Average Error: 28.7 → 8.7
Time: 3.2s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -3.98282351915828164 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt[3]{\left(e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1\right) \cdot \left(e^{a \cdot x} - 1\right)}}{\sqrt[3]{e^{a \cdot x} + 1}} \cdot \sqrt[3]{e^{a \cdot x} - 1}\\ \mathbf{elif}\;a \cdot x \le 2.1553377009136073 \cdot 10^{-16}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -3.98282351915828164 \cdot 10^{-7}:\\
\;\;\;\;\frac{\sqrt[3]{\left(e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1\right) \cdot \left(e^{a \cdot x} - 1\right)}}{\sqrt[3]{e^{a \cdot x} + 1}} \cdot \sqrt[3]{e^{a \cdot x} - 1}\\

\mathbf{elif}\;a \cdot x \le 2.1553377009136073 \cdot 10^{-16}:\\
\;\;\;\;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)\\

\mathbf{else}:\\
\;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\

\end{array}
double f(double a, double x) {
        double r121461 = a;
        double r121462 = x;
        double r121463 = r121461 * r121462;
        double r121464 = exp(r121463);
        double r121465 = 1.0;
        double r121466 = r121464 - r121465;
        return r121466;
}
↓
double f(double a, double x) {
        double r121467 = a;
        double r121468 = x;
        double r121469 = r121467 * r121468;
        double r121470 = -3.9828235191582816e-07;
        bool r121471 = r121469 <= r121470;
        double r121472 = exp(r121469);
        double r121473 = r121472 * r121472;
        double r121474 = 1.0;
        double r121475 = r121474 * r121474;
        double r121476 = r121473 - r121475;
        double r121477 = r121472 - r121474;
        double r121478 = r121476 * r121477;
        double r121479 = cbrt(r121478);
        double r121480 = r121472 + r121474;
        double r121481 = cbrt(r121480);
        double r121482 = r121479 / r121481;
        double r121483 = cbrt(r121477);
        double r121484 = r121482 * r121483;
        double r121485 = 2.1553377009136073e-16;
        bool r121486 = r121469 <= r121485;
        double r121487 = 0.5;
        double r121488 = 2.0;
        double r121489 = pow(r121467, r121488);
        double r121490 = r121487 * r121489;
        double r121491 = r121490 * r121468;
        double r121492 = r121467 + r121491;
        double r121493 = r121468 * r121492;
        double r121494 = 0.16666666666666666;
        double r121495 = 3.0;
        double r121496 = pow(r121467, r121495);
        double r121497 = pow(r121468, r121495);
        double r121498 = r121496 * r121497;
        double r121499 = r121494 * r121498;
        double r121500 = r121493 + r121499;
        double r121501 = exp(r121477);
        double r121502 = log(r121501);
        double r121503 = r121486 ? r121500 : r121502;
        double r121504 = r121471 ? r121484 : r121503;
        return r121504;
}

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
Herbie8.7
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* a x) < -3.9828235191582816e-07

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube0.1

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

    4. Simplified0.1

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

    6. Applied add-cube-cbrt0.2

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

    7. Applied unpow-prod-down0.2

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

    8. Applied cbrt-prod0.2

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

    9. Simplified0.1

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

    10. Simplified0.1

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

    12. Applied flip--0.1

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

    13. Applied associate-*r/0.1

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

    14. Applied cbrt-div0.1

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

    15. Simplified0.1

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

    if -3.9828235191582816e-07 < (* a x) < 2.1553377009136073e-16

    1. Initial program 44.9

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

    2. Taylor expanded around 0 13.0

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

    3. Simplified13.0

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

    if 2.1553377009136073e-16 < (* a x)

    1. Initial program 17.4

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

    3. Applied add-log-exp17.4

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

    4. Applied add-log-exp24.5

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

    5. Applied diff-log24.9

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

    6. Simplified24.5

      \[\leadsto \log \color{blue}{\left(e^{e^{a \cdot x} - 1}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -3.98282351915828164 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt[3]{\left(e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1\right) \cdot \left(e^{a \cdot x} - 1\right)}}{\sqrt[3]{e^{a \cdot x} + 1}} \cdot \sqrt[3]{e^{a \cdot x} - 1}\\ \mathbf{elif}\;a \cdot x \le 2.1553377009136073 \cdot 10^{-16}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 
(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))