Hyperbolic arcsine

Average Error: 52.4 → 0.2

Time: 17.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.0611438585736817:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} - \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{8}}{\left(x \cdot x\right) \cdot x}\right)\right)\\ \mathbf{elif}\;x \le 0.9494345950728007:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \frac{3}{40} + \left(x - \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{1}{6}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\frac{\frac{1}{2}}{x} + \left(\frac{\frac{\frac{-1}{8}}{x \cdot x}}{x} + x\right)\right) + x\right)\\ \end{array}\]

C

double f(double x) {
        double r6167930 = x;
        double r6167931 = r6167930 * r6167930;
        double r6167932 = 1.0;
        double r6167933 = r6167931 + r6167932;
        double r6167934 = sqrt(r6167933);
        double r6167935 = r6167930 + r6167934;
        double r6167936 = log(r6167935);
        return r6167936;
}

↓

double f(double x) {
        double r6167937 = x;
        double r6167938 = -1.0611438585736817;
        bool r6167939 = r6167937 <= r6167938;
        double r6167940 = -0.0625;
        double r6167941 = r6167937 * r6167937;
        double r6167942 = r6167941 * r6167937;
        double r6167943 = r6167941 * r6167942;
        double r6167944 = r6167940 / r6167943;
        double r6167945 = 0.5;
        double r6167946 = r6167945 / r6167937;
        double r6167947 = 0.125;
        double r6167948 = r6167947 / r6167942;
        double r6167949 = r6167946 - r6167948;
        double r6167950 = r6167944 - r6167949;
        double r6167951 = log(r6167950);
        double r6167952 = 0.9494345950728007;
        bool r6167953 = r6167937 <= r6167952;
        double r6167954 = 0.075;
        double r6167955 = r6167943 * r6167954;
        double r6167956 = 0.16666666666666666;
        double r6167957 = r6167942 * r6167956;
        double r6167958 = r6167937 - r6167957;
        double r6167959 = r6167955 + r6167958;
        double r6167960 = -0.125;
        double r6167961 = r6167960 / r6167941;
        double r6167962 = r6167961 / r6167937;
        double r6167963 = r6167962 + r6167937;
        double r6167964 = r6167946 + r6167963;
        double r6167965 = r6167964 + r6167937;
        double r6167966 = log(r6167965);
        double r6167967 = r6167953 ? r6167959 : r6167966;
        double r6167968 = r6167939 ? r6167951 : r6167967;
        return r6167968;
}

Error

Bits error versus x

Target

Original	52.4
Target	44.6
Herbie	0.2

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

Derivation

1. Split input into 3 regimes

2. if x < -1.0611438585736817

   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. Simplified 0.2

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

   if -1.0611438585736817 < x < 0.9494345950728007

   1. Initial program 58.6

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

   2. Taylor expanded around 0 0.1

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

   3. Simplified 0.1

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

   if 0.9494345950728007 < x

   1. Initial program 30.9

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

   2. Taylor expanded around inf 0.2

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

   3. Simplified 0.2

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0611438585736817:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} - \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{8}}{\left(x \cdot x\right) \cdot x}\right)\right)\\ \mathbf{elif}\;x \le 0.9494345950728007:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \frac{3}{40} + \left(x - \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{1}{6}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\frac{\frac{1}{2}}{x} + \left(\frac{\frac{\frac{-1}{8}}{x \cdot x}}{x} + x\right)\right) + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 
(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)))))