Average Error: 4.3 → 0.9
Time: 1.1m
Precision: 64
\[\sqrt{\frac{e^{2.0 \cdot x} - 1.0}{e^{x} - 1.0}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.6879468583117964 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\frac{\sqrt{1.0} + \sqrt{e^{2.0 \cdot x}}}{\left(\left(\frac{e^{x} - 1.0}{\sqrt{e^{2.0 \cdot x}} - \sqrt{1.0}}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\sqrt{2.0}}, 0.5, \sqrt{2.0}\right) + \left(0.25 - \frac{0.125}{2.0}\right) \cdot \frac{x}{\frac{\sqrt{2.0}}{x}}\\ \end{array}\]
\sqrt{\frac{e^{2.0 \cdot x} - 1.0}{e^{x} - 1.0}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.6879468583117964 \cdot 10^{-16}:\\
\;\;\;\;\sqrt{\frac{\sqrt{1.0} + \sqrt{e^{2.0 \cdot x}}}{\left(\left(\frac{e^{x} - 1.0}{\sqrt{e^{2.0 \cdot x}} - \sqrt{1.0}}\right)\right)}}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{\sqrt{2.0}}, 0.5, \sqrt{2.0}\right) + \left(0.25 - \frac{0.125}{2.0}\right) \cdot \frac{x}{\frac{\sqrt{2.0}}{x}}\\

\end{array}
double f(double x) {
        double r2493017 = 2.0;
        double r2493018 = x;
        double r2493019 = r2493017 * r2493018;
        double r2493020 = exp(r2493019);
        double r2493021 = 1.0;
        double r2493022 = r2493020 - r2493021;
        double r2493023 = exp(r2493018);
        double r2493024 = r2493023 - r2493021;
        double r2493025 = r2493022 / r2493024;
        double r2493026 = sqrt(r2493025);
        return r2493026;
}
↓
double f(double x) {
        double r2493027 = x;
        double r2493028 = -1.6879468583117964e-16;
        bool r2493029 = r2493027 <= r2493028;
        double r2493030 = 1.0;
        double r2493031 = sqrt(r2493030);
        double r2493032 = 2.0;
        double r2493033 = r2493032 * r2493027;
        double r2493034 = exp(r2493033);
        double r2493035 = sqrt(r2493034);
        double r2493036 = r2493031 + r2493035;
        double r2493037 = exp(r2493027);
        double r2493038 = r2493037 - r2493030;
        double r2493039 = r2493035 - r2493031;
        double r2493040 = r2493038 / r2493039;
        double r2493041 = /* ERROR: no posit support in C */;
        double r2493042 = /* ERROR: no posit support in C */;
        double r2493043 = r2493036 / r2493042;
        double r2493044 = sqrt(r2493043);
        double r2493045 = sqrt(r2493032);
        double r2493046 = r2493027 / r2493045;
        double r2493047 = 0.5;
        double r2493048 = fma(r2493046, r2493047, r2493045);
        double r2493049 = 0.25;
        double r2493050 = 0.125;
        double r2493051 = r2493050 / r2493032;
        double r2493052 = r2493049 - r2493051;
        double r2493053 = r2493045 / r2493027;
        double r2493054 = r2493027 / r2493053;
        double r2493055 = r2493052 * r2493054;
        double r2493056 = r2493048 + r2493055;
        double r2493057 = r2493029 ? r2493044 : r2493056;
        return r2493057;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -1.6879468583117964e-16

    1. Initial program 0.8

      \[\sqrt{\frac{e^{2.0 \cdot x} - 1.0}{e^{x} - 1.0}}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt0.8

      \[\leadsto \sqrt{\frac{e^{2.0 \cdot x} - \color{blue}{\sqrt{1.0} \cdot \sqrt{1.0}}}{e^{x} - 1.0}}\]

    4. Applied add-sqr-sqrt0.7

      \[\leadsto \sqrt{\frac{\color{blue}{\sqrt{e^{2.0 \cdot x}} \cdot \sqrt{e^{2.0 \cdot x}}} - \sqrt{1.0} \cdot \sqrt{1.0}}{e^{x} - 1.0}}\]

    5. Applied difference-of-squares0.3

      \[\leadsto \sqrt{\frac{\color{blue}{\left(\sqrt{e^{2.0 \cdot x}} + \sqrt{1.0}\right) \cdot \left(\sqrt{e^{2.0 \cdot x}} - \sqrt{1.0}\right)}}{e^{x} - 1.0}}\]

    6. Applied associate-/l*0.3

      \[\leadsto \sqrt{\color{blue}{\frac{\sqrt{e^{2.0 \cdot x}} + \sqrt{1.0}}{\frac{e^{x} - 1.0}{\sqrt{e^{2.0 \cdot x}} - \sqrt{1.0}}}}}\]
    7. Using strategy rm

    8. Applied insert-posit160.1

      \[\leadsto \sqrt{\frac{\sqrt{e^{2.0 \cdot x}} + \sqrt{1.0}}{\color{blue}{\left(\left(\frac{e^{x} - 1.0}{\sqrt{e^{2.0 \cdot x}} - \sqrt{1.0}}\right)\right)}}}\]

    if -1.6879468583117964e-16 < x

    1. Initial program 37.6

      \[\sqrt{\frac{e^{2.0 \cdot x} - 1.0}{e^{x} - 1.0}}\]

    2. Taylor expanded around 0 8.4

      \[\leadsto \color{blue}{\left(\sqrt{2.0} + \left(0.5 \cdot \frac{x}{\sqrt{2.0}} + 0.25 \cdot \frac{{x}^{2}}{\sqrt{2.0}}\right)\right) - 0.125 \cdot \frac{{x}^{2}}{{\left(\sqrt{2.0}\right)}^{3}}}\]

    3. Simplified8.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{2.0}}, 0.5, \sqrt{2.0}\right) + \frac{x}{\frac{\sqrt{2.0}}{x}} \cdot \left(0.25 - \frac{0.125}{2.0}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.6879468583117964 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\frac{\sqrt{1.0} + \sqrt{e^{2.0 \cdot x}}}{\left(\left(\frac{e^{x} - 1.0}{\sqrt{e^{2.0 \cdot x}} - \sqrt{1.0}}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\sqrt{2.0}}, 0.5, \sqrt{2.0}\right) + \left(0.25 - \frac{0.125}{2.0}\right) \cdot \frac{x}{\frac{\sqrt{2.0}}{x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))