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

Percentage Accurate: 100.0% → 100.0%

Time: 4.1s

Alternatives: 6

Speedup: 1.2×

• 42.8% 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, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

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

Alternative 2: 72.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -800000000000:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-40}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-107}:\\ \;\;\;\;x + x\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-84}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-69}:\\ \;\;\;\;x + x\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+38}:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -800000000000.0)
   (* x x)
   (if (<= x -2.8e-40)
     (* y y)
     (if (<= x -4.2e-107)
       (+ x x)
       (if (<= x 1.75e-84)
         (* y y)
         (if (<= x 4.2e-69) (+ x x) (if (<= x 6e+38) (* y y) (* x x))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -800000000000.0) {
		tmp = x * x;
	} else if (x <= -2.8e-40) {
		tmp = y * y;
	} else if (x <= -4.2e-107) {
		tmp = x + x;
	} else if (x <= 1.75e-84) {
		tmp = y * y;
	} else if (x <= 4.2e-69) {
		tmp = x + x;
	} else if (x <= 6e+38) {
		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 <= (-800000000000.0d0)) then
        tmp = x * x
    else if (x <= (-2.8d-40)) then
        tmp = y * y
    else if (x <= (-4.2d-107)) then
        tmp = x + x
    else if (x <= 1.75d-84) then
        tmp = y * y
    else if (x <= 4.2d-69) then
        tmp = x + x
    else if (x <= 6d+38) 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 <= -800000000000.0) {
		tmp = x * x;
	} else if (x <= -2.8e-40) {
		tmp = y * y;
	} else if (x <= -4.2e-107) {
		tmp = x + x;
	} else if (x <= 1.75e-84) {
		tmp = y * y;
	} else if (x <= 4.2e-69) {
		tmp = x + x;
	} else if (x <= 6e+38) {
		tmp = y * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -800000000000.0:
		tmp = x * x
	elif x <= -2.8e-40:
		tmp = y * y
	elif x <= -4.2e-107:
		tmp = x + x
	elif x <= 1.75e-84:
		tmp = y * y
	elif x <= 4.2e-69:
		tmp = x + x
	elif x <= 6e+38:
		tmp = y * y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -800000000000.0)
		tmp = Float64(x * x);
	elseif (x <= -2.8e-40)
		tmp = Float64(y * y);
	elseif (x <= -4.2e-107)
		tmp = Float64(x + x);
	elseif (x <= 1.75e-84)
		tmp = Float64(y * y);
	elseif (x <= 4.2e-69)
		tmp = Float64(x + x);
	elseif (x <= 6e+38)
		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 <= -800000000000.0)
		tmp = x * x;
	elseif (x <= -2.8e-40)
		tmp = y * y;
	elseif (x <= -4.2e-107)
		tmp = x + x;
	elseif (x <= 1.75e-84)
		tmp = y * y;
	elseif (x <= 4.2e-69)
		tmp = x + x;
	elseif (x <= 6e+38)
		tmp = y * y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -800000000000.0], N[(x * x), $MachinePrecision], If[LessEqual[x, -2.8e-40], N[(y * y), $MachinePrecision], If[LessEqual[x, -4.2e-107], N[(x + x), $MachinePrecision], If[LessEqual[x, 1.75e-84], N[(y * y), $MachinePrecision], If[LessEqual[x, 4.2e-69], N[(x + x), $MachinePrecision], If[LessEqual[x, 6e+38], N[(y * y), $MachinePrecision], N[(x * x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8e11 or 6.0000000000000002e38 < x

   1. Initial program 99.0%

      \[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
   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 91.1%

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

   9. Simplified 91.1%

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

   if -8e11 < x < -2.8e-40 or -4.1999999999999998e-107 < x < 1.7500000000000001e-84 or 4.1999999999999999e-69 < x < 6.0000000000000002e38

   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 75.0%

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

      1. unpow2 75.0%

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

   6. Simplified 75.0%

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

   7. Taylor expanded in x around 0 73.6%

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

   9. Simplified 73.6%

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

   if -2.8e-40 < x < -4.1999999999999998e-107 or 1.7500000000000001e-84 < x < 4.1999999999999999e-69

   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 81.3%

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

   9. Simplified 81.3%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -800000000000:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-40}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-107}:\\ \;\;\;\;x + x\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-84}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-69}:\\ \;\;\;\;x + x\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+38}:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 3: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6800000000 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;y \cdot y + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -6800000000.0) (not (<= x 2.0)))
   (+ (* y y) (* x x))
   (+ (* y y) (+ x x))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -6800000000.0) || !(x <= 2.0)) {
		tmp = (y * y) + (x * x);
	} 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 <= (-6800000000.0d0)) .or. (.not. (x <= 2.0d0))) then
        tmp = (y * y) + (x * x)
    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 <= -6800000000.0) || !(x <= 2.0)) {
		tmp = (y * y) + (x * x);
	} else {
		tmp = (y * y) + (x + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -6800000000.0) or not (x <= 2.0):
		tmp = (y * y) + (x * x)
	else:
		tmp = (y * y) + (x + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -6800000000.0) || !(x <= 2.0))
		tmp = Float64(Float64(y * y) + Float64(x * x));
	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 <= -6800000000.0) || ~((x <= 2.0)))
		tmp = (y * y) + (x * x);
	else
		tmp = (y * y) + (x + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -6800000000.0], N[Not[LessEqual[x, 2.0]], $MachinePrecision]], N[(N[(y * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] + N[(x + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.8e9 or 2 < x

   1. Initial program 99.1%

      \[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
   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 99.7%

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

      1. unpow2 99.7%

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

   6. Simplified 99.7%

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

   if -6.8e9 < 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 99.5%

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

      1. count-2 99.5%

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

   6. Simplified 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6800000000 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;y \cdot y + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + \left(x + x\right)\\ \end{array} \]

Alternative 4: 62.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{-279}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-160}:\\ \;\;\;\;x + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 6e-279) (* x x) (if (<= y 2.7e-160) (+ x x) (+ (* y y) (* x x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6e-279) {
		tmp = x * x;
	} else if (y <= 2.7e-160) {
		tmp = x + x;
	} 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 (y <= 6d-279) then
        tmp = x * x
    else if (y <= 2.7d-160) then
        tmp = x + x
    else
        tmp = (y * y) + (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 6e-279) {
		tmp = x * x;
	} else if (y <= 2.7e-160) {
		tmp = x + x;
	} else {
		tmp = (y * y) + (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 6e-279:
		tmp = x * x
	elif y <= 2.7e-160:
		tmp = x + x
	else:
		tmp = (y * y) + (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6e-279)
		tmp = Float64(x * x);
	elseif (y <= 2.7e-160)
		tmp = Float64(x + x);
	else
		tmp = Float64(Float64(y * y) + Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6e-279)
		tmp = x * x;
	elseif (y <= 2.7e-160)
		tmp = x + x;
	else
		tmp = (y * y) + (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6e-279], N[(x * x), $MachinePrecision], If[LessEqual[y, 2.7e-160], N[(x + x), $MachinePrecision], N[(N[(y * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.9999999999999999e-279

   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 76.7%

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

      1. unpow2 76.7%

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

   6. Simplified 76.7%

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

   7. Taylor expanded in x around inf 40.6%

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

   9. Simplified 40.6%

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

   if 5.9999999999999999e-279 < y < 2.7000000000000001e-160

   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 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 90.9%

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

      1. count-2 90.9%

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

   6. Simplified 90.9%

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

   7. Taylor expanded in x around inf 90.9%

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

   9. Simplified 90.9%

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

   if 2.7000000000000001e-160 < y

   1. Initial program 99.1%

      \[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
   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 96.1%

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

      1. unpow2 96.1%

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

   6. Simplified 96.1%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{-279}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-160}:\\ \;\;\;\;x + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + x \cdot x\\ \end{array} \]

Alternative 5: 72.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+19}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{+38}:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.2e+19) (* x x) (if (<= x 9.6e+38) (* y y) (* x x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.2e+19) {
		tmp = x * x;
	} else if (x <= 9.6e+38) {
		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 <= (-4.2d+19)) then
        tmp = x * x
    else if (x <= 9.6d+38) 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 <= -4.2e+19) {
		tmp = x * x;
	} else if (x <= 9.6e+38) {
		tmp = y * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.2e+19:
		tmp = x * x
	elif x <= 9.6e+38:
		tmp = y * y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.2e+19)
		tmp = Float64(x * x);
	elseif (x <= 9.6e+38)
		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 <= -4.2e+19)
		tmp = x * x;
	elseif (x <= 9.6e+38)
		tmp = y * y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.2e+19], N[(x * x), $MachinePrecision], If[LessEqual[x, 9.6e+38], N[(y * y), $MachinePrecision], N[(x * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.2e19 or 9.60000000000000069e38 < x

   1. Initial program 99.0%

      \[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
   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 91.1%

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

   9. Simplified 91.1%

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

   if -4.2e19 < x < 9.60000000000000069e38

   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 69.4%

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

      1. unpow2 69.4%

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

   6. Simplified 69.4%

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

   7. Taylor expanded in x around 0 68.1%

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

   9. Simplified 68.1%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+19}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{+38}:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 6: 42.5% 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 99.6%

   \[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
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 81.9%

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

   1. unpow2 81.9%

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

6. Simplified 81.9%

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

7. Taylor expanded in x around inf 40.4%

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

9. Simplified 40.4%

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

10. Final simplification 40.4%

    \[\leadsto x \cdot x \]

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 2023285 
(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)))