Average Error: 52.5 → 0.1
Time: 20.1s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0472174035704698:\\ \;\;\;\;\log \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \left(\frac{\frac{1}{2}}{x} + \frac{\frac{1}{16}}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.006429410549232899:\\ \;\;\;\;(\left(x \cdot \frac{-1}{6}\right) \cdot \left(x \cdot x\right) + \left((\frac{3}{40} \cdot \left({x}^{5}\right) + x)_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \log \left(\left(\sqrt[3]{\sqrt{1^2 + x^2}^* + x} \cdot \sqrt[3]{\sqrt{1^2 + x^2}^* + x}\right) \cdot \sqrt[3]{\sqrt{1^2 + x^2}^* + x}\right) + \log \left(\sqrt{\sqrt{1^2 + x^2}^* + x}\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0472174035704698:\\
\;\;\;\;\log \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \left(\frac{\frac{1}{2}}{x} + \frac{\frac{1}{16}}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 0.006429410549232899:\\
\;\;\;\;(\left(x \cdot \frac{-1}{6}\right) \cdot \left(x \cdot x\right) + \left((\frac{3}{40} \cdot \left({x}^{5}\right) + x)_*\right))_*\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{2} \cdot \log \left(\left(\sqrt[3]{\sqrt{1^2 + x^2}^* + x} \cdot \sqrt[3]{\sqrt{1^2 + x^2}^* + x}\right) \cdot \sqrt[3]{\sqrt{1^2 + x^2}^* + x}\right) + \log \left(\sqrt{\sqrt{1^2 + x^2}^* + x}\right)\\

\end{array}
double f(double x) {
        double r18337647 = x;
        double r18337648 = r18337647 * r18337647;
        double r18337649 = 1.0;
        double r18337650 = r18337648 + r18337649;
        double r18337651 = sqrt(r18337650);
        double r18337652 = r18337647 + r18337651;
        double r18337653 = log(r18337652);
        return r18337653;
}
↓
double f(double x) {
        double r18337654 = x;
        double r18337655 = -1.0472174035704698;
        bool r18337656 = r18337654 <= r18337655;
        double r18337657 = 0.125;
        double r18337658 = r18337657 / r18337654;
        double r18337659 = r18337654 * r18337654;
        double r18337660 = r18337658 / r18337659;
        double r18337661 = 0.5;
        double r18337662 = r18337661 / r18337654;
        double r18337663 = 0.0625;
        double r18337664 = 5.0;
        double r18337665 = pow(r18337654, r18337664);
        double r18337666 = r18337663 / r18337665;
        double r18337667 = r18337662 + r18337666;
        double r18337668 = r18337660 - r18337667;
        double r18337669 = log(r18337668);
        double r18337670 = 0.006429410549232899;
        bool r18337671 = r18337654 <= r18337670;
        double r18337672 = -0.16666666666666666;
        double r18337673 = r18337654 * r18337672;
        double r18337674 = 0.075;
        double r18337675 = fma(r18337674, r18337665, r18337654);
        double r18337676 = fma(r18337673, r18337659, r18337675);
        double r18337677 = 1.0;
        double r18337678 = hypot(r18337677, r18337654);
        double r18337679 = r18337678 + r18337654;
        double r18337680 = cbrt(r18337679);
        double r18337681 = r18337680 * r18337680;
        double r18337682 = r18337681 * r18337680;
        double r18337683 = log(r18337682);
        double r18337684 = r18337661 * r18337683;
        double r18337685 = sqrt(r18337679);
        double r18337686 = log(r18337685);
        double r18337687 = r18337684 + r18337686;
        double r18337688 = r18337671 ? r18337676 : r18337687;
        double r18337689 = r18337656 ? r18337669 : r18337688;
        return r18337689;
}

Error

Bits error versus x

Target

Original52.5
Target44.8
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \lt 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0472174035704698

    1. Initial program 61.7

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

    2. Simplified60.9

      \[\leadsto \color{blue}{\log \left(x + \sqrt{1^2 + x^2}^*\right)}\]

    3. Taylor expanded around -inf 0.2

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

    4. Simplified0.2

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

    if -1.0472174035704698 < x < 0.006429410549232899

    1. Initial program 58.8

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

    2. Simplified58.8

      \[\leadsto \color{blue}{\log \left(x + \sqrt{1^2 + x^2}^*\right)}\]

    3. Taylor expanded around 0 0.1

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

    4. Simplified0.1

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

    if 0.006429410549232899 < x

    1. Initial program 31.0

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

    2. Simplified0.1

      \[\leadsto \color{blue}{\log \left(x + \sqrt{1^2 + x^2}^*\right)}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt0.1

      \[\leadsto \log \color{blue}{\left(\sqrt{x + \sqrt{1^2 + x^2}^*} \cdot \sqrt{x + \sqrt{1^2 + x^2}^*}\right)}\]

    5. Applied log-prod0.1

      \[\leadsto \color{blue}{\log \left(\sqrt{x + \sqrt{1^2 + x^2}^*}\right) + \log \left(\sqrt{x + \sqrt{1^2 + x^2}^*}\right)}\]
    6. Using strategy rm

    7. Applied pow10.1

      \[\leadsto \log \left(\sqrt{x + \sqrt{1^2 + x^2}^*}\right) + \log \left(\sqrt{\color{blue}{{\left(x + \sqrt{1^2 + x^2}^*\right)}^{1}}}\right)\]

    8. Applied sqrt-pow10.1

      \[\leadsto \log \left(\sqrt{x + \sqrt{1^2 + x^2}^*}\right) + \log \color{blue}{\left({\left(x + \sqrt{1^2 + x^2}^*\right)}^{\left(\frac{1}{2}\right)}\right)}\]

    9. Applied log-pow0.1

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

    10. Simplified0.1

      \[\leadsto \log \left(\sqrt{x + \sqrt{1^2 + x^2}^*}\right) + \color{blue}{\frac{1}{2}} \cdot \log \left(x + \sqrt{1^2 + x^2}^*\right)\]
    11. Using strategy rm

    12. Applied add-cube-cbrt0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0472174035704698:\\ \;\;\;\;\log \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \left(\frac{\frac{1}{2}}{x} + \frac{\frac{1}{16}}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.006429410549232899:\\ \;\;\;\;(\left(x \cdot \frac{-1}{6}\right) \cdot \left(x \cdot x\right) + \left((\frac{3}{40} \cdot \left({x}^{5}\right) + x)_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \log \left(\left(\sqrt[3]{\sqrt{1^2 + x^2}^* + x} \cdot \sqrt[3]{\sqrt{1^2 + x^2}^* + x}\right) \cdot \sqrt[3]{\sqrt{1^2 + x^2}^* + x}\right) + \log \left(\sqrt{\sqrt{1^2 + x^2}^* + x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(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)))))