Average Error: 53.0 → 0.2
Time: 6.0s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0296783040954165:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.0011756572267579168:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\left|\sqrt[3]{\mathsf{hypot}\left(x, \sqrt{1}\right)}\right| \cdot \sqrt{\sqrt[3]{\mathsf{hypot}\left(x, \sqrt{1}\right)}}, \sqrt{\mathsf{hypot}\left(x, \sqrt{1}\right)}, x\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0296783040954165:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 0.0011756572267579168:\\
\;\;\;\;\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(\mathsf{fma}\left(\left|\sqrt[3]{\mathsf{hypot}\left(x, \sqrt{1}\right)}\right| \cdot \sqrt{\sqrt[3]{\mathsf{hypot}\left(x, \sqrt{1}\right)}}, \sqrt{\mathsf{hypot}\left(x, \sqrt{1}\right)}, x\right)\right)\\

\end{array}
double f(double x) {
        double r156924 = x;
        double r156925 = r156924 * r156924;
        double r156926 = 1.0;
        double r156927 = r156925 + r156926;
        double r156928 = sqrt(r156927);
        double r156929 = r156924 + r156928;
        double r156930 = log(r156929);
        return r156930;
}
↓
double f(double x) {
        double r156931 = x;
        double r156932 = -1.0296783040954165;
        bool r156933 = r156931 <= r156932;
        double r156934 = 0.125;
        double r156935 = 3.0;
        double r156936 = pow(r156931, r156935);
        double r156937 = r156934 / r156936;
        double r156938 = 0.5;
        double r156939 = r156938 / r156931;
        double r156940 = 0.0625;
        double r156941 = -r156940;
        double r156942 = 5.0;
        double r156943 = pow(r156931, r156942);
        double r156944 = r156941 / r156943;
        double r156945 = r156939 - r156944;
        double r156946 = r156937 - r156945;
        double r156947 = log(r156946);
        double r156948 = 0.0011756572267579168;
        bool r156949 = r156931 <= r156948;
        double r156950 = 1.0;
        double r156951 = sqrt(r156950);
        double r156952 = log(r156951);
        double r156953 = r156931 / r156951;
        double r156954 = r156952 + r156953;
        double r156955 = 0.16666666666666666;
        double r156956 = pow(r156951, r156935);
        double r156957 = r156936 / r156956;
        double r156958 = r156955 * r156957;
        double r156959 = r156954 - r156958;
        double r156960 = hypot(r156931, r156951);
        double r156961 = cbrt(r156960);
        double r156962 = fabs(r156961);
        double r156963 = sqrt(r156961);
        double r156964 = r156962 * r156963;
        double r156965 = sqrt(r156960);
        double r156966 = fma(r156964, r156965, r156931);
        double r156967 = log(r156966);
        double r156968 = r156949 ? r156959 : r156967;
        double r156969 = r156933 ? r156947 : r156968;
        return r156969;
}

Error

Bits error versus x

Target

Original53.0
Target45.3
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 < -1.0296783040954165

    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 -1.0296783040954165 < x < 0.0011756572267579168

    1. Initial program 58.7

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

    2. Taylor expanded around 0 0.2

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

    if 0.0011756572267579168 < x

    1. Initial program 31.8

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

    3. Applied add-log-exp31.8

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

    4. Simplified0.1

      \[\leadsto \log \color{blue}{\left(\mathsf{hypot}\left(x, \sqrt{1}\right) + x\right)}\]
    5. Using strategy rm

    6. Applied add-sqr-sqrt0.1

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

    7. Applied fma-def0.1

      \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(\sqrt{\mathsf{hypot}\left(x, \sqrt{1}\right)}, \sqrt{\mathsf{hypot}\left(x, \sqrt{1}\right)}, x\right)\right)}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt0.1

      \[\leadsto \log \left(\mathsf{fma}\left(\sqrt{\color{blue}{\left(\sqrt[3]{\mathsf{hypot}\left(x, \sqrt{1}\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(x, \sqrt{1}\right)}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(x, \sqrt{1}\right)}}}, \sqrt{\mathsf{hypot}\left(x, \sqrt{1}\right)}, x\right)\right)\]

    10. Applied sqrt-prod0.1

      \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{\sqrt{\sqrt[3]{\mathsf{hypot}\left(x, \sqrt{1}\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(x, \sqrt{1}\right)}} \cdot \sqrt{\sqrt[3]{\mathsf{hypot}\left(x, \sqrt{1}\right)}}}, \sqrt{\mathsf{hypot}\left(x, \sqrt{1}\right)}, x\right)\right)\]

    11. Simplified0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0296783040954165:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.0011756572267579168:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\left|\sqrt[3]{\mathsf{hypot}\left(x, \sqrt{1}\right)}\right| \cdot \sqrt{\sqrt[3]{\mathsf{hypot}\left(x, \sqrt{1}\right)}}, \sqrt{\mathsf{hypot}\left(x, \sqrt{1}\right)}, x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +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)))))