sqrt D (should all be same)

Percentage Accurate: 53.7% → 99.4%

Time: 5.3s

Alternatives: 2

Speedup: 2.0×

Specification

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot {x}^{2}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (* 2.0 (pow x 2.0))))

C

double code(double x) {
	return sqrt((2.0 * pow(x, 2.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((2.0d0 * (x ** 2.0d0)))
end function

Java

public static double code(double x) {
	return Math.sqrt((2.0 * Math.pow(x, 2.0)));
}

Python

def code(x):
	return math.sqrt((2.0 * math.pow(x, 2.0)))

Julia

function code(x)
	return sqrt(Float64(2.0 * (x ^ 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((2.0 * (x ^ 2.0)));
end

Wolfram

code[x_] := N[Sqrt[N[(2.0 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 53.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot {x}^{2}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (* 2.0 (pow x 2.0))))

C

double code(double x) {
	return sqrt((2.0 * pow(x, 2.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((2.0d0 * (x ** 2.0d0)))
end function

Java

public static double code(double x) {
	return Math.sqrt((2.0 * Math.pow(x, 2.0)));
}

Python

def code(x):
	return math.sqrt((2.0 * math.pow(x, 2.0)))

Julia

function code(x)
	return sqrt(Float64(2.0 * (x ^ 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((2.0 * (x ^ 2.0)));
end

Wolfram

code[x_] := N[Sqrt[N[(2.0 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \sqrt{2 \cdot x} \cdot \sqrt{x} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (* (sqrt (* 2.0 x)) (sqrt x)))

C

x = abs(x);
double code(double x) {
	return sqrt((2.0 * x)) * sqrt(x);
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((2.0d0 * x)) * sqrt(x)
end function

Java

x = Math.abs(x);
public static double code(double x) {
	return Math.sqrt((2.0 * x)) * Math.sqrt(x);
}

Python

x = abs(x)
def code(x):
	return math.sqrt((2.0 * x)) * math.sqrt(x)

Julia

x = abs(x)
function code(x)
	return Float64(sqrt(Float64(2.0 * x)) * sqrt(x))
end

MATLAB

x = abs(x)
function tmp = code(x)
	tmp = sqrt((2.0 * x)) * sqrt(x);
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := N[(N[Sqrt[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.7%

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

   1. unpow2 56.7%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(x \cdot x\right)}} \]

   2. sqr-neg 56.7%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(-x\right) \cdot \left(-x\right)\right)}} \]

   3. unpow2 56.7%

      \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(-x\right)}^{2}}} \]

   4. unpow2 56.7%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(-x\right) \cdot \left(-x\right)\right)}} \]

   5. sqr-neg 56.7%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(x \cdot x\right)}} \]

3. Simplified 56.7%

   \[\leadsto \color{blue}{\sqrt{2 \cdot \left(x \cdot x\right)}} \]
4. Step-by-step derivation

   1. associate-*r* 56.7%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot x\right) \cdot x}} \]

   2. sqrt-prod 47.0%

      \[\leadsto \color{blue}{\sqrt{2 \cdot x} \cdot \sqrt{x}} \]

5. Applied egg-rr 47.0%

   \[\leadsto \color{blue}{\sqrt{2 \cdot x} \cdot \sqrt{x}} \]

6. Final simplification 47.0%

   \[\leadsto \sqrt{2 \cdot x} \cdot \sqrt{x} \]

Alternative 2: 99.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ x \cdot \sqrt{2} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (* x (sqrt 2.0)))

C

x = abs(x);
double code(double x) {
	return x * sqrt(2.0);
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = x * sqrt(2.0d0)
end function

Java

x = Math.abs(x);
public static double code(double x) {
	return x * Math.sqrt(2.0);
}

Python

x = abs(x)
def code(x):
	return x * math.sqrt(2.0)

Julia

x = abs(x)
function code(x)
	return Float64(x * sqrt(2.0))
end

MATLAB

x = abs(x)
function tmp = code(x)
	tmp = x * sqrt(2.0);
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.7%

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

   1. unpow2 56.7%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(x \cdot x\right)}} \]

   2. sqr-neg 56.7%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(-x\right) \cdot \left(-x\right)\right)}} \]

   3. unpow2 56.7%

      \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(-x\right)}^{2}}} \]

   4. unpow2 56.7%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(-x\right) \cdot \left(-x\right)\right)}} \]

   5. sqr-neg 56.7%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(x \cdot x\right)}} \]

3. Simplified 56.7%

   \[\leadsto \color{blue}{\sqrt{2 \cdot \left(x \cdot x\right)}} \]
4. Step-by-step derivation

   1. sqrt-prod 56.5%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{x \cdot x}} \]

   2. sqrt-prod 46.8%

      \[\leadsto \sqrt{2} \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \]

   3. add-sqr-sqrt 48.1%

      \[\leadsto \sqrt{2} \cdot \color{blue}{x} \]

5. Applied egg-rr 48.1%

   \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]

6. Final simplification 48.1%

   \[\leadsto x \cdot \sqrt{2} \]

Reproduce

herbie shell --seed 2023278 
(FPCore (x)
  :name "sqrt D (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))