Average Error: 52.5 → 0.2
Time: 16.0s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0772372773411123:\\ \;\;\;\;\log \left(\sqrt{\frac{\frac{-1}{16}}{{x}^{5}} + \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} + \frac{\frac{-1}{2}}{x}\right)}\right) + \log \left(\sqrt{\frac{\frac{-1}{16}}{{x}^{5}} + \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} + \frac{\frac{-1}{2}}{x}\right)}\right)\\ \mathbf{elif}\;x \le 0.9655399906543994:\\ \;\;\;\;\left(\frac{3}{40} \cdot {x}^{5} + \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{6}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{x}{\frac{1}{2}} + \left(\frac{\frac{1}{2}}{x} - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0772372773411123:\\
\;\;\;\;\log \left(\sqrt{\frac{\frac{-1}{16}}{{x}^{5}} + \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} + \frac{\frac{-1}{2}}{x}\right)}\right) + \log \left(\sqrt{\frac{\frac{-1}{16}}{{x}^{5}} + \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} + \frac{\frac{-1}{2}}{x}\right)}\right)\\

\mathbf{elif}\;x \le 0.9655399906543994:\\
\;\;\;\;\left(\frac{3}{40} \cdot {x}^{5} + \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{6}\right) + x\\

\mathbf{else}:\\
\;\;\;\;\log \left(\frac{x}{\frac{1}{2}} + \left(\frac{\frac{1}{2}}{x} - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\right)\\

\end{array}
double f(double x) {
        double r4972032 = x;
        double r4972033 = r4972032 * r4972032;
        double r4972034 = 1.0;
        double r4972035 = r4972033 + r4972034;
        double r4972036 = sqrt(r4972035);
        double r4972037 = r4972032 + r4972036;
        double r4972038 = log(r4972037);
        return r4972038;
}
↓
double f(double x) {
        double r4972039 = x;
        double r4972040 = -1.0772372773411123;
        bool r4972041 = r4972039 <= r4972040;
        double r4972042 = -0.0625;
        double r4972043 = 5.0;
        double r4972044 = pow(r4972039, r4972043);
        double r4972045 = r4972042 / r4972044;
        double r4972046 = 0.125;
        double r4972047 = r4972046 / r4972039;
        double r4972048 = r4972039 * r4972039;
        double r4972049 = r4972047 / r4972048;
        double r4972050 = -0.5;
        double r4972051 = r4972050 / r4972039;
        double r4972052 = r4972049 + r4972051;
        double r4972053 = r4972045 + r4972052;
        double r4972054 = sqrt(r4972053);
        double r4972055 = log(r4972054);
        double r4972056 = r4972055 + r4972055;
        double r4972057 = 0.9655399906543994;
        bool r4972058 = r4972039 <= r4972057;
        double r4972059 = 0.075;
        double r4972060 = r4972059 * r4972044;
        double r4972061 = r4972048 * r4972039;
        double r4972062 = -0.16666666666666666;
        double r4972063 = r4972061 * r4972062;
        double r4972064 = r4972060 + r4972063;
        double r4972065 = r4972064 + r4972039;
        double r4972066 = 0.5;
        double r4972067 = r4972039 / r4972066;
        double r4972068 = r4972066 / r4972039;
        double r4972069 = r4972068 - r4972049;
        double r4972070 = r4972067 + r4972069;
        double r4972071 = log(r4972070);
        double r4972072 = r4972058 ? r4972065 : r4972071;
        double r4972073 = r4972041 ? r4972056 : r4972072;
        return r4972073;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.5
Target45.0
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0772372773411123

    1. Initial program 61.8

      \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]

    2. Taylor expanded around -inf 0.2

      \[\leadsto \log \color{blue}{\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}} - \left(\frac{1}{16} \cdot \frac{1}{{x}^{5}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\]

    3. Simplified0.2

      \[\leadsto \log \color{blue}{\left(\frac{\frac{-1}{16}}{{x}^{5}} + \left(\frac{\frac{-1}{2}}{x} + \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\right)}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt0.2

      \[\leadsto \log \color{blue}{\left(\sqrt{\frac{\frac{-1}{16}}{{x}^{5}} + \left(\frac{\frac{-1}{2}}{x} + \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)} \cdot \sqrt{\frac{\frac{-1}{16}}{{x}^{5}} + \left(\frac{\frac{-1}{2}}{x} + \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)}\right)}\]

    6. Applied log-prod0.2

      \[\leadsto \color{blue}{\log \left(\sqrt{\frac{\frac{-1}{16}}{{x}^{5}} + \left(\frac{\frac{-1}{2}}{x} + \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)}\right) + \log \left(\sqrt{\frac{\frac{-1}{16}}{{x}^{5}} + \left(\frac{\frac{-1}{2}}{x} + \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)}\right)}\]

    if -1.0772372773411123 < x < 0.9655399906543994

    1. Initial program 58.5

      \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]

    2. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{\left(x + \frac{3}{40} \cdot {x}^{5}\right) - \frac{1}{6} \cdot {x}^{3}}\]

    3. Simplified0.2

      \[\leadsto \color{blue}{x + \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{6} + \frac{3}{40} \cdot {x}^{5}\right)}\]

    if 0.9655399906543994 < x

    1. Initial program 30.6

      \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]

    2. Taylor expanded around inf 0.2

      \[\leadsto \log \color{blue}{\left(\left(2 \cdot x + \frac{1}{2} \cdot \frac{1}{x}\right) - \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)}\]

    3. Simplified0.2

      \[\leadsto \log \color{blue}{\left(\frac{x}{\frac{1}{2}} + \left(\frac{\frac{1}{2}}{x} - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0772372773411123:\\ \;\;\;\;\log \left(\sqrt{\frac{\frac{-1}{16}}{{x}^{5}} + \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} + \frac{\frac{-1}{2}}{x}\right)}\right) + \log \left(\sqrt{\frac{\frac{-1}{16}}{{x}^{5}} + \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} + \frac{\frac{-1}{2}}{x}\right)}\right)\\ \mathbf{elif}\;x \le 0.9655399906543994:\\ \;\;\;\;\left(\frac{3}{40} \cdot {x}^{5} + \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{6}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{x}{\frac{1}{2}} + \left(\frac{\frac{1}{2}}{x} - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 
(FPCore (x)
  :name "Hyperbolic arcsine"

  :herbie-target
  (if (< x 0) (log (/ -1 (- x (sqrt (+ (* x x) 1))))) (log (+ x (sqrt (+ (* x x) 1)))))

  (log (+ x (sqrt (+ (* x x) 1)))))