Average Error: 52.7 → 0.2
Time: 32.2s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.014477909773237040980120582389645278454:\\ \;\;\;\;\log \left(\frac{\frac{0.125}{x}}{x \cdot x} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.8909481582438814051272402139147743582726:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x \cdot x}{1} \cdot x}{\sqrt{1}}, \frac{-1}{6}, \frac{x}{\sqrt{1}} + \log \left(\sqrt{1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(x, 2, \frac{0.5}{x} - \frac{\frac{0.125}{x}}{x \cdot x}\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.014477909773237040980120582389645278454:\\
\;\;\;\;\log \left(\frac{\frac{0.125}{x}}{x \cdot x} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 0.8909481582438814051272402139147743582726:\\
\;\;\;\;\mathsf{fma}\left(\frac{\frac{x \cdot x}{1} \cdot x}{\sqrt{1}}, \frac{-1}{6}, \frac{x}{\sqrt{1}} + \log \left(\sqrt{1}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\mathsf{fma}\left(x, 2, \frac{0.5}{x} - \frac{\frac{0.125}{x}}{x \cdot x}\right)\right)\\

\end{array}
double f(double x) {
        double r4576094 = x;
        double r4576095 = r4576094 * r4576094;
        double r4576096 = 1.0;
        double r4576097 = r4576095 + r4576096;
        double r4576098 = sqrt(r4576097);
        double r4576099 = r4576094 + r4576098;
        double r4576100 = log(r4576099);
        return r4576100;
}
↓
double f(double x) {
        double r4576101 = x;
        double r4576102 = -1.014477909773237;
        bool r4576103 = r4576101 <= r4576102;
        double r4576104 = 0.125;
        double r4576105 = r4576104 / r4576101;
        double r4576106 = r4576101 * r4576101;
        double r4576107 = r4576105 / r4576106;
        double r4576108 = 0.5;
        double r4576109 = r4576108 / r4576101;
        double r4576110 = 0.0625;
        double r4576111 = 5.0;
        double r4576112 = pow(r4576101, r4576111);
        double r4576113 = r4576110 / r4576112;
        double r4576114 = r4576109 + r4576113;
        double r4576115 = r4576107 - r4576114;
        double r4576116 = log(r4576115);
        double r4576117 = 0.8909481582438814;
        bool r4576118 = r4576101 <= r4576117;
        double r4576119 = 1.0;
        double r4576120 = r4576106 / r4576119;
        double r4576121 = r4576120 * r4576101;
        double r4576122 = sqrt(r4576119);
        double r4576123 = r4576121 / r4576122;
        double r4576124 = -0.16666666666666666;
        double r4576125 = r4576101 / r4576122;
        double r4576126 = log(r4576122);
        double r4576127 = r4576125 + r4576126;
        double r4576128 = fma(r4576123, r4576124, r4576127);
        double r4576129 = 2.0;
        double r4576130 = r4576109 - r4576107;
        double r4576131 = fma(r4576101, r4576129, r4576130);
        double r4576132 = log(r4576131);
        double r4576133 = r4576118 ? r4576128 : r4576132;
        double r4576134 = r4576103 ? r4576116 : r4576133;
        return r4576134;
}

Error

Bits error versus x

Target

Original52.7
Target44.9
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.014477909773237

    1. Initial program 63.0

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

    2. Simplified63.0

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

    3. Taylor expanded around -inf 0.2

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

    4. Simplified0.2

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

    if -1.014477909773237 < x < 0.8909481582438814

    1. Initial program 58.4

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

    2. Simplified58.4

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

    3. Taylor expanded around 0 0.3

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

    4. Simplified0.3

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

    if 0.8909481582438814 < x

    1. Initial program 31.4

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

    2. Simplified31.4

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

    3. Taylor expanded around inf 0.1

      \[\leadsto \log \color{blue}{\left(\left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right) - 0.125 \cdot \frac{1}{{x}^{3}}\right)}\]

    4. Simplified0.1

      \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(x, 2, \frac{0.5}{x} - \frac{\frac{0.125}{x}}{x \cdot x}\right)\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.014477909773237040980120582389645278454:\\ \;\;\;\;\log \left(\frac{\frac{0.125}{x}}{x \cdot x} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.8909481582438814051272402139147743582726:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x \cdot x}{1} \cdot x}{\sqrt{1}}, \frac{-1}{6}, \frac{x}{\sqrt{1}} + \log \left(\sqrt{1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(x, 2, \frac{0.5}{x} - \frac{\frac{0.125}{x}}{x \cdot x}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arcsine"

  :herbie-target
  (if (< x 0.0) (log (/ -1.0 (- x (sqrt (+ (* x x) 1.0))))) (log (+ x (sqrt (+ (* x x) 1.0)))))

  (log (+ x (sqrt (+ (* x x) 1.0)))))