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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.0378325736233511:\\ \;\;\;\;\log \left((\left((\left(\frac{\frac{1}{8}}{x}\right) \cdot \left(\frac{1}{x}\right) + \left(-\frac{1}{2}\right))_*\right) \cdot \left(\frac{1}{x}\right) + \left(\frac{-\frac{1}{16}}{{x}^{5}}\right))_*\right)\\ \mathbf{elif}\;x \le 0.006982629770630236:\\ \;\;\;\;\left({x}^{5} \cdot \frac{3}{40} + x\right) - {x}^{3} \cdot \frac{1}{6}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{1^2 + x^2}^* + x\right)\\ \end{array}\]

Original	53.1
Target	45.4
Herbie	0.1

Derivation

Split input into 3 regimes
if x < -1.0378325736233511
1. Initial program 61.9
  \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
2. Initial simplification61.1
  \[\leadsto \log \left(x + \sqrt{1^2 + x^2}^*\right)\]
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
  \[\leadsto \log \color{blue}{\left((\left((\left(\frac{\frac{1}{8}}{x}\right) \cdot \left(\frac{1}{x}\right) + \left(-\frac{1}{2}\right))_*\right) \cdot \left(\frac{1}{x}\right) + \left(\frac{-\frac{1}{16}}{{x}^{5}}\right))_*\right)}\]
if -1.0378325736233511 < x < 0.006982629770630236
1. Initial program 59.0
  \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
2. Initial simplification59.0
  \[\leadsto \log \left(x + \sqrt{1^2 + x^2}^*\right)\]
3. Taylor expanded around 0 0.1
  \[\leadsto \color{blue}{\left(x + \frac{3}{40} \cdot {x}^{5}\right) - \frac{1}{6} \cdot {x}^{3}}\]
if 0.006982629770630236 < x
1. Initial program 32.0
  \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
2. Initial simplification0.1
  \[\leadsto \log \left(x + \sqrt{1^2 + x^2}^*\right)\]
Recombined 3 regimes into one program.
Final simplification0.1
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0378325736233511:\\ \;\;\;\;\log \left((\left((\left(\frac{\frac{1}{8}}{x}\right) \cdot \left(\frac{1}{x}\right) + \left(-\frac{1}{2}\right))_*\right) \cdot \left(\frac{1}{x}\right) + \left(\frac{-\frac{1}{16}}{{x}^{5}}\right))_*\right)\\ \mathbf{elif}\;x \le 0.006982629770630236:\\ \;\;\;\;\left({x}^{5} \cdot \frac{3}{40} + x\right) - {x}^{3} \cdot \frac{1}{6}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{1^2 + x^2}^* + x\right)\\ \end{array}\]

Runtime

herbie shell --seed 2018220 +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)))))

Error

Target

Derivation

`if x < -1.0378325736233511`

`if -1.0378325736233511 < x < 0.006982629770630236`

`if 0.006982629770630236 < x`

Runtime