Average Error: 53.0 → 0.2
Time: 1.3min
Precision: binary64
Cost: 1794
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.042983533250168:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \leq 0.9609921084103406:\\ \;\;\;\;\left(x + \left(0.075 \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{5}\right) \cdot {\left(\sqrt[3]{x}\right)}^{5}\right) + {x}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\frac{0.5}{x} + \left(x + x\right)\right) - \frac{0.125}{{x}^{3}}\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.042983533250168:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \leq 0.9609921084103406:\\
\;\;\;\;\left(x + \left(0.075 \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{5}\right) \cdot {\left(\sqrt[3]{x}\right)}^{5}\right) + {x}^{3} \cdot -0.16666666666666666\\

\mathbf{else}:\\
\;\;\;\;\log \left(\left(\frac{0.5}{x} + \left(x + x\right)\right) - \frac{0.125}{{x}^{3}}\right)\\

\end{array}
(FPCore (x) :precision binary64 (log (+ x (sqrt (+ (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -1.042983533250168)
   (log (- (/ 0.125 (pow x 3.0)) (+ (/ 0.5 x) (/ 0.0625 (pow x 5.0)))))
   (if (<= x 0.9609921084103406)
     (+
      (+ x (* (* 0.075 (pow (* (cbrt x) (cbrt x)) 5.0)) (pow (cbrt x) 5.0)))
      (* (pow x 3.0) -0.16666666666666666))
     (log (- (+ (/ 0.5 x) (+ x x)) (/ 0.125 (pow x 3.0)))))))
double code(double x) {
	return log(x + sqrt((x * x) + 1.0));
}
↓
double code(double x) {
	double tmp;
	if (x <= -1.042983533250168) {
		tmp = log((0.125 / pow(x, 3.0)) - ((0.5 / x) + (0.0625 / pow(x, 5.0))));
	} else if (x <= 0.9609921084103406) {
		tmp = (x + ((0.075 * pow((cbrt(x) * cbrt(x)), 5.0)) * pow(cbrt(x), 5.0))) + (pow(x, 3.0) * -0.16666666666666666);
	} else {
		tmp = log(((0.5 / x) + (x + x)) - (0.125 / pow(x, 3.0)));
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.0
Target45.5
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x < 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]
Alternative 1
Accuracy8.5
Cost2947
\[\begin{array}{l} \mathbf{if}\;x + \sqrt{x \cdot x + 1} \leq 0.39449055362280006:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{elif}\;x + \sqrt{x \cdot x + 1} \leq 1.0000000000000462:\\ \;\;\;\;\left(x + \left(0.075 \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{5}\right) \cdot {\left(\sqrt[3]{x}\right)}^{5}\right) + {x}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;x + \sqrt{x \cdot x + 1} \leq 6.325716558569126 \cdot 10^{+147}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \end{array}\]
Alternative 2
Accuracy16.5
Cost2690
\[\begin{array}{l} \mathbf{if}\;x + \sqrt{x \cdot x + 1} \leq 0.39449055362280006:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{elif}\;x + \sqrt{x \cdot x + 1} \leq 1.0000000000000462:\\ \;\;\;\;\left(x + \left(0.075 \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{5}\right) \cdot {\left(\sqrt[3]{x}\right)}^{5}\right) + {x}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]
Alternative 3
Accuracy24.2
Cost2049
\[\begin{array}{l} \mathbf{if}\;x + \sqrt{x \cdot x + 1} \leq 1.0000000000000462:\\ \;\;\;\;\left(x + \left(0.075 \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{5}\right) \cdot {\left(\sqrt[3]{x}\right)}^{5}\right) + {x}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]
Alternative 4
Accuracy23.9
Cost1601
\[\begin{array}{l} \mathbf{if}\;x \leq 0.007387916484323772:\\ \;\;\;\;\left(x + \left(0.075 \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{5}\right) \cdot {\left(\sqrt[3]{x}\right)}^{5}\right) + {x}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]
Alternative 5
Accuracy38.5
Cost1668
\[\begin{array}{l} \mathbf{if}\;x \leq 2.1455432006512 \cdot 10^{-310} \lor \neg \left(x \leq 0.007387916484323772\right):\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + 0.075 \cdot {x}^{5}\right) - {\left(\sqrt[3]{x} \cdot e^{\log \left(\sqrt[3]{x}\right)}\right)}^{3} \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array}\]
Alternative 6
Accuracy47.9
Cost1344
\[\left(x + 0.075 \cdot {x}^{5}\right) - {\left(\sqrt[3]{x} \cdot e^{\log \left(\sqrt[3]{x}\right)}\right)}^{3} \cdot \left(x \cdot 0.16666666666666666\right)\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.042983533250168

    1. Initial program 63.1

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

    2. Taylor expanded around -inf 0.1

      \[\leadsto \log \color{blue}{\left(0.125 \cdot \frac{1}{{x}^{3}} - \left(0.5 \cdot \frac{1}{x} + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right)}\]

    3. Simplified0.1

      \[\leadsto \log \color{blue}{\left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)}\]

    if -1.042983533250168 < x < 0.96099210841034055

    1. Initial program 58.6

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

    2. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{\left(x + 0.075 \cdot {x}^{5}\right) - 0.16666666666666666 \cdot {x}^{3}}\]

    3. Simplified0.2

      \[\leadsto \color{blue}{\left(x + 0.075 \cdot {x}^{5}\right) - {x}^{3} \cdot 0.16666666666666666}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt_binary64_18180.2

      \[\leadsto \left(x + 0.075 \cdot {\color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)}}^{5}\right) - {x}^{3} \cdot 0.16666666666666666\]

    6. Applied unpow-prod-down_binary64_18620.2

      \[\leadsto \left(x + 0.075 \cdot \color{blue}{\left({\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{5} \cdot {\left(\sqrt[3]{x}\right)}^{5}\right)}\right) - {x}^{3} \cdot 0.16666666666666666\]

    7. Applied associate-*r*_binary64_17230.2

      \[\leadsto \left(x + \color{blue}{\left(0.075 \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{5}\right) \cdot {\left(\sqrt[3]{x}\right)}^{5}}\right) - {x}^{3} \cdot 0.16666666666666666\]
    8. Using strategy rm

    9. Applied sub-neg_binary64_17760.2

      \[\leadsto \color{blue}{\left(x + \left(0.075 \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{5}\right) \cdot {\left(\sqrt[3]{x}\right)}^{5}\right) + \left(-{x}^{3} \cdot 0.16666666666666666\right)}\]

    10. Simplified0.2

      \[\leadsto \left(x + \left(0.075 \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{5}\right) \cdot {\left(\sqrt[3]{x}\right)}^{5}\right) + \color{blue}{{x}^{3} \cdot -0.16666666666666666}\]

    if 0.96099210841034055 < x

    1. Initial program 31.8

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

    2. Taylor expanded around inf 0.3

      \[\leadsto \log \color{blue}{\left(\left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right) - 0.125 \cdot \frac{1}{{x}^{3}}\right)}\]

    3. Simplified0.3

      \[\leadsto \log \color{blue}{\left(\left(\frac{0.5}{x} + \left(x + x\right)\right) - \frac{0.125}{{x}^{3}}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.042983533250168:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \leq 0.9609921084103406:\\ \;\;\;\;\left(x + \left(0.075 \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{5}\right) \cdot {\left(\sqrt[3]{x}\right)}^{5}\right) + {x}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\frac{0.5}{x} + \left(x + x\right)\right) - \frac{0.125}{{x}^{3}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020322 
(FPCore (x)
  :name "Hyperbolic arcsine"
  :precision binary64

  :herbie-target
  (if (< x 0.0) (log (/ -1.0 (- x (sqrt (+ (* x x) 1.0))))) (log (+ x (sqrt (+ (* x x) 1.0)))))

  (log (+ x (sqrt (+ (* x x) 1.0)))))