Average Error: 52.9 → 0.1
Time: 7.1s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0196250072526523:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.00100680770020326426:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 \cdot \left(\mathsf{hypot}\left(x, \sqrt{1}\right) + x\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0196250072526523:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 0.00100680770020326426:\\
\;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\log \left(1 \cdot \left(\mathsf{hypot}\left(x, \sqrt{1}\right) + x\right)\right)\\

\end{array}
double f(double x) {
        double r186163 = x;
        double r186164 = r186163 * r186163;
        double r186165 = 1.0;
        double r186166 = r186164 + r186165;
        double r186167 = sqrt(r186166);
        double r186168 = r186163 + r186167;
        double r186169 = log(r186168);
        return r186169;
}
↓
double f(double x) {
        double r186170 = x;
        double r186171 = -1.0196250072526523;
        bool r186172 = r186170 <= r186171;
        double r186173 = 0.125;
        double r186174 = 3.0;
        double r186175 = pow(r186170, r186174);
        double r186176 = r186173 / r186175;
        double r186177 = 0.5;
        double r186178 = r186177 / r186170;
        double r186179 = 0.0625;
        double r186180 = -r186179;
        double r186181 = 5.0;
        double r186182 = pow(r186170, r186181);
        double r186183 = r186180 / r186182;
        double r186184 = r186178 - r186183;
        double r186185 = r186176 - r186184;
        double r186186 = log(r186185);
        double r186187 = 0.0010068077002032643;
        bool r186188 = r186170 <= r186187;
        double r186189 = 1.0;
        double r186190 = sqrt(r186189);
        double r186191 = log(r186190);
        double r186192 = r186170 / r186190;
        double r186193 = r186191 + r186192;
        double r186194 = 0.16666666666666666;
        double r186195 = pow(r186190, r186174);
        double r186196 = r186175 / r186195;
        double r186197 = r186194 * r186196;
        double r186198 = r186193 - r186197;
        double r186199 = 1.0;
        double r186200 = hypot(r186170, r186190);
        double r186201 = r186200 + r186170;
        double r186202 = r186199 * r186201;
        double r186203 = log(r186202);
        double r186204 = r186188 ? r186198 : r186203;
        double r186205 = r186172 ? r186186 : r186204;
        return r186205;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.9
Target45.3
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0196250072526523

    1. Initial program 62.9

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

    2. Taylor expanded around -inf 0.2

      \[\leadsto \log \color{blue}{\left(0.125 \cdot \frac{1}{{x}^{3}} - \left(0.5 \cdot \frac{1}{x} + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right)}\]

    3. Simplified0.2

      \[\leadsto \log \color{blue}{\left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)}\]

    if -1.0196250072526523 < x < 0.0010068077002032643

    1. Initial program 59.1

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

    2. Taylor expanded around 0 0.1

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

    if 0.0010068077002032643 < x

    1. Initial program 31.2

      \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
    2. Using strategy rm

    3. Applied *-un-lft-identity31.2

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

    4. Applied *-un-lft-identity31.2

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

    5. Applied distribute-lft-out31.2

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

    6. Simplified0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0196250072526523:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.00100680770020326426:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 \cdot \left(\mathsf{hypot}\left(x, \sqrt{1}\right) + x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020003 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arcsine"
  :precision binary64

  :herbie-target
  (if (< x 0.0) (log (/ -1 (- x (sqrt (+ (* x x) 1))))) (log (+ x (sqrt (+ (* x x) 1)))))

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