Numeric.Log:$clog1p from log-domain-0.10.2.1, A

Percentage Accurate: 100.0% → 100.0%

Time: 2.3s

Alternatives: 7

Speedup: 1.2×

• 43.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot 2 + x \cdot x\right) + y \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (+ (* x 2.0) (* x x)) (* y y)))

C

double code(double x, double y) {
	return ((x * 2.0) + (x * x)) + (y * y);
}

Fortran

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

Java

public static double code(double x, double y) {
	return ((x * 2.0) + (x * x)) + (y * y);
}

Python

def code(x, y):
	return ((x * 2.0) + (x * x)) + (y * y)

Julia

function code(x, y)
	return Float64(Float64(Float64(x * 2.0) + Float64(x * x)) + Float64(y * y))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * 2.0) + (x * x)) + (y * y);
end

Wolfram

code[x_, y_] := N[(N[(N[(x * 2.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot 2 + x \cdot x\right) + y \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (+ (* x 2.0) (* x x)) (* y y)))

C

double code(double x, double y) {
	return ((x * 2.0) + (x * x)) + (y * y);
}

Fortran

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

Java

public static double code(double x, double y) {
	return ((x * 2.0) + (x * x)) + (y * y);
}

Python

def code(x, y):
	return ((x * 2.0) + (x * x)) + (y * y)

Julia

function code(x, y)
	return Float64(Float64(Float64(x * 2.0) + Float64(x * x)) + Float64(y * y))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * 2.0) + (x * x)) + (y * y);
end

Wolfram

code[x_, y_] := N[(N[(N[(x * 2.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot 2 + x \cdot x\right) + y \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (+ (* x 2.0) (* x x)) (* y y)))

C

double code(double x, double y) {
	return ((x * 2.0) + (x * x)) + (y * y);
}

Fortran

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

Java

public static double code(double x, double y) {
	return ((x * 2.0) + (x * x)) + (y * y);
}

Python

def code(x, y):
	return ((x * 2.0) + (x * x)) + (y * y)

Julia

function code(x, y)
	return Float64(Float64(Float64(x * 2.0) + Float64(x * x)) + Float64(y * y))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * 2.0) + (x * x)) + (y * y);
end

Wolfram

code[x_, y_] := N[(N[(N[(x * 2.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]

2. Final simplification 100.0%

   \[\leadsto \left(x \cdot 2 + x \cdot x\right) + y \cdot y \]

Alternative 2: 72.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+20}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-72}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq 1.36 \cdot 10^{-18}:\\ \;\;\;\;x + x\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+33}:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7e+20)
   (* x x)
   (if (<= x 1.05e-72)
     (* y y)
     (if (<= x 1.36e-18) (+ x x) (if (<= x 2.4e+33) (* y y) (* x x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7e+20) {
		tmp = x * x;
	} else if (x <= 1.05e-72) {
		tmp = y * y;
	} else if (x <= 1.36e-18) {
		tmp = x + x;
	} else if (x <= 2.4e+33) {
		tmp = y * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-7d+20)) then
        tmp = x * x
    else if (x <= 1.05d-72) then
        tmp = y * y
    else if (x <= 1.36d-18) then
        tmp = x + x
    else if (x <= 2.4d+33) then
        tmp = y * y
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -7e+20) {
		tmp = x * x;
	} else if (x <= 1.05e-72) {
		tmp = y * y;
	} else if (x <= 1.36e-18) {
		tmp = x + x;
	} else if (x <= 2.4e+33) {
		tmp = y * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7e+20:
		tmp = x * x
	elif x <= 1.05e-72:
		tmp = y * y
	elif x <= 1.36e-18:
		tmp = x + x
	elif x <= 2.4e+33:
		tmp = y * y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7e+20)
		tmp = Float64(x * x);
	elseif (x <= 1.05e-72)
		tmp = Float64(y * y);
	elseif (x <= 1.36e-18)
		tmp = Float64(x + x);
	elseif (x <= 2.4e+33)
		tmp = Float64(y * y);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7e+20)
		tmp = x * x;
	elseif (x <= 1.05e-72)
		tmp = y * y;
	elseif (x <= 1.36e-18)
		tmp = x + x;
	elseif (x <= 2.4e+33)
		tmp = y * y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7e+20], N[(x * x), $MachinePrecision], If[LessEqual[x, 1.05e-72], N[(y * y), $MachinePrecision], If[LessEqual[x, 1.36e-18], N[(x + x), $MachinePrecision], If[LessEqual[x, 2.4e+33], N[(y * y), $MachinePrecision], N[(x * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7e20 or 2.4e33 < x

   1. Initial program 100.0%

      \[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
   2. Step-by-step derivation

      1. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{{x}^{2}} + y \cdot y \]
   5. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto \color{blue}{x \cdot x} + y \cdot y \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot x} + y \cdot y \]

   7. Taylor expanded in x around inf 84.1%

      \[\leadsto \color{blue}{{x}^{2}} \]

   9. Simplified 84.1%

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

   if -7e20 < x < 1.05e-72 or 1.3600000000000001e-18 < x < 2.4e33

   1. Initial program 100.0%

      \[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
   2. Step-by-step derivation

      1. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 70.7%

      \[\leadsto \color{blue}{{x}^{2}} + y \cdot y \]
   5. Step-by-step derivation

      1. unpow2 70.7%

         \[\leadsto \color{blue}{x \cdot x} + y \cdot y \]

   6. Simplified 70.7%

      \[\leadsto \color{blue}{x \cdot x} + y \cdot y \]

   7. Taylor expanded in x around 0 70.1%

      \[\leadsto \color{blue}{{y}^{2}} \]

   9. Simplified 70.1%

      \[\leadsto \color{blue}{y \cdot y} \]

   if 1.05e-72 < x < 1.3600000000000001e-18

   1. Initial program 100.0%

      \[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
   2. Step-by-step derivation

      1. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{2 \cdot x} + y \cdot y \]
   5. Step-by-step derivation

      1. count-2 100.0%

         \[\leadsto \color{blue}{\left(x + x\right)} + y \cdot y \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\left(x + x\right)} + y \cdot y \]

   7. Taylor expanded in x around inf 73.8%

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

   9. Simplified 73.8%

      \[\leadsto \color{blue}{x + x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+20}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-72}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq 1.36 \cdot 10^{-18}:\\ \;\;\;\;x + x\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+33}:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 3: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1320000 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;x \cdot x + y \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1320000.0) (not (<= x 2.0)))
   (+ (* x x) (* y y))
   (+ (* y y) (+ x x))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1320000.0) || !(x <= 2.0)) {
		tmp = (x * x) + (y * y);
	} else {
		tmp = (y * y) + (x + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1320000.0d0)) .or. (.not. (x <= 2.0d0))) then
        tmp = (x * x) + (y * y)
    else
        tmp = (y * y) + (x + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1320000.0) || !(x <= 2.0)) {
		tmp = (x * x) + (y * y);
	} else {
		tmp = (y * y) + (x + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1320000.0) or not (x <= 2.0):
		tmp = (x * x) + (y * y)
	else:
		tmp = (y * y) + (x + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1320000.0) || !(x <= 2.0))
		tmp = Float64(Float64(x * x) + Float64(y * y));
	else
		tmp = Float64(Float64(y * y) + Float64(x + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1320000.0) || ~((x <= 2.0)))
		tmp = (x * x) + (y * y);
	else
		tmp = (y * y) + (x + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1320000.0], N[Not[LessEqual[x, 2.0]], $MachinePrecision]], N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] + N[(x + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.32e6 or 2 < x

   1. Initial program 100.0%

      \[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
   2. Step-by-step derivation

      1. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{{x}^{2}} + y \cdot y \]
   5. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto \color{blue}{x \cdot x} + y \cdot y \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot x} + y \cdot y \]

   if -1.32e6 < x < 2

   1. Initial program 100.0%

      \[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
   2. Step-by-step derivation

      1. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 98.7%

      \[\leadsto \color{blue}{2 \cdot x} + y \cdot y \]
   5. Step-by-step derivation

      1. count-2 98.7%

         \[\leadsto \color{blue}{\left(x + x\right)} + y \cdot y \]

   6. Simplified 98.7%

      \[\leadsto \color{blue}{\left(x + x\right)} + y \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1320000 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;x \cdot x + y \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + \left(x + x\right)\\ \end{array} \]

Alternative 4: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ y \cdot y + x \cdot \left(x + 2\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (* y y) (* x (+ x 2.0))))

C

double code(double x, double y) {
	return (y * y) + (x * (x + 2.0));
}

Fortran

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

Java

public static double code(double x, double y) {
	return (y * y) + (x * (x + 2.0));
}

Python

def code(x, y):
	return (y * y) + (x * (x + 2.0))

Julia

function code(x, y)
	return Float64(Float64(y * y) + Float64(x * Float64(x + 2.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = (y * y) + (x * (x + 2.0));
end

Wolfram

code[x_, y_] := N[(N[(y * y), $MachinePrecision] + N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
2. Step-by-step derivation

   1. distribute-lft-out 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto y \cdot y + x \cdot \left(x + 2\right) \]

Alternative 5: 72.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+22}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+33}:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.2e+22) (* x x) (if (<= x 2.5e+33) (* y y) (* x x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.2e+22) {
		tmp = x * x;
	} else if (x <= 2.5e+33) {
		tmp = y * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5.2d+22)) then
        tmp = x * x
    else if (x <= 2.5d+33) then
        tmp = y * y
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5.2e+22) {
		tmp = x * x;
	} else if (x <= 2.5e+33) {
		tmp = y * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.2e+22:
		tmp = x * x
	elif x <= 2.5e+33:
		tmp = y * y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.2e+22)
		tmp = Float64(x * x);
	elseif (x <= 2.5e+33)
		tmp = Float64(y * y);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.2e+22)
		tmp = x * x;
	elseif (x <= 2.5e+33)
		tmp = y * y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.2e+22], N[(x * x), $MachinePrecision], If[LessEqual[x, 2.5e+33], N[(y * y), $MachinePrecision], N[(x * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.2e22 or 2.49999999999999986e33 < x

   1. Initial program 100.0%

      \[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
   2. Step-by-step derivation

      1. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{{x}^{2}} + y \cdot y \]
   5. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto \color{blue}{x \cdot x} + y \cdot y \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot x} + y \cdot y \]

   7. Taylor expanded in x around inf 84.1%

      \[\leadsto \color{blue}{{x}^{2}} \]

   9. Simplified 84.1%

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

   if -5.2e22 < x < 2.49999999999999986e33

   1. Initial program 100.0%

      \[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
   2. Step-by-step derivation

      1. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 67.5%

      \[\leadsto \color{blue}{{x}^{2}} + y \cdot y \]
   5. Step-by-step derivation

      1. unpow2 67.5%

         \[\leadsto \color{blue}{x \cdot x} + y \cdot y \]

   6. Simplified 67.5%

      \[\leadsto \color{blue}{x \cdot x} + y \cdot y \]

   7. Taylor expanded in x around 0 66.7%

      \[\leadsto \color{blue}{{y}^{2}} \]

   9. Simplified 66.7%

      \[\leadsto \color{blue}{y \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+22}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+33}:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 6: 81.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ x \cdot x + y \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (* x x) (* y y)))

C

double code(double x, double y) {
	return (x * x) + (y * y);
}

Fortran

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

Java

public static double code(double x, double y) {
	return (x * x) + (y * y);
}

Python

def code(x, y):
	return (x * x) + (y * y)

Julia

function code(x, y)
	return Float64(Float64(x * x) + Float64(y * y))
end

MATLAB

function tmp = code(x, y)
	tmp = (x * x) + (y * y);
end

Wolfram

code[x_, y_] := N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
2. Step-by-step derivation

   1. distribute-lft-out 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around inf 83.4%

   \[\leadsto \color{blue}{{x}^{2}} + y \cdot y \]
5. Step-by-step derivation

   1. unpow2 83.4%

      \[\leadsto \color{blue}{x \cdot x} + y \cdot y \]

6. Simplified 83.4%

   \[\leadsto \color{blue}{x \cdot x} + y \cdot y \]

7. Final simplification 83.4%

   \[\leadsto x \cdot x + y \cdot y \]

Alternative 7: 42.0% accurate, 3.7× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (* x x))

C

double code(double x, double y) {
	return x * x;
}

Fortran

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

Java

public static double code(double x, double y) {
	return x * x;
}

Python

def code(x, y):
	return x * x

Julia

function code(x, y)
	return Float64(x * x)
end

MATLAB

function tmp = code(x, y)
	tmp = x * x;
end

Wolfram

code[x_, y_] := N[(x * x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
2. Step-by-step derivation

   1. distribute-lft-out 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around inf 83.4%

   \[\leadsto \color{blue}{{x}^{2}} + y \cdot y \]
5. Step-by-step derivation

   1. unpow2 83.4%

      \[\leadsto \color{blue}{x \cdot x} + y \cdot y \]

6. Simplified 83.4%

   \[\leadsto \color{blue}{x \cdot x} + y \cdot y \]

7. Taylor expanded in x around inf 42.9%

   \[\leadsto \color{blue}{{x}^{2}} \]

9. Simplified 42.9%

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

10. Final simplification 42.9%

    \[\leadsto x \cdot x \]

Developer target: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ y \cdot y + \left(2 \cdot x + x \cdot x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (* y y) (+ (* 2.0 x) (* x x))))

C

double code(double x, double y) {
	return (y * y) + ((2.0 * x) + (x * x));
}

Fortran

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

Java

public static double code(double x, double y) {
	return (y * y) + ((2.0 * x) + (x * x));
}

Python

def code(x, y):
	return (y * y) + ((2.0 * x) + (x * x))

Julia

function code(x, y)
	return Float64(Float64(y * y) + Float64(Float64(2.0 * x) + Float64(x * x)))
end

MATLAB

function tmp = code(x, y)
	tmp = (y * y) + ((2.0 * x) + (x * x));
end

Wolfram

code[x_, y_] := N[(N[(y * y), $MachinePrecision] + N[(N[(2.0 * x), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023196 
(FPCore (x y)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, A"
  :precision binary64

  :herbie-target
  (+ (* y y) (+ (* 2.0 x) (* x x)))

  (+ (+ (* x 2.0) (* x x)) (* y y)))