Average Error: 29.4 → 0.9
Time: 17.7s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 179.02536135484516:\\ \;\;\;\;\frac{\frac{{\left(\mathsf{fma}\left(0.66666666666666674, {x}^{3}, 2\right)\right)}^{3} - {\left(1 \cdot {x}^{2}\right)}^{3}}{\mathsf{fma}\left(1 \cdot 1, {x}^{4}, \mathsf{fma}\left(1, {x}^{2}, \mathsf{fma}\left({x}^{3}, 0.66666666666666674, 2\right)\right) \cdot \mathsf{fma}\left(0.66666666666666674, {x}^{3}, 2\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}\right)}{2}\\ \end{array}\]
\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 179.02536135484516:\\
\;\;\;\;\frac{\frac{{\left(\mathsf{fma}\left(0.66666666666666674, {x}^{3}, 2\right)\right)}^{3} - {\left(1 \cdot {x}^{2}\right)}^{3}}{\mathsf{fma}\left(1 \cdot 1, {x}^{4}, \mathsf{fma}\left(1, {x}^{2}, \mathsf{fma}\left({x}^{3}, 0.66666666666666674, 2\right)\right) \cdot \mathsf{fma}\left(0.66666666666666674, {x}^{3}, 2\right)\right)}}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r56646 = 1.0;
        double r56647 = eps;
        double r56648 = r56646 / r56647;
        double r56649 = r56646 + r56648;
        double r56650 = r56646 - r56647;
        double r56651 = x;
        double r56652 = r56650 * r56651;
        double r56653 = -r56652;
        double r56654 = exp(r56653);
        double r56655 = r56649 * r56654;
        double r56656 = r56648 - r56646;
        double r56657 = r56646 + r56647;
        double r56658 = r56657 * r56651;
        double r56659 = -r56658;
        double r56660 = exp(r56659);
        double r56661 = r56656 * r56660;
        double r56662 = r56655 - r56661;
        double r56663 = 2.0;
        double r56664 = r56662 / r56663;
        return r56664;
}
↓
double f(double x, double eps) {
        double r56665 = x;
        double r56666 = 179.02536135484516;
        bool r56667 = r56665 <= r56666;
        double r56668 = 0.6666666666666667;
        double r56669 = 3.0;
        double r56670 = pow(r56665, r56669);
        double r56671 = 2.0;
        double r56672 = fma(r56668, r56670, r56671);
        double r56673 = pow(r56672, r56669);
        double r56674 = 1.0;
        double r56675 = 2.0;
        double r56676 = pow(r56665, r56675);
        double r56677 = r56674 * r56676;
        double r56678 = pow(r56677, r56669);
        double r56679 = r56673 - r56678;
        double r56680 = r56674 * r56674;
        double r56681 = 4.0;
        double r56682 = pow(r56665, r56681);
        double r56683 = fma(r56670, r56668, r56671);
        double r56684 = fma(r56674, r56676, r56683);
        double r56685 = r56684 * r56672;
        double r56686 = fma(r56680, r56682, r56685);
        double r56687 = r56679 / r56686;
        double r56688 = r56687 / r56671;
        double r56689 = eps;
        double r56690 = r56674 / r56689;
        double r56691 = r56674 + r56690;
        double r56692 = r56674 - r56689;
        double r56693 = r56692 * r56665;
        double r56694 = -r56693;
        double r56695 = exp(r56694);
        double r56696 = r56690 - r56674;
        double r56697 = r56674 + r56689;
        double r56698 = r56697 * r56665;
        double r56699 = exp(r56698);
        double r56700 = r56696 / r56699;
        double r56701 = -r56700;
        double r56702 = fma(r56691, r56695, r56701);
        double r56703 = r56702 / r56671;
        double r56704 = r56667 ? r56688 : r56703;
        return r56704;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 179.02536135484516

    1. Initial program 38.9

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

    2. Taylor expanded around 0 1.2

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

    3. Simplified1.2

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

    5. Applied flip3--1.2

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

    6. Simplified1.2

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

    if 179.02536135484516 < x

    1. Initial program 0.1

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

    3. Applied fma-neg0.1

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

    4. Simplified0.1

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 179.02536135484516:\\ \;\;\;\;\frac{\frac{{\left(\mathsf{fma}\left(0.66666666666666674, {x}^{3}, 2\right)\right)}^{3} - {\left(1 \cdot {x}^{2}\right)}^{3}}{\mathsf{fma}\left(1 \cdot 1, {x}^{4}, \mathsf{fma}\left(1, {x}^{2}, \mathsf{fma}\left({x}^{3}, 0.66666666666666674, 2\right)\right) \cdot \mathsf{fma}\left(0.66666666666666674, {x}^{3}, 2\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +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))