Hyperbolic arcsine

Average Error: 52.7 → 0.2

Time: 19.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.0529948069434159:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \left(\frac{\frac{1}{8}}{\left(x \cdot x\right) \cdot x} - \frac{\frac{1}{2}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.9556310302201441:\\ \;\;\;\;x + \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{3}{40} - \frac{1}{6}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{\frac{1}{2}}{x} + \left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + x\right)\right)\right)\\ \end{array}\]

C

double f(double x) {
        double r6724179 = x;
        double r6724180 = r6724179 * r6724179;
        double r6724181 = 1.0;
        double r6724182 = r6724180 + r6724181;
        double r6724183 = sqrt(r6724182);
        double r6724184 = r6724179 + r6724183;
        double r6724185 = log(r6724184);
        return r6724185;
}

↓

double f(double x) {
        double r6724186 = x;
        double r6724187 = -1.0529948069434159;
        bool r6724188 = r6724186 <= r6724187;
        double r6724189 = -0.0625;
        double r6724190 = r6724186 * r6724186;
        double r6724191 = r6724190 * r6724186;
        double r6724192 = r6724190 * r6724191;
        double r6724193 = r6724189 / r6724192;
        double r6724194 = 0.125;
        double r6724195 = r6724194 / r6724191;
        double r6724196 = 0.5;
        double r6724197 = r6724196 / r6724186;
        double r6724198 = r6724195 - r6724197;
        double r6724199 = r6724193 + r6724198;
        double r6724200 = log(r6724199);
        double r6724201 = 0.9556310302201441;
        bool r6724202 = r6724186 <= r6724201;
        double r6724203 = 0.075;
        double r6724204 = r6724190 * r6724203;
        double r6724205 = 0.16666666666666666;
        double r6724206 = r6724204 - r6724205;
        double r6724207 = r6724191 * r6724206;
        double r6724208 = r6724186 + r6724207;
        double r6724209 = -0.125;
        double r6724210 = r6724209 / r6724191;
        double r6724211 = r6724210 + r6724186;
        double r6724212 = r6724197 + r6724211;
        double r6724213 = r6724186 + r6724212;
        double r6724214 = log(r6724213);
        double r6724215 = r6724202 ? r6724208 : r6724214;
        double r6724216 = r6724188 ? r6724200 : r6724215;
        return r6724216;
}

Error

Bits error versus x

Target

Original	52.7
Target	44.9
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 < -1.0529948069434159

   1. Initial program 61.8

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

   2. Taylor expanded around -inf 0.1

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

   3. Simplified 0.1

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

   if -1.0529948069434159 < x < 0.9556310302201441

   1. Initial program 58.4

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

   2. Taylor expanded around 0 0.3

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

   3. Simplified 0.3

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

   4. Taylor expanded around -inf 0.3

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

   5. Simplified 0.3

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

   if 0.9556310302201441 < x

   1. Initial program 32.1

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

   2. Taylor expanded around inf 0.1

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

   3. Simplified 0.1

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0529948069434159:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \left(\frac{\frac{1}{8}}{\left(x \cdot x\right) \cdot x} - \frac{\frac{1}{2}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.9556310302201441:\\ \;\;\;\;x + \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{3}{40} - \frac{1}{6}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{\frac{1}{2}}{x} + \left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + x\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 
(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)))))