Average Error: 53.0 → 0.3
Time: 5.0s
Precision: binary64
Cost: 7938
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.3299041834197318:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.0253859810357775:\\ \;\;\;\;\left(x - \left(x \cdot x\right) \cdot \left(x \cdot 0.16666666666666666\right)\right) + {x}^{5} \cdot 0.075\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{-1 - \frac{0.25}{x \cdot x}}{\frac{-0.5}{x}}\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.3299041834197318:\\
\;\;\;\;\log \left(\frac{-0.5}{x}\right)\\

\mathbf{elif}\;x \leq 1.0253859810357775:\\
\;\;\;\;\left(x - \left(x \cdot x\right) \cdot \left(x \cdot 0.16666666666666666\right)\right) + {x}^{5} \cdot 0.075\\

\mathbf{else}:\\
\;\;\;\;\log \left(\frac{-1 - \frac{0.25}{x \cdot x}}{\frac{-0.5}{x}}\right)\\

\end{array}
(FPCore (x) :precision binary64 (log (+ x (sqrt (+ (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -1.3299041834197318)
   (log (/ -0.5 x))
   (if (<= x 1.0253859810357775)
     (+ (- x (* (* x x) (* x 0.16666666666666666))) (* (pow x 5.0) 0.075))
     (log (/ (- -1.0 (/ 0.25 (* x x))) (/ -0.5 x))))))
double code(double x) {
	return log(x + sqrt((x * x) + 1.0));
}
↓
double code(double x) {
	double tmp;
	if (x <= -1.3299041834197318) {
		tmp = log(-0.5 / x);
	} else if (x <= 1.0253859810357775) {
		tmp = (x - ((x * x) * (x * 0.16666666666666666))) + (pow(x, 5.0) * 0.075);
	} else {
		tmp = log((-1.0 - (0.25 / (x * x))) / (-0.5 / x));
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.0
Target45.1
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x < 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Alternatives

Alternative 1
Error0.4
Cost7746
\[\begin{array}{l} \mathbf{if}\;x \leq -1.284479305234417:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.9589061090684858:\\ \;\;\;\;x - 0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{-1 - \frac{0.25}{x \cdot x}}{\frac{-0.5}{x}}\right)\\ \end{array}\]
Alternative 2
Error0.4
Cost7490
\[\begin{array}{l} \mathbf{if}\;x \leq -1.284479305234417:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.9589061090684858:\\ \;\;\;\;x - 0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array}\]
Alternative 3
Error0.5
Cost7426
\[\begin{array}{l} \mathbf{if}\;x \leq -1.284479305234417:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.2763914277996573:\\ \;\;\;\;x - 0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array}\]
Alternative 4
Error0.6
Cost7234
\[\begin{array}{l} \mathbf{if}\;x \leq -1.2390544270491022:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.2763914277996573:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array}\]
Alternative 5
Error14.4
Cost7234
\[\begin{array}{l} \mathbf{if}\;x \leq -1.0119300361225283:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.2763914277996573:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array}\]
Alternative 6
Error27.7
Cost706
\[\begin{array}{l} \mathbf{if}\;x \leq -1.0119300361225283:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 0.9947640300347544:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
Alternative 7
Error57.8
Cost385
\[\begin{array}{l} \mathbf{if}\;x \leq -4.74488972669753 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
Alternative 8
Error59.3
Cost385
\[\begin{array}{l} \mathbf{if}\;x \leq 1.1080017980008641 \cdot 10^{-154}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
Alternative 9
Error60.3
Cost64
\[1\]

Error

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3299041834197318

    1. Initial program 63.0

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

    2. Taylor expanded around -inf 0.5

      \[\leadsto \log \color{blue}{\left(\frac{-0.5}{x}\right)}\]

    3. Simplified0.5

      \[\leadsto \color{blue}{\log \left(\frac{-0.5}{x}\right)}\]

    if -1.3299041834197318 < x < 1.0253859810357775

    1. Initial program 58.5

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

    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

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

    5. Applied unpow3_binary64_21900.2

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

    6. Applied associate-*l*_binary64_20650.2

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

    7. Simplified0.2

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

    if 1.0253859810357775 < x

    1. Initial program 31.2

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

    2. Taylor expanded around inf 0.4

      \[\leadsto \log \left(x + \color{blue}{\left(x + 0.5 \cdot \frac{1}{x}\right)}\right)\]

    3. Simplified0.4

      \[\leadsto \log \left(x + \color{blue}{\left(x + \frac{0.5}{x}\right)}\right)\]
    4. Using strategy rm

    5. Applied flip-+_binary64_209863.0

      \[\leadsto \log \color{blue}{\left(\frac{x \cdot x - \left(x + \frac{0.5}{x}\right) \cdot \left(x + \frac{0.5}{x}\right)}{x - \left(x + \frac{0.5}{x}\right)}\right)}\]

    6. Simplified63.0

      \[\leadsto \log \left(\frac{\color{blue}{-1 - \frac{0.25}{x \cdot x}}}{x - \left(x + \frac{0.5}{x}\right)}\right)\]

    7. Simplified0.4

      \[\leadsto \log \left(\frac{-1 - \frac{0.25}{x \cdot x}}{\color{blue}{\frac{-0.5}{x}}}\right)\]

    8. Simplified0.4

      \[\leadsto \color{blue}{\log \left(\frac{-1 - \frac{0.25}{x \cdot x}}{\frac{-0.5}{x}}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3299041834197318:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.0253859810357775:\\ \;\;\;\;\left(x - \left(x \cdot x\right) \cdot \left(x \cdot 0.16666666666666666\right)\right) + {x}^{5} \cdot 0.075\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{-1 - \frac{0.25}{x \cdot x}}{\frac{-0.5}{x}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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)))))