Hyperbolic arcsine

Average Error: 53.0 → 0.3

Time: 11.2s

Precision: binary64

Cost: 20488

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;{x}^{3} \cdot -0.16666666666666666 + \left({x}^{5} \cdot 0.075 + \left(x + -0.044642857142857144 \cdot {x}^{7}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{0.5}{x} + \left(x + x\right)\right)\\ \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log (+ x (sqrt (+ (* x x) 1.0)))))

↓

(FPCore (x)
 :precision binary64
 (if (<= x -1.3)
   (log (/ -0.5 x))
   (if (<= x 1.05)
     (+
      (* (pow x 3.0) -0.16666666666666666)
      (+ (* (pow x 5.0) 0.075) (+ x (* -0.044642857142857144 (pow x 7.0)))))
     (log (+ (/ 0.5 x) (+ x x))))))

C

double code(double x) {
	return log((x + sqrt(((x * x) + 1.0))));
}

↓

double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.05) {
		tmp = (pow(x, 3.0) * -0.16666666666666666) + ((pow(x, 5.0) * 0.075) + (x + (-0.044642857142857144 * pow(x, 7.0))));
	} else {
		tmp = log(((0.5 / x) + (x + x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log((x + sqrt(((x * x) + 1.0d0))))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.3d0)) then
        tmp = log(((-0.5d0) / x))
    else if (x <= 1.05d0) then
        tmp = ((x ** 3.0d0) * (-0.16666666666666666d0)) + (((x ** 5.0d0) * 0.075d0) + (x + ((-0.044642857142857144d0) * (x ** 7.0d0))))
    else
        tmp = log(((0.5d0 / x) + (x + x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	return Math.log((x + Math.sqrt(((x * x) + 1.0))));
}

↓

public static double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 1.05) {
		tmp = (Math.pow(x, 3.0) * -0.16666666666666666) + ((Math.pow(x, 5.0) * 0.075) + (x + (-0.044642857142857144 * Math.pow(x, 7.0))));
	} else {
		tmp = Math.log(((0.5 / x) + (x + x)));
	}
	return tmp;
}

Python

def code(x):
	return math.log((x + math.sqrt(((x * x) + 1.0))))

↓

def code(x):
	tmp = 0
	if x <= -1.3:
		tmp = math.log((-0.5 / x))
	elif x <= 1.05:
		tmp = (math.pow(x, 3.0) * -0.16666666666666666) + ((math.pow(x, 5.0) * 0.075) + (x + (-0.044642857142857144 * math.pow(x, 7.0))))
	else:
		tmp = math.log(((0.5 / x) + (x + x)))
	return tmp

Julia

function code(x)
	return log(Float64(x + sqrt(Float64(Float64(x * x) + 1.0))))
end

↓

function code(x)
	tmp = 0.0
	if (x <= -1.3)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.05)
		tmp = Float64(Float64((x ^ 3.0) * -0.16666666666666666) + Float64(Float64((x ^ 5.0) * 0.075) + Float64(x + Float64(-0.044642857142857144 * (x ^ 7.0)))));
	else
		tmp = log(Float64(Float64(0.5 / x) + Float64(x + x)));
	end
	return tmp
end

MATLAB

function tmp = code(x)
	tmp = log((x + sqrt(((x * x) + 1.0))));
end

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.3)
		tmp = log((-0.5 / x));
	elseif (x <= 1.05)
		tmp = ((x ^ 3.0) * -0.16666666666666666) + (((x ^ 5.0) * 0.075) + (x + (-0.044642857142857144 * (x ^ 7.0))));
	else
		tmp = log(((0.5 / x) + (x + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := N[Log[N[(x + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[x_] := If[LessEqual[x, -1.3], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.05], N[(N[(N[Power[x, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + N[(N[(N[Power[x, 5.0], $MachinePrecision] * 0.075), $MachinePrecision] + N[(x + N[(-0.044642857142857144 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(0.5 / x), $MachinePrecision] + N[(x + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Target

Original	53.0
Target	45.6
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if x < -1.30000000000000004

   1. Initial program 63.1

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

   2. Taylor expanded in x around -inf 0.4

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

   if -1.30000000000000004 < x < 1.05000000000000004

   1. Initial program 58.7

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

   2. Taylor expanded in x around 0 0.2

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(-0.044642857142857144 \cdot {x}^{7} + x\right)\right)} \]

   3. Simplified 0.2

      \[\leadsto \color{blue}{{x}^{3} \cdot -0.16666666666666666 + \left({x}^{5} \cdot 0.075 + \left(x + -0.044642857142857144 \cdot {x}^{7}\right)\right)} \]
      Proof

      [Start] 0.2	\[ -0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(-0.044642857142857144 \cdot {x}^{7} + x\right)\right) \]
      rational.json-simplify-2 [=>] 0.2	\[ \color{blue}{{x}^{3} \cdot -0.16666666666666666} + \left(0.075 \cdot {x}^{5} + \left(-0.044642857142857144 \cdot {x}^{7} + x\right)\right) \]
      rational.json-simplify-2 [=>] 0.2	\[ {x}^{3} \cdot -0.16666666666666666 + \left(\color{blue}{{x}^{5} \cdot 0.075} + \left(-0.044642857142857144 \cdot {x}^{7} + x\right)\right) \]
      rational.json-simplify-1 [=>] 0.2	\[ {x}^{3} \cdot -0.16666666666666666 + \left({x}^{5} \cdot 0.075 + \color{blue}{\left(x + -0.044642857142857144 \cdot {x}^{7}\right)}\right) \]

   if 1.05000000000000004 < x

   1. Initial program 31.9

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

   2. Taylor expanded in x around inf 0.3

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

   3. Applied egg-rr 0.3

      \[\leadsto \log \color{blue}{\left(\left(\left(x + x\right) + \frac{0.5}{x}\right) - 0\right)} \]

   4. Simplified 0.3

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

      [Start] 0.3	\[ \log \left(\left(\left(x + x\right) + \frac{0.5}{x}\right) - 0\right) \]
      rational.json-simplify-5 [=>] 0.3	\[ \log \color{blue}{\left(\left(x + x\right) + \frac{0.5}{x}\right)} \]
      rational.json-simplify-1 [=>] 0.3	\[ \log \color{blue}{\left(\frac{0.5}{x} + \left(x + x\right)\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;{x}^{3} \cdot -0.16666666666666666 + \left({x}^{5} \cdot 0.075 + \left(x + -0.044642857142857144 \cdot {x}^{7}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{0.5}{x} + \left(x + x\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	13768

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.02:\\ \;\;\;\;\left(x + {x}^{3} \cdot -0.16666666666666666\right) + {x}^{5} \cdot 0.075\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{0.5}{x} + \left(x + x\right)\right)\\ \end{array} \]

Alternative 2
Error	0.3
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3} + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{0.5}{x} + \left(x + x\right)\right)\\ \end{array} \]

Alternative 3
Error	0.4
Cost	7048

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3} + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternative 4
Error	0.6
Cost	6856

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternative 5
Error	26.2
Cost	6724

\[\begin{array}{l} \mathbf{if}\;x \leq 1.58:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 1\right)\\ \end{array} \]

Alternative 6
Error	15.2
Cost	6724

\[\begin{array}{l} \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternative 7
Error	30.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023064 
(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)))))