Average Error: 29.6 → 9.6
Time: 4.2s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -48.69806950283927449163456913083791732788:\\ \;\;\;\;\frac{\frac{\frac{{\left({\left(e^{\left(a \cdot x\right) \cdot 3}\right)}^{3}\right)}^{3} - {\left({\left({1}^{3}\right)}^{3}\right)}^{3}}{\left({\left(e^{\left(a \cdot x\right) \cdot 3}\right)}^{6} + {\left({1}^{3}\right)}^{6}\right) + {\left(e^{\left(a \cdot x\right) \cdot 3}\right)}^{3} \cdot {\left({1}^{3}\right)}^{3}}}{\left({\left(e^{a \cdot x}\right)}^{6} + e^{\left(a \cdot x\right) \cdot 3} \cdot {1}^{3}\right) + {1}^{6}}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \mathbf{elif}\;a \cdot x \le 1.399335336201219789742515322647377985716 \cdot 10^{-29}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{\log \left({\left(e^{\left(a \cdot x\right) \cdot 3}\right)}^{3} - {\left({1}^{3}\right)}^{3}\right)}}{\left({\left(e^{a \cdot x}\right)}^{6} + e^{\left(a \cdot x\right) \cdot 3} \cdot {1}^{3}\right) + {1}^{6}}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -48.69806950283927449163456913083791732788:\\
\;\;\;\;\frac{\frac{\frac{{\left({\left(e^{\left(a \cdot x\right) \cdot 3}\right)}^{3}\right)}^{3} - {\left({\left({1}^{3}\right)}^{3}\right)}^{3}}{\left({\left(e^{\left(a \cdot x\right) \cdot 3}\right)}^{6} + {\left({1}^{3}\right)}^{6}\right) + {\left(e^{\left(a \cdot x\right) \cdot 3}\right)}^{3} \cdot {\left({1}^{3}\right)}^{3}}}{\left({\left(e^{a \cdot x}\right)}^{6} + e^{\left(a \cdot x\right) \cdot 3} \cdot {1}^{3}\right) + {1}^{6}}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\

\mathbf{elif}\;a \cdot x \le 1.399335336201219789742515322647377985716 \cdot 10^{-29}:\\
\;\;\;\;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}:\\
\;\;\;\;\frac{\frac{e^{\log \left({\left(e^{\left(a \cdot x\right) \cdot 3}\right)}^{3} - {\left({1}^{3}\right)}^{3}\right)}}{\left({\left(e^{a \cdot x}\right)}^{6} + e^{\left(a \cdot x\right) \cdot 3} \cdot {1}^{3}\right) + {1}^{6}}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\

\end{array}
double f(double a, double x) {
        double r68475 = a;
        double r68476 = x;
        double r68477 = r68475 * r68476;
        double r68478 = exp(r68477);
        double r68479 = 1.0;
        double r68480 = r68478 - r68479;
        return r68480;
}
↓
double f(double a, double x) {
        double r68481 = a;
        double r68482 = x;
        double r68483 = r68481 * r68482;
        double r68484 = -48.698069502839274;
        bool r68485 = r68483 <= r68484;
        double r68486 = 3.0;
        double r68487 = r68483 * r68486;
        double r68488 = exp(r68487);
        double r68489 = pow(r68488, r68486);
        double r68490 = pow(r68489, r68486);
        double r68491 = 1.0;
        double r68492 = pow(r68491, r68486);
        double r68493 = pow(r68492, r68486);
        double r68494 = pow(r68493, r68486);
        double r68495 = r68490 - r68494;
        double r68496 = 6.0;
        double r68497 = pow(r68488, r68496);
        double r68498 = pow(r68492, r68496);
        double r68499 = r68497 + r68498;
        double r68500 = r68489 * r68493;
        double r68501 = r68499 + r68500;
        double r68502 = r68495 / r68501;
        double r68503 = exp(r68483);
        double r68504 = pow(r68503, r68496);
        double r68505 = r68488 * r68492;
        double r68506 = r68504 + r68505;
        double r68507 = pow(r68491, r68496);
        double r68508 = r68506 + r68507;
        double r68509 = r68502 / r68508;
        double r68510 = r68503 + r68491;
        double r68511 = r68503 * r68510;
        double r68512 = r68491 * r68491;
        double r68513 = r68511 + r68512;
        double r68514 = r68509 / r68513;
        double r68515 = 1.3993353362012198e-29;
        bool r68516 = r68483 <= r68515;
        double r68517 = 0.5;
        double r68518 = 2.0;
        double r68519 = pow(r68481, r68518);
        double r68520 = r68517 * r68519;
        double r68521 = r68520 * r68482;
        double r68522 = r68481 + r68521;
        double r68523 = r68482 * r68522;
        double r68524 = 0.16666666666666666;
        double r68525 = pow(r68481, r68486);
        double r68526 = pow(r68482, r68486);
        double r68527 = r68525 * r68526;
        double r68528 = r68524 * r68527;
        double r68529 = r68523 + r68528;
        double r68530 = r68489 - r68493;
        double r68531 = log(r68530);
        double r68532 = exp(r68531);
        double r68533 = r68532 / r68508;
        double r68534 = r68533 / r68513;
        double r68535 = r68516 ? r68529 : r68534;
        double r68536 = r68485 ? r68514 : r68535;
        return r68536;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.6
Target0.2
Herbie9.6
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* a x) < -48.698069502839274

    1. Initial program 0

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

    3. Applied flip3--0

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

    4. Simplified0

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

    6. Applied pow-exp0

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

    8. Applied flip3--0

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

    9. Simplified0

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

    11. Applied flip3--0

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

    12. Simplified0

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

    if -48.698069502839274 < (* a x) < 1.3993353362012198e-29

    1. Initial program 44.6

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

    2. Taylor expanded around 0 13.1

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

    3. Simplified13.1

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

    if 1.3993353362012198e-29 < (* a x)

    1. Initial program 36.2

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

    3. Applied flip3--36.4

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

    4. Simplified36.4

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

    6. Applied pow-exp35.6

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

    8. Applied flip3--36.6

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

    9. Simplified36.7

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

    11. Applied add-exp-log36.7

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

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -48.69806950283927449163456913083791732788:\\ \;\;\;\;\frac{\frac{\frac{{\left({\left(e^{\left(a \cdot x\right) \cdot 3}\right)}^{3}\right)}^{3} - {\left({\left({1}^{3}\right)}^{3}\right)}^{3}}{\left({\left(e^{\left(a \cdot x\right) \cdot 3}\right)}^{6} + {\left({1}^{3}\right)}^{6}\right) + {\left(e^{\left(a \cdot x\right) \cdot 3}\right)}^{3} \cdot {\left({1}^{3}\right)}^{3}}}{\left({\left(e^{a \cdot x}\right)}^{6} + e^{\left(a \cdot x\right) \cdot 3} \cdot {1}^{3}\right) + {1}^{6}}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \mathbf{elif}\;a \cdot x \le 1.399335336201219789742515322647377985716 \cdot 10^{-29}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{\log \left({\left(e^{\left(a \cdot x\right) \cdot 3}\right)}^{3} - {\left({1}^{3}\right)}^{3}\right)}}{\left({\left(e^{a \cdot x}\right)}^{6} + e^{\left(a \cdot x\right) \cdot 3} \cdot {1}^{3}\right) + {1}^{6}}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019346 
(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))