sqrt times

Average Error: 0.5 → 0.5

Time: 29.9s

Precision: binary64

Cost: 26952

Math

\[\sqrt{x - 1} \cdot \sqrt{x} \]

↓

\[\begin{array}{l} t_0 := \sqrt{x + -1} \cdot \sqrt{x}\\ \mathbf{if}\;t_0 \ne 0:\\ \;\;\;\;\frac{-1}{\begin{array}{l} \mathbf{if}\;-1 + x \ne 0:\\ \;\;\;\;{\left(-1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{-1 + x}}\\ \end{array} \cdot \frac{-1}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (sqrt (- x 1.0)) (sqrt x)))

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (sqrt (+ x -1.0)) (sqrt x))))
   (if (!= t_0 0.0)
     (/
      -1.0
      (*
       (if (!= (+ -1.0 x) 0.0) (pow (+ -1.0 x) -0.5) (/ 1.0 (sqrt (+ -1.0 x))))
       (/ -1.0 (sqrt x))))
     t_0)))

C

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

↓

double code(double x) {
	double t_0 = sqrt((x + -1.0)) * sqrt(x);
	double tmp_1;
	if (t_0 != 0.0) {
		double tmp_2;
		if ((-1.0 + x) != 0.0) {
			tmp_2 = pow((-1.0 + x), -0.5);
		} else {
			tmp_2 = 1.0 / sqrt((-1.0 + x));
		}
		tmp_1 = -1.0 / (tmp_2 * (-1.0 / sqrt(x)));
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

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

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = sqrt((x + (-1.0d0))) * sqrt(x)
    if (t_0 /= 0.0d0) then
        if (((-1.0d0) + x) /= 0.0d0) then
            tmp_2 = ((-1.0d0) + x) ** (-0.5d0)
        else
            tmp_2 = 1.0d0 / sqrt(((-1.0d0) + x))
        end if
        tmp_1 = (-1.0d0) / (tmp_2 * ((-1.0d0) / sqrt(x)))
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

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

↓

public static double code(double x) {
	double t_0 = Math.sqrt((x + -1.0)) * Math.sqrt(x);
	double tmp_1;
	if (t_0 != 0.0) {
		double tmp_2;
		if ((-1.0 + x) != 0.0) {
			tmp_2 = Math.pow((-1.0 + x), -0.5);
		} else {
			tmp_2 = 1.0 / Math.sqrt((-1.0 + x));
		}
		tmp_1 = -1.0 / (tmp_2 * (-1.0 / Math.sqrt(x)));
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

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

↓

def code(x):
	t_0 = math.sqrt((x + -1.0)) * math.sqrt(x)
	tmp_1 = 0
	if t_0 != 0.0:
		tmp_2 = 0
		if (-1.0 + x) != 0.0:
			tmp_2 = math.pow((-1.0 + x), -0.5)
		else:
			tmp_2 = 1.0 / math.sqrt((-1.0 + x))
		tmp_1 = -1.0 / (tmp_2 * (-1.0 / math.sqrt(x)))
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(x)
	return Float64(sqrt(Float64(x - 1.0)) * sqrt(x))
end

↓

function code(x)
	t_0 = Float64(sqrt(Float64(x + -1.0)) * sqrt(x))
	tmp_1 = 0.0
	if (t_0 != 0.0)
		tmp_2 = 0.0
		if (Float64(-1.0 + x) != 0.0)
			tmp_2 = Float64(-1.0 + x) ^ -0.5;
		else
			tmp_2 = Float64(1.0 / sqrt(Float64(-1.0 + x)));
		end
		tmp_1 = Float64(-1.0 / Float64(tmp_2 * Float64(-1.0 / sqrt(x))));
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

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

↓

function tmp_4 = code(x)
	t_0 = sqrt((x + -1.0)) * sqrt(x);
	tmp_2 = 0.0;
	if (t_0 ~= 0.0)
		tmp_3 = 0.0;
		if ((-1.0 + x) ~= 0.0)
			tmp_3 = (-1.0 + x) ^ -0.5;
		else
			tmp_3 = 1.0 / sqrt((-1.0 + x));
		end
		tmp_2 = -1.0 / (tmp_3 * (-1.0 / sqrt(x)));
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

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

↓

code[x_] := Block[{t$95$0 = N[(N[Sqrt[N[(x + -1.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[Unequal[t$95$0, 0.0], N[(-1.0 / N[(If[Unequal[N[(-1.0 + x), $MachinePrecision], 0.0], N[Power[N[(-1.0 + x), $MachinePrecision], -0.5], $MachinePrecision], N[(1.0 / N[Sqrt[N[(-1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] * N[(-1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Initial program 0.5

   \[\sqrt{x - 1} \cdot \sqrt{x} \]

2. Applied egg-rr 42.9

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\sqrt{x + -1} \cdot \sqrt{x}\right)}^{3}}} \]

3. Applied egg-rr 0.5

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\sqrt{x + -1} \cdot \sqrt{x} \ne 0:\\ \;\;\;\;\frac{1}{{\left(\sqrt{x + -1} \cdot \sqrt{x}\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + -1} \cdot \sqrt{x}\\ } \end{array}} \]

4. Simplified 0.5

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\sqrt{x + -1} \cdot \sqrt{x} \ne 0:\\ \;\;\;\;\frac{-1}{\frac{-1}{\sqrt{x + -1} \cdot \sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + -1} \cdot \sqrt{x}\\ } \end{array}} \]
   Proof

5. Applied egg-rr 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + -1} \cdot \sqrt{x} \ne 0:\\ \;\;\;\;\frac{-1}{\color{blue}{\frac{1}{\sqrt{-1 + x}} \cdot \frac{-1}{\sqrt{x}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + -1} \cdot \sqrt{x}\\ \end{array} \]

6. Applied egg-rr 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + -1} \cdot \sqrt{x} \ne 0:\\ \;\;\;\;\frac{-1}{\color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;-1 + x \ne 0:\\ \;\;\;\;{\left(-1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{-1 + x}}\\ } \end{array}} \cdot \frac{-1}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + -1} \cdot \sqrt{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	13120

\[\sqrt{x - 1} \cdot \sqrt{x} \]

Reproduce

herbie shell --seed 2023010 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1.0)) (sqrt x)))