Average Error: 4.5 → 0.8
Time: 21.8s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -9.47195806647322485446235246220325620925 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(\sqrt{e^{2 \cdot x}}, e^{\log \left(\sqrt{e^{2 \cdot x}}\right)}, -1\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \mathsf{fma}\left(0.4999999999999997779553950749686919152737, x, 1\right) + 2}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -9.47195806647322485446235246220325620925 \cdot 10^{-6}:\\
\;\;\;\;\sqrt{\frac{\mathsf{fma}\left(\sqrt{e^{2 \cdot x}}, e^{\log \left(\sqrt{e^{2 \cdot x}}\right)}, -1\right)}{e^{x} - 1}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{x \cdot \mathsf{fma}\left(0.4999999999999997779553950749686919152737, x, 1\right) + 2}\\

\end{array}
double f(double x) {
        double r1106370 = 2.0;
        double r1106371 = x;
        double r1106372 = r1106370 * r1106371;
        double r1106373 = exp(r1106372);
        double r1106374 = 1.0;
        double r1106375 = r1106373 - r1106374;
        double r1106376 = exp(r1106371);
        double r1106377 = r1106376 - r1106374;
        double r1106378 = r1106375 / r1106377;
        double r1106379 = sqrt(r1106378);
        return r1106379;
}
↓
double f(double x) {
        double r1106380 = x;
        double r1106381 = -9.471958066473225e-06;
        bool r1106382 = r1106380 <= r1106381;
        double r1106383 = 2.0;
        double r1106384 = r1106383 * r1106380;
        double r1106385 = exp(r1106384);
        double r1106386 = sqrt(r1106385);
        double r1106387 = log(r1106386);
        double r1106388 = exp(r1106387);
        double r1106389 = 1.0;
        double r1106390 = -r1106389;
        double r1106391 = fma(r1106386, r1106388, r1106390);
        double r1106392 = exp(r1106380);
        double r1106393 = r1106392 - r1106389;
        double r1106394 = r1106391 / r1106393;
        double r1106395 = sqrt(r1106394);
        double r1106396 = 0.4999999999999998;
        double r1106397 = fma(r1106396, r1106380, r1106389);
        double r1106398 = r1106380 * r1106397;
        double r1106399 = r1106398 + r1106383;
        double r1106400 = sqrt(r1106399);
        double r1106401 = r1106382 ? r1106395 : r1106400;
        return r1106401;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -9.471958066473225e-06

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.1

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

    4. Applied fma-neg0.0

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

    6. Applied add-exp-log0.0

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

    if -9.471958066473225e-06 < x

    1. Initial program 34.5

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

    3. Applied add-sqr-sqrt31.9

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

    4. Applied fma-neg26.5

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

    6. Applied add-exp-log26.5

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

    7. Taylor expanded around 0 6.5

      \[\leadsto \sqrt{\color{blue}{1 \cdot x + \left(0.4999999999999997779553950749686919152737 \cdot {x}^{2} + 2\right)}}\]

    8. Simplified6.5

      \[\leadsto \sqrt{\color{blue}{2 + x \cdot \mathsf{fma}\left(0.4999999999999997779553950749686919152737, x, 1\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9.47195806647322485446235246220325620925 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(\sqrt{e^{2 \cdot x}}, e^{\log \left(\sqrt{e^{2 \cdot x}}\right)}, -1\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \mathsf{fma}\left(0.4999999999999997779553950749686919152737, x, 1\right) + 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))