Average Error: 4.7 → 0.8
Time: 27.5s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.54501292903996873 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.54501292903996873 \cdot 10^{-10}:\\
\;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\

\end{array}
double f(double x) {
        double r25567 = 2.0;
        double r25568 = x;
        double r25569 = r25567 * r25568;
        double r25570 = exp(r25569);
        double r25571 = 1.0;
        double r25572 = r25570 - r25571;
        double r25573 = exp(r25568);
        double r25574 = r25573 - r25571;
        double r25575 = r25572 / r25574;
        double r25576 = sqrt(r25575);
        return r25576;
}
↓
double f(double x) {
        double r25577 = x;
        double r25578 = -4.5450129290399687e-10;
        bool r25579 = r25577 <= r25578;
        double r25580 = 2.0;
        double r25581 = r25580 * r25577;
        double r25582 = exp(r25581);
        double r25583 = 1.0;
        double r25584 = r25582 - r25583;
        double r25585 = r25577 + r25577;
        double r25586 = exp(r25585);
        double r25587 = r25583 * r25583;
        double r25588 = r25586 - r25587;
        double r25589 = r25584 / r25588;
        double r25590 = exp(r25577);
        double r25591 = r25590 + r25583;
        double r25592 = r25589 * r25591;
        double r25593 = sqrt(r25592);
        double r25594 = 0.5;
        double r25595 = 2.0;
        double r25596 = pow(r25577, r25595);
        double r25597 = fma(r25583, r25577, r25580);
        double r25598 = fma(r25594, r25596, r25597);
        double r25599 = sqrt(r25598);
        double r25600 = r25579 ? r25593 : r25599;
        return r25600;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -4.5450129290399687e-10

    1. Initial program 0.3

      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
    2. Using strategy rm

    3. Applied flip--0.2

      \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}}\]

    4. Applied associate-/r/0.2

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

    5. Simplified0.0

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

    if -4.5450129290399687e-10 < x

    1. Initial program 38.3

      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]

    2. Taylor expanded around 0 7.0

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

    3. Simplified7.0

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.54501292903996873 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))