Average Error: 52.7 → 0.3
Time: 15.7s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0840635159626253:\\ \;\;\;\;\log \left(\frac{\frac{-1}{2}}{x} - \left(\frac{\frac{1}{16}}{{x}^{5}} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\\ \mathbf{elif}\;x \le 0.9540314055762552:\\ \;\;\;\;\left(x - \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{6}\right) + \frac{3}{40} \cdot {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\frac{1}{2}}{x} + \left(\left(x + x\right) - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0840635159626253:\\
\;\;\;\;\log \left(\frac{\frac{-1}{2}}{x} - \left(\frac{\frac{1}{16}}{{x}^{5}} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\\

\mathbf{elif}\;x \le 0.9540314055762552:\\
\;\;\;\;\left(x - \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{6}\right) + \frac{3}{40} \cdot {x}^{5}\\

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

\end{array}
double f(double x) {
        double r5433037 = x;
        double r5433038 = r5433037 * r5433037;
        double r5433039 = 1.0;
        double r5433040 = r5433038 + r5433039;
        double r5433041 = sqrt(r5433040);
        double r5433042 = r5433037 + r5433041;
        double r5433043 = log(r5433042);
        return r5433043;
}
↓
double f(double x) {
        double r5433044 = x;
        double r5433045 = -1.0840635159626253;
        bool r5433046 = r5433044 <= r5433045;
        double r5433047 = -0.5;
        double r5433048 = r5433047 / r5433044;
        double r5433049 = 0.0625;
        double r5433050 = 5.0;
        double r5433051 = pow(r5433044, r5433050);
        double r5433052 = r5433049 / r5433051;
        double r5433053 = 0.125;
        double r5433054 = r5433044 * r5433044;
        double r5433055 = r5433044 * r5433054;
        double r5433056 = r5433053 / r5433055;
        double r5433057 = r5433052 - r5433056;
        double r5433058 = r5433048 - r5433057;
        double r5433059 = log(r5433058);
        double r5433060 = 0.9540314055762552;
        bool r5433061 = r5433044 <= r5433060;
        double r5433062 = 0.16666666666666666;
        double r5433063 = r5433055 * r5433062;
        double r5433064 = r5433044 - r5433063;
        double r5433065 = 0.075;
        double r5433066 = r5433065 * r5433051;
        double r5433067 = r5433064 + r5433066;
        double r5433068 = 0.5;
        double r5433069 = r5433068 / r5433044;
        double r5433070 = r5433044 + r5433044;
        double r5433071 = r5433070 - r5433056;
        double r5433072 = r5433069 + r5433071;
        double r5433073 = log(r5433072);
        double r5433074 = r5433061 ? r5433067 : r5433073;
        double r5433075 = r5433046 ? r5433059 : r5433074;
        return r5433075;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.7
Target44.8
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x \lt 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0840635159626253

    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

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

    if -1.0840635159626253 < x < 0.9540314055762552

    1. Initial program 58.4

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

    2. Taylor expanded around 0 0.3

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

    3. Simplified0.3

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

    if 0.9540314055762552 < x

    1. Initial program 32.3

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

    2. Taylor expanded around inf 0.4

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

    3. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0840635159626253:\\ \;\;\;\;\log \left(\frac{\frac{-1}{2}}{x} - \left(\frac{\frac{1}{16}}{{x}^{5}} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\\ \mathbf{elif}\;x \le 0.9540314055762552:\\ \;\;\;\;\left(x - \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{6}\right) + \frac{3}{40} \cdot {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\frac{1}{2}}{x} + \left(\left(x + x\right) - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(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)))))