Average Error: 29.5 → 1.0
Time: 6.6s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.2315305524276785:\\ \;\;\;\;\frac{{\left(\left(\sqrt[3]{0.33333333333333337 \cdot {x}^{3}} \cdot \sqrt[3]{0.33333333333333337 \cdot {x}^{3}}\right) \cdot \sqrt[3]{0.33333333333333337 \cdot {x}^{3}} + 1\right)}^{3} - {\left(0.5 \cdot {x}^{2}\right)}^{3}}{\left(1 \cdot 1 + 1 \cdot {\left(\sqrt[3]{0.33333333333333337 \cdot {x}^{3}}\right)}^{3}\right) + \left(\left({x}^{2} \cdot \left(0.5 \cdot \left(\left(0.5 \cdot {x}^{2} + 1\right) + {\left(\sqrt[3]{0.33333333333333337 \cdot {x}^{3}}\right)}^{3}\right)\right) + {\left(\sqrt[3]{0.33333333333333337 \cdot {x}^{3}}\right)}^{3} \cdot 1\right) + {\left(\sqrt[3]{0.33333333333333337 \cdot {x}^{3}}\right)}^{6}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\right) + \frac{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]
\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
↓
\begin{array}{l}
\mathbf{if}\;x \le 1.2315305524276785:\\
\;\;\;\;\frac{{\left(\left(\sqrt[3]{0.33333333333333337 \cdot {x}^{3}} \cdot \sqrt[3]{0.33333333333333337 \cdot {x}^{3}}\right) \cdot \sqrt[3]{0.33333333333333337 \cdot {x}^{3}} + 1\right)}^{3} - {\left(0.5 \cdot {x}^{2}\right)}^{3}}{\left(1 \cdot 1 + 1 \cdot {\left(\sqrt[3]{0.33333333333333337 \cdot {x}^{3}}\right)}^{3}\right) + \left(\left({x}^{2} \cdot \left(0.5 \cdot \left(\left(0.5 \cdot {x}^{2} + 1\right) + {\left(\sqrt[3]{0.33333333333333337 \cdot {x}^{3}}\right)}^{3}\right)\right) + {\left(\sqrt[3]{0.33333333333333337 \cdot {x}^{3}}\right)}^{3} \cdot 1\right) + {\left(\sqrt[3]{0.33333333333333337 \cdot {x}^{3}}\right)}^{6}\right)}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\right) + \frac{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\

\end{array}
double f(double x, double eps) {
        double r33487 = 1.0;
        double r33488 = eps;
        double r33489 = r33487 / r33488;
        double r33490 = r33487 + r33489;
        double r33491 = r33487 - r33488;
        double r33492 = x;
        double r33493 = r33491 * r33492;
        double r33494 = -r33493;
        double r33495 = exp(r33494);
        double r33496 = r33490 * r33495;
        double r33497 = r33489 - r33487;
        double r33498 = r33487 + r33488;
        double r33499 = r33498 * r33492;
        double r33500 = -r33499;
        double r33501 = exp(r33500);
        double r33502 = r33497 * r33501;
        double r33503 = r33496 - r33502;
        double r33504 = 2.0;
        double r33505 = r33503 / r33504;
        return r33505;
}
↓
double f(double x, double eps) {
        double r33506 = x;
        double r33507 = 1.2315305524276785;
        bool r33508 = r33506 <= r33507;
        double r33509 = 0.33333333333333337;
        double r33510 = 3.0;
        double r33511 = pow(r33506, r33510);
        double r33512 = r33509 * r33511;
        double r33513 = cbrt(r33512);
        double r33514 = r33513 * r33513;
        double r33515 = r33514 * r33513;
        double r33516 = 1.0;
        double r33517 = r33515 + r33516;
        double r33518 = pow(r33517, r33510);
        double r33519 = 0.5;
        double r33520 = 2.0;
        double r33521 = pow(r33506, r33520);
        double r33522 = r33519 * r33521;
        double r33523 = pow(r33522, r33510);
        double r33524 = r33518 - r33523;
        double r33525 = r33516 * r33516;
        double r33526 = pow(r33513, r33510);
        double r33527 = r33516 * r33526;
        double r33528 = r33525 + r33527;
        double r33529 = r33522 + r33516;
        double r33530 = r33529 + r33526;
        double r33531 = r33519 * r33530;
        double r33532 = r33521 * r33531;
        double r33533 = r33526 * r33516;
        double r33534 = r33532 + r33533;
        double r33535 = 6.0;
        double r33536 = pow(r33513, r33535);
        double r33537 = r33534 + r33536;
        double r33538 = r33528 + r33537;
        double r33539 = r33524 / r33538;
        double r33540 = eps;
        double r33541 = r33516 / r33540;
        double r33542 = r33516 + r33541;
        double r33543 = r33516 - r33540;
        double r33544 = r33543 * r33506;
        double r33545 = exp(r33544);
        double r33546 = r33542 / r33545;
        double r33547 = 2.0;
        double r33548 = r33546 / r33547;
        double r33549 = r33516 + r33540;
        double r33550 = r33549 * r33506;
        double r33551 = exp(r33550);
        double r33552 = r33541 / r33551;
        double r33553 = r33552 / r33547;
        double r33554 = r33548 - r33553;
        double r33555 = r33516 / r33551;
        double r33556 = r33555 / r33547;
        double r33557 = r33554 + r33556;
        double r33558 = r33508 ? r33539 : r33557;
        return r33558;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < 1.2315305524276785

    1. Initial program 38.8

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]

    2. Simplified38.8

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

    3. Taylor expanded around 0 1.1

      \[\leadsto \color{blue}{\left(0.33333333333333337 \cdot {x}^{3} + 1\right) - 0.5 \cdot {x}^{2}}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt1.1

      \[\leadsto \left(\color{blue}{\left(\sqrt[3]{0.33333333333333337 \cdot {x}^{3}} \cdot \sqrt[3]{0.33333333333333337 \cdot {x}^{3}}\right) \cdot \sqrt[3]{0.33333333333333337 \cdot {x}^{3}}} + 1\right) - 0.5 \cdot {x}^{2}\]
    6. Using strategy rm

    7. Applied flip3--1.1

      \[\leadsto \color{blue}{\frac{{\left(\left(\sqrt[3]{0.33333333333333337 \cdot {x}^{3}} \cdot \sqrt[3]{0.33333333333333337 \cdot {x}^{3}}\right) \cdot \sqrt[3]{0.33333333333333337 \cdot {x}^{3}} + 1\right)}^{3} - {\left(0.5 \cdot {x}^{2}\right)}^{3}}{\left(\left(\sqrt[3]{0.33333333333333337 \cdot {x}^{3}} \cdot \sqrt[3]{0.33333333333333337 \cdot {x}^{3}}\right) \cdot \sqrt[3]{0.33333333333333337 \cdot {x}^{3}} + 1\right) \cdot \left(\left(\sqrt[3]{0.33333333333333337 \cdot {x}^{3}} \cdot \sqrt[3]{0.33333333333333337 \cdot {x}^{3}}\right) \cdot \sqrt[3]{0.33333333333333337 \cdot {x}^{3}} + 1\right) + \left(\left(0.5 \cdot {x}^{2}\right) \cdot \left(0.5 \cdot {x}^{2}\right) + \left(\left(\sqrt[3]{0.33333333333333337 \cdot {x}^{3}} \cdot \sqrt[3]{0.33333333333333337 \cdot {x}^{3}}\right) \cdot \sqrt[3]{0.33333333333333337 \cdot {x}^{3}} + 1\right) \cdot \left(0.5 \cdot {x}^{2}\right)\right)}}\]

    8. Simplified1.1

      \[\leadsto \frac{{\left(\left(\sqrt[3]{0.33333333333333337 \cdot {x}^{3}} \cdot \sqrt[3]{0.33333333333333337 \cdot {x}^{3}}\right) \cdot \sqrt[3]{0.33333333333333337 \cdot {x}^{3}} + 1\right)}^{3} - {\left(0.5 \cdot {x}^{2}\right)}^{3}}{\color{blue}{\left(1 \cdot 1 + 1 \cdot {\left(\sqrt[3]{0.33333333333333337 \cdot {x}^{3}}\right)}^{3}\right) + \left(\left({x}^{2} \cdot \left(0.5 \cdot \left(\left(0.5 \cdot {x}^{2} + 1\right) + {\left(\sqrt[3]{0.33333333333333337 \cdot {x}^{3}}\right)}^{3}\right)\right) + {\left(\sqrt[3]{0.33333333333333337 \cdot {x}^{3}}\right)}^{3} \cdot 1\right) + {\left(\sqrt[3]{0.33333333333333337 \cdot {x}^{3}}\right)}^{6}\right)}}\]

    if 1.2315305524276785 < x

    1. Initial program 0.8

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]

    2. Simplified0.8

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

    4. Applied div-sub0.8

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

    5. Applied div-sub0.8

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

    6. Applied associate--r-0.7

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.2315305524276785:\\ \;\;\;\;\frac{{\left(\left(\sqrt[3]{0.33333333333333337 \cdot {x}^{3}} \cdot \sqrt[3]{0.33333333333333337 \cdot {x}^{3}}\right) \cdot \sqrt[3]{0.33333333333333337 \cdot {x}^{3}} + 1\right)}^{3} - {\left(0.5 \cdot {x}^{2}\right)}^{3}}{\left(1 \cdot 1 + 1 \cdot {\left(\sqrt[3]{0.33333333333333337 \cdot {x}^{3}}\right)}^{3}\right) + \left(\left({x}^{2} \cdot \left(0.5 \cdot \left(\left(0.5 \cdot {x}^{2} + 1\right) + {\left(\sqrt[3]{0.33333333333333337 \cdot {x}^{3}}\right)}^{3}\right)\right) + {\left(\sqrt[3]{0.33333333333333337 \cdot {x}^{3}}\right)}^{3} \cdot 1\right) + {\left(\sqrt[3]{0.33333333333333337 \cdot {x}^{3}}\right)}^{6}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\right) + \frac{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  :precision binary64
  (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2))