Average Error: 29.8 → 9.8
Time: 3.6s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -3.6128960162996795 \cdot 10^{-4}:\\ \;\;\;\;\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \frac{e^{a \cdot x} - 1}{\sqrt{\left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}\right) \cdot \sqrt[3]{e^{a \cdot x}}} + \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -3.6128960162996795 \cdot 10^{-4}:\\
\;\;\;\;\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \frac{e^{a \cdot x} - 1}{\sqrt{\left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}\right) \cdot \sqrt[3]{e^{a \cdot x}}} + \sqrt{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 r124351 = a;
        double r124352 = x;
        double r124353 = r124351 * r124352;
        double r124354 = exp(r124353);
        double r124355 = 1.0;
        double r124356 = r124354 - r124355;
        return r124356;
}
↓
double f(double a, double x) {
        double r124357 = a;
        double r124358 = x;
        double r124359 = r124357 * r124358;
        double r124360 = -0.00036128960162996795;
        bool r124361 = r124359 <= r124360;
        double r124362 = exp(r124359);
        double r124363 = sqrt(r124362);
        double r124364 = 1.0;
        double r124365 = sqrt(r124364);
        double r124366 = r124363 + r124365;
        double r124367 = r124362 - r124364;
        double r124368 = cbrt(r124362);
        double r124369 = r124368 * r124368;
        double r124370 = r124369 * r124368;
        double r124371 = sqrt(r124370);
        double r124372 = r124371 + r124365;
        double r124373 = r124367 / r124372;
        double r124374 = r124366 * r124373;
        double r124375 = 0.5;
        double r124376 = 2.0;
        double r124377 = pow(r124357, r124376);
        double r124378 = r124375 * r124377;
        double r124379 = r124378 * r124358;
        double r124380 = r124357 + r124379;
        double r124381 = r124358 * r124380;
        double r124382 = 0.16666666666666666;
        double r124383 = 3.0;
        double r124384 = pow(r124357, r124383);
        double r124385 = pow(r124358, r124383);
        double r124386 = r124384 * r124385;
        double r124387 = r124382 * r124386;
        double r124388 = r124381 + r124387;
        double r124389 = r124361 ? r124374 : r124388;
        return r124389;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.8
Target0.2
Herbie9.8
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.00036128960162996795

    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-sqrt0.0

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

    5. Applied difference-of-squares0.0

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

    7. Applied flip--0.0

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

    8. Simplified0.0

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

    10. Applied add-cube-cbrt0.0

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

    if -0.00036128960162996795 < (* a x)

    1. Initial program 44.5

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

    2. Taylor expanded around 0 14.6

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

    3. Simplified14.6

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -3.6128960162996795 \cdot 10^{-4}:\\ \;\;\;\;\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \frac{e^{a \cdot x} - 1}{\sqrt{\left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}\right) \cdot \sqrt[3]{e^{a \cdot x}}} + \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 
(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))