sqrt times

Percentage Accurate: 99.2% → 100.0%

Time: 5.7s

Alternatives: 7

Speedup: 29.3×

Specification

Math

\[\begin{array}{l} \\ \sqrt{x - 1} \cdot \sqrt{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{x - 1} \cdot \sqrt{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 78000:\\ \;\;\;\;\sqrt{x \cdot \left(x + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -0.5\right) - \frac{0.125}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 78000.0) (sqrt (* x (+ x -1.0))) (- (+ x -0.5) (/ 0.125 x))))

C

double code(double x) {
	double tmp;
	if (x <= 78000.0) {
		tmp = sqrt((x * (x + -1.0)));
	} else {
		tmp = (x + -0.5) - (0.125 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 78000.0d0) then
        tmp = sqrt((x * (x + (-1.0d0))))
    else
        tmp = (x + (-0.5d0)) - (0.125d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 78000.0) {
		tmp = Math.sqrt((x * (x + -1.0)));
	} else {
		tmp = (x + -0.5) - (0.125 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 78000.0:
		tmp = math.sqrt((x * (x + -1.0)))
	else:
		tmp = (x + -0.5) - (0.125 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 78000.0)
		tmp = sqrt(Float64(x * Float64(x + -1.0)));
	else
		tmp = Float64(Float64(x + -0.5) - Float64(0.125 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 78000.0)
		tmp = sqrt((x * (x + -1.0)));
	else
		tmp = (x + -0.5) - (0.125 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 78000.0], N[Sqrt[N[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(x + -0.5), $MachinePrecision] - N[(0.125 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 78000

   1. Initial program 99.6%

      \[\sqrt{x - 1} \cdot \sqrt{x} \]
   2. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \color{blue}{\sqrt{x} \cdot \sqrt{x - 1}} \]

      2. sqrt-unprod 99.8%

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

      3. sub-neg 99.8%

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

      4. metadata-eval 99.8%

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

   3. Applied egg-rr 99.8%

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

   if 78000 < x

   1. Initial program 99.2%

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

   2. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{x - \left(0.5 + 0.125 \cdot \frac{1}{x}\right)} \]
   3. Step-by-step derivation

      1. associate--r+ 100.0%

         \[\leadsto \color{blue}{\left(x - 0.5\right) - 0.125 \cdot \frac{1}{x}} \]

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-*r/ 100.0%

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

      5. metadata-eval 100.0%

         \[\leadsto \left(x + -0.5\right) - \frac{\color{blue}{0.125}}{x} \]

   4. Simplified 100.0%

      \[\leadsto \color{blue}{\left(x + -0.5\right) - \frac{0.125}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 78000:\\ \;\;\;\;\sqrt{x \cdot \left(x + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -0.5\right) - \frac{0.125}{x}\\ \end{array} \]

Alternative 2: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{x + -1} \cdot \sqrt{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

2. Final simplification 99.2%

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

Alternative 3: 99.5% accurate, 15.8× speedup

Math

\[\begin{array}{l} \\ x + \left(\frac{-0.125}{x} - \left(\frac{0.0625}{x \cdot x} + 0.5\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ x (- (/ -0.125 x) (+ (/ 0.0625 (* x x)) 0.5))))

C

double code(double x) {
	return x + ((-0.125 / x) - ((0.0625 / (x * x)) + 0.5));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x + (((-0.125d0) / x) - ((0.0625d0 / (x * x)) + 0.5d0))
end function

Java

public static double code(double x) {
	return x + ((-0.125 / x) - ((0.0625 / (x * x)) + 0.5));
}

Python

def code(x):
	return x + ((-0.125 / x) - ((0.0625 / (x * x)) + 0.5))

Julia

function code(x)
	return Float64(x + Float64(Float64(-0.125 / x) - Float64(Float64(0.0625 / Float64(x * x)) + 0.5)))
end

MATLAB

function tmp = code(x)
	tmp = x + ((-0.125 / x) - ((0.0625 / (x * x)) + 0.5));
end

Wolfram

code[x_] := N[(x + N[(N[(-0.125 / x), $MachinePrecision] - N[(N[(0.0625 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

2. Taylor expanded in x around inf 98.2%

   \[\leadsto \color{blue}{x - \left(0.5 + \left(0.0625 \cdot \frac{1}{{x}^{2}} + 0.125 \cdot \frac{1}{x}\right)\right)} \]
3. Step-by-step derivation

   1. sub-neg 98.2%

      \[\leadsto \color{blue}{x + \left(-\left(0.5 + \left(0.0625 \cdot \frac{1}{{x}^{2}} + 0.125 \cdot \frac{1}{x}\right)\right)\right)} \]

   2. neg-sub0 98.2%

      \[\leadsto x + \color{blue}{\left(0 - \left(0.5 + \left(0.0625 \cdot \frac{1}{{x}^{2}} + 0.125 \cdot \frac{1}{x}\right)\right)\right)} \]

   3. associate-+r+ 98.2%

      \[\leadsto x + \left(0 - \color{blue}{\left(\left(0.5 + 0.0625 \cdot \frac{1}{{x}^{2}}\right) + 0.125 \cdot \frac{1}{x}\right)}\right) \]

   4. +-commutative 98.2%

      \[\leadsto x + \left(0 - \color{blue}{\left(0.125 \cdot \frac{1}{x} + \left(0.5 + 0.0625 \cdot \frac{1}{{x}^{2}}\right)\right)}\right) \]

   5. associate--r+ 98.2%

      \[\leadsto x + \color{blue}{\left(\left(0 - 0.125 \cdot \frac{1}{x}\right) - \left(0.5 + 0.0625 \cdot \frac{1}{{x}^{2}}\right)\right)} \]

   6. neg-sub0 98.2%

      \[\leadsto x + \left(\color{blue}{\left(-0.125 \cdot \frac{1}{x}\right)} - \left(0.5 + 0.0625 \cdot \frac{1}{{x}^{2}}\right)\right) \]

   7. associate-*r/ 98.2%

      \[\leadsto x + \left(\left(-\color{blue}{\frac{0.125 \cdot 1}{x}}\right) - \left(0.5 + 0.0625 \cdot \frac{1}{{x}^{2}}\right)\right) \]

   8. metadata-eval 98.2%

      \[\leadsto x + \left(\left(-\frac{\color{blue}{0.125}}{x}\right) - \left(0.5 + 0.0625 \cdot \frac{1}{{x}^{2}}\right)\right) \]

   9. distribute-neg-frac 98.2%

      \[\leadsto x + \left(\color{blue}{\frac{-0.125}{x}} - \left(0.5 + 0.0625 \cdot \frac{1}{{x}^{2}}\right)\right) \]

   10. metadata-eval 98.2%

       \[\leadsto x + \left(\frac{\color{blue}{-0.125}}{x} - \left(0.5 + 0.0625 \cdot \frac{1}{{x}^{2}}\right)\right) \]

   11. associate-*r/ 98.2%

       \[\leadsto x + \left(\frac{-0.125}{x} - \left(0.5 + \color{blue}{\frac{0.0625 \cdot 1}{{x}^{2}}}\right)\right) \]

   12. metadata-eval 98.2%

       \[\leadsto x + \left(\frac{-0.125}{x} - \left(0.5 + \frac{\color{blue}{0.0625}}{{x}^{2}}\right)\right) \]

   13. unpow2 98.2%

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

4. Simplified 98.2%

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

5. Final simplification 98.2%

   \[\leadsto x + \left(\frac{-0.125}{x} - \left(\frac{0.0625}{x \cdot x} + 0.5\right)\right) \]

Alternative 4: 99.5% accurate, 15.8× speedup

Math

\[\begin{array}{l} \\ \left(x + -0.5\right) - \left(\frac{0.125}{x} + \frac{0.0625}{x \cdot x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (+ x -0.5) (+ (/ 0.125 x) (/ 0.0625 (* x x)))))

C

double code(double x) {
	return (x + -0.5) - ((0.125 / x) + (0.0625 / (x * x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x + (-0.5d0)) - ((0.125d0 / x) + (0.0625d0 / (x * x)))
end function

Java

public static double code(double x) {
	return (x + -0.5) - ((0.125 / x) + (0.0625 / (x * x)));
}

Python

def code(x):
	return (x + -0.5) - ((0.125 / x) + (0.0625 / (x * x)))

Julia

function code(x)
	return Float64(Float64(x + -0.5) - Float64(Float64(0.125 / x) + Float64(0.0625 / Float64(x * x))))
end

MATLAB

function tmp = code(x)
	tmp = (x + -0.5) - ((0.125 / x) + (0.0625 / (x * x)));
end

Wolfram

code[x_] := N[(N[(x + -0.5), $MachinePrecision] - N[(N[(0.125 / x), $MachinePrecision] + N[(0.0625 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

2. Taylor expanded in x around inf 98.2%

   \[\leadsto \color{blue}{x - \left(0.5 + \left(0.0625 \cdot \frac{1}{{x}^{2}} + 0.125 \cdot \frac{1}{x}\right)\right)} \]
3. Step-by-step derivation

   1. associate--r+ 98.2%

      \[\leadsto \color{blue}{\left(x - 0.5\right) - \left(0.0625 \cdot \frac{1}{{x}^{2}} + 0.125 \cdot \frac{1}{x}\right)} \]

   2. sub-neg 98.2%

      \[\leadsto \color{blue}{\left(x + \left(-0.5\right)\right)} - \left(0.0625 \cdot \frac{1}{{x}^{2}} + 0.125 \cdot \frac{1}{x}\right) \]

   3. metadata-eval 98.2%

      \[\leadsto \left(x + \color{blue}{-0.5}\right) - \left(0.0625 \cdot \frac{1}{{x}^{2}} + 0.125 \cdot \frac{1}{x}\right) \]

   4. +-commutative 98.2%

      \[\leadsto \left(x + -0.5\right) - \color{blue}{\left(0.125 \cdot \frac{1}{x} + 0.0625 \cdot \frac{1}{{x}^{2}}\right)} \]

   5. associate-*r/ 98.2%

      \[\leadsto \left(x + -0.5\right) - \left(\color{blue}{\frac{0.125 \cdot 1}{x}} + 0.0625 \cdot \frac{1}{{x}^{2}}\right) \]

   6. metadata-eval 98.2%

      \[\leadsto \left(x + -0.5\right) - \left(\frac{\color{blue}{0.125}}{x} + 0.0625 \cdot \frac{1}{{x}^{2}}\right) \]

   7. associate-*r/ 98.2%

      \[\leadsto \left(x + -0.5\right) - \left(\frac{0.125}{x} + \color{blue}{\frac{0.0625 \cdot 1}{{x}^{2}}}\right) \]

   8. metadata-eval 98.2%

      \[\leadsto \left(x + -0.5\right) - \left(\frac{0.125}{x} + \frac{\color{blue}{0.0625}}{{x}^{2}}\right) \]

   9. unpow2 98.2%

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

4. Simplified 98.2%

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

5. Final simplification 98.2%

   \[\leadsto \left(x + -0.5\right) - \left(\frac{0.125}{x} + \frac{0.0625}{x \cdot x}\right) \]

Alternative 5: 99.4% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \left(x + -0.5\right) - \frac{0.125}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (+ x -0.5) (/ 0.125 x)))

C

double code(double x) {
	return (x + -0.5) - (0.125 / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x + (-0.5d0)) - (0.125d0 / x)
end function

Java

public static double code(double x) {
	return (x + -0.5) - (0.125 / x);
}

Python

def code(x):
	return (x + -0.5) - (0.125 / x)

Julia

function code(x)
	return Float64(Float64(x + -0.5) - Float64(0.125 / x))
end

MATLAB

function tmp = code(x)
	tmp = (x + -0.5) - (0.125 / x);
end

Wolfram

code[x_] := N[(N[(x + -0.5), $MachinePrecision] - N[(0.125 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

2. Taylor expanded in x around inf 98.0%

   \[\leadsto \color{blue}{x - \left(0.5 + 0.125 \cdot \frac{1}{x}\right)} \]
3. Step-by-step derivation

   1. associate--r+ 98.0%

      \[\leadsto \color{blue}{\left(x - 0.5\right) - 0.125 \cdot \frac{1}{x}} \]

   2. sub-neg 98.0%

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

   3. metadata-eval 98.0%

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

   4. associate-*r/ 98.0%

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

   5. metadata-eval 98.0%

      \[\leadsto \left(x + -0.5\right) - \frac{\color{blue}{0.125}}{x} \]

4. Simplified 98.0%

   \[\leadsto \color{blue}{\left(x + -0.5\right) - \frac{0.125}{x}} \]

5. Final simplification 98.0%

   \[\leadsto \left(x + -0.5\right) - \frac{0.125}{x} \]

Alternative 6: 99.1% accurate, 68.3× speedup

Math

\[\begin{array}{l} \\ x - 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- x 0.5))

C

double code(double x) {
	return x - 0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x - 0.5d0
end function

Java

public static double code(double x) {
	return x - 0.5;
}

Python

def code(x):
	return x - 0.5

Julia

function code(x)
	return Float64(x - 0.5)
end

MATLAB

function tmp = code(x)
	tmp = x - 0.5;
end

Wolfram

code[x_] := N[(x - 0.5), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

2. Taylor expanded in x around inf 97.7%

   \[\leadsto \color{blue}{x - 0.5} \]

3. Final simplification 97.7%

   \[\leadsto x - 0.5 \]

Alternative 7: 98.1% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x
end function

Java

public static double code(double x) {
	return x;
}

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

function tmp = code(x)
	tmp = x;
end

Wolfram

code[x_] := x

Derivation

1. Initial program 99.2%

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

2. Taylor expanded in x around inf 97.0%

   \[\leadsto \color{blue}{x} \]

3. Final simplification 97.0%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023283 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1.0)) (sqrt x)))