied pow-prod-down4.2

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

    7. Applied distribute-lft-in4.2

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

    8. Simplified4.2

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

    9. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le \frac{-7138742798571315}{36893488147419103232}:\\ \;\;\;\;\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{a \cdot x}} - \sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x + \frac{{\left(a \cdot x\right)}^{2}}{2}\right) + \frac{{\left(a \cdot x\right)}^{3}}{6}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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))