Average Error: 30.1 → 1.0
Time: 10.9s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 95.541640315829881:\\ \;\;\;\;\frac{\left(\sqrt[3]{0.66666666666666674 \cdot {x}^{3}} \cdot \sqrt[3]{0.66666666666666674 \cdot {x}^{3}}\right) \cdot \sqrt[3]{0.66666666666666674 \cdot {x}^{3}} + \left(2 - 1 \cdot {x}^{2}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(\sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}} \cdot \sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}}\right) \cdot \sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]
\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 95.541640315829881:\\
\;\;\;\;\frac{\left(\sqrt[3]{0.66666666666666674 \cdot {x}^{3}} \cdot \sqrt[3]{0.66666666666666674 \cdot {x}^{3}}\right) \cdot \sqrt[3]{0.66666666666666674 \cdot {x}^{3}} + \left(2 - 1 \cdot {x}^{2}\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(\sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}} \cdot \sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}}\right) \cdot \sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\

\end{array}
double f(double x, double eps) {
        double r30494 = 1.0;
        double r30495 = eps;
        double r30496 = r30494 / r30495;
        double r30497 = r30494 + r30496;
        double r30498 = r30494 - r30495;
        double r30499 = x;
        double r30500 = r30498 * r30499;
        double r30501 = -r30500;
        double r30502 = exp(r30501);
        double r30503 = r30497 * r30502;
        double r30504 = r30496 - r30494;
        double r30505 = r30494 + r30495;
        double r30506 = r30505 * r30499;
        double r30507 = -r30506;
        double r30508 = exp(r30507);
        double r30509 = r30504 * r30508;
        double r30510 = r30503 - r30509;
        double r30511 = 2.0;
        double r30512 = r30510 / r30511;
        return r30512;
}
↓
double f(double x, double eps) {
        double r30513 = x;
        double r30514 = 95.54164031582988;
        bool r30515 = r30513 <= r30514;
        double r30516 = 0.6666666666666667;
        double r30517 = 3.0;
        double r30518 = pow(r30513, r30517);
        double r30519 = r30516 * r30518;
        double r30520 = cbrt(r30519);
        double r30521 = r30520 * r30520;
        double r30522 = r30521 * r30520;
        double r30523 = 2.0;
        double r30524 = 1.0;
        double r30525 = 2.0;
        double r30526 = pow(r30513, r30525);
        double r30527 = r30524 * r30526;
        double r30528 = r30523 - r30527;
        double r30529 = r30522 + r30528;
        double r30530 = r30529 / r30523;
        double r30531 = eps;
        double r30532 = r30524 / r30531;
        double r30533 = r30524 + r30532;
        double r30534 = r30524 - r30531;
        double r30535 = r30534 * r30513;
        double r30536 = -r30535;
        double r30537 = exp(r30536);
        double r30538 = cbrt(r30537);
        double r30539 = r30538 * r30538;
        double r30540 = r30539 * r30538;
        double r30541 = r30533 * r30540;
        double r30542 = r30532 - r30524;
        double r30543 = r30524 + r30531;
        double r30544 = r30543 * r30513;
        double r30545 = -r30544;
        double r30546 = exp(r30545);
        double r30547 = r30542 * r30546;
        double r30548 = r30541 - r30547;
        double r30549 = r30548 / r30523;
        double r30550 = r30515 ? r30530 : r30549;
        return r30550;
}

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 < 95.54164031582988

    1. Initial program 39.6

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

    2. Taylor expanded around 0 1.3

      \[\leadsto \frac{\color{blue}{\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}}{2}\]

    3. Simplified1.3

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.66666666666666674, {x}^{3}, 2\right) - 1 \cdot {x}^{2}}}{2}\]
    4. Using strategy rm

    5. Applied fma-udef1.3

      \[\leadsto \frac{\color{blue}{\left(0.66666666666666674 \cdot {x}^{3} + 2\right)} - 1 \cdot {x}^{2}}{2}\]

    6. Applied associate--l+1.3

      \[\leadsto \frac{\color{blue}{0.66666666666666674 \cdot {x}^{3} + \left(2 - 1 \cdot {x}^{2}\right)}}{2}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt1.3

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{0.66666666666666674 \cdot {x}^{3}} \cdot \sqrt[3]{0.66666666666666674 \cdot {x}^{3}}\right) \cdot \sqrt[3]{0.66666666666666674 \cdot {x}^{3}}} + \left(2 - 1 \cdot {x}^{2}\right)}{2}\]

    if 95.54164031582988 < x

    1. Initial program 0.3

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

    3. Applied add-cube-cbrt0.3

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 95.541640315829881:\\ \;\;\;\;\frac{\left(\sqrt[3]{0.66666666666666674 \cdot {x}^{3}} \cdot \sqrt[3]{0.66666666666666674 \cdot {x}^{3}}\right) \cdot \sqrt[3]{0.66666666666666674 \cdot {x}^{3}} + \left(2 - 1 \cdot {x}^{2}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(\sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}} \cdot \sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}}\right) \cdot \sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(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))