Average Error: 30.1 → 1.0
Time: 10.0s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 95.541640315829881:\\ \;\;\;\;\frac{\left(\sqrt[3]{0.66666666666666674 \cdot {x}^{3}} \cdot \sqrt[3]{0.66666666666666674 \cdot {x}^{3}}\right) \cdot \sqrt[3]{0.66666666666666674 \cdot {x}^{3}} + \left(2 - 1 \cdot {x}^{2}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]
\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 \frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\

\end{array}
double f(double x, double eps) {
        double r31343 = 1.0;
        double r31344 = eps;
        double r31345 = r31343 / r31344;
        double r31346 = r31343 + r31345;
        double r31347 = r31343 - r31344;
        double r31348 = x;
        double r31349 = r31347 * r31348;
        double r31350 = -r31349;
        double r31351 = exp(r31350);
        double r31352 = r31346 * r31351;
        double r31353 = r31345 - r31343;
        double r31354 = r31343 + r31344;
        double r31355 = r31354 * r31348;
        double r31356 = -r31355;
        double r31357 = exp(r31356);
        double r31358 = r31353 * r31357;
        double r31359 = r31352 - r31358;
        double r31360 = 2.0;
        double r31361 = r31359 / r31360;
        return r31361;
}
↓
double f(double x, double eps) {
        double r31362 = x;
        double r31363 = 95.54164031582988;
        bool r31364 = r31362 <= r31363;
        double r31365 = 0.6666666666666667;
        double r31366 = 3.0;
        double r31367 = pow(r31362, r31366);
        double r31368 = r31365 * r31367;
        double r31369 = cbrt(r31368);
        double r31370 = r31369 * r31369;
        double r31371 = r31370 * r31369;
        double r31372 = 2.0;
        double r31373 = 1.0;
        double r31374 = 2.0;
        double r31375 = pow(r31362, r31374);
        double r31376 = r31373 * r31375;
        double r31377 = r31372 - r31376;
        double r31378 = r31371 + r31377;
        double r31379 = r31378 / r31372;
        double r31380 = eps;
        double r31381 = r31373 / r31380;
        double r31382 = r31373 + r31381;
        double r31383 = 1.0;
        double r31384 = r31373 - r31380;
        double r31385 = r31384 * r31362;
        double r31386 = exp(r31385);
        double r31387 = r31383 / r31386;
        double r31388 = r31382 * r31387;
        double r31389 = r31381 - r31373;
        double r31390 = r31373 + r31380;
        double r31391 = r31390 * r31362;
        double r31392 = -r31391;
        double r31393 = exp(r31392);
        double r31394 = r31389 * r31393;
        double r31395 = r31388 - r31394;
        double r31396 = r31395 / r31372;
        double r31397 = r31364 ? r31379 : r31396;
        return r31397;
}

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. Using strategy rm

    4. Applied associate--l+1.3

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

    6. 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 exp-neg0.3

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

  4. Final simplification1.0

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

Reproduce

herbie shell --seed 2020047 
(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))