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

Percentage Accurate: 99.9% → 100.0%

Time: 2.9s

Alternatives: 8

Speedup: 1.2×

• 43.8% 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: 99.9% 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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. fma-def 100.0%

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

   3. +-commutative 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(x, x + 2, y \cdot y\right) \]

Alternative 2: 82.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 7.5 \cdot 10^{-100}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{elif}\;y \cdot y \leq 17:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;y \cdot y \leq 1.1 \cdot 10^{+62}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 7.5e-100)
   (* x (+ x 2.0))
   (if (<= (* y y) 17.0) (* y y) (if (<= (* y y) 1.1e+62) (* x x) (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 7.5e-100) {
		tmp = x * (x + 2.0);
	} else if ((y * y) <= 17.0) {
		tmp = y * y;
	} else if ((y * y) <= 1.1e+62) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * y) <= 7.5d-100) then
        tmp = x * (x + 2.0d0)
    else if ((y * y) <= 17.0d0) then
        tmp = y * y
    else if ((y * y) <= 1.1d+62) then
        tmp = x * x
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 7.5e-100) {
		tmp = x * (x + 2.0);
	} else if ((y * y) <= 17.0) {
		tmp = y * y;
	} else if ((y * y) <= 1.1e+62) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 7.5e-100:
		tmp = x * (x + 2.0)
	elif (y * y) <= 17.0:
		tmp = y * y
	elif (y * y) <= 1.1e+62:
		tmp = x * x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 7.5e-100)
		tmp = Float64(x * Float64(x + 2.0));
	elseif (Float64(y * y) <= 17.0)
		tmp = Float64(y * y);
	elseif (Float64(y * y) <= 1.1e+62)
		tmp = Float64(x * x);
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 7.5e-100)
		tmp = x * (x + 2.0);
	elseif ((y * y) <= 17.0)
		tmp = y * y;
	elseif ((y * y) <= 1.1e+62)
		tmp = x * x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 7.5e-100], N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y * y), $MachinePrecision], 17.0], N[(y * y), $MachinePrecision], If[LessEqual[N[(y * y), $MachinePrecision], 1.1e+62], N[(x * x), $MachinePrecision], N[(y * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y y) < 7.50000000000000015e-100

   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 y around 0 95.3%

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

   if 7.50000000000000015e-100 < (*.f64 y y) < 17 or 1.10000000000000007e62 < (*.f64 y y)

   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 80.6%

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

      1. unpow2 80.6%

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

   6. Simplified 80.6%

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

   if 17 < (*.f64 y y) < 1.10000000000000007e62

   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 y around 0 65.7%

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

   5. Taylor expanded in x around inf 66.1%

      \[\leadsto \color{blue}{{x}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 66.1%

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

   7. Simplified 66.1%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 7.5 \cdot 10^{-100}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{elif}\;y \cdot y \leq 17:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;y \cdot y \leq 1.1 \cdot 10^{+62}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 3: 71.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{+17}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-81}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-10}:\\ \;\;\;\;x + x\\ \mathbf{elif}\;x \leq 3700000000000:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -9.8e+17)
   (* x x)
   (if (<= x 2.2e-81)
     (* y y)
     (if (<= x 6.6e-10) (+ x x) (if (<= x 3700000000000.0) (* y y) (* x x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9.8e+17) {
		tmp = x * x;
	} else if (x <= 2.2e-81) {
		tmp = y * y;
	} else if (x <= 6.6e-10) {
		tmp = x + x;
	} else if (x <= 3700000000000.0) {
		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 <= (-9.8d+17)) then
        tmp = x * x
    else if (x <= 2.2d-81) then
        tmp = y * y
    else if (x <= 6.6d-10) then
        tmp = x + x
    else if (x <= 3700000000000.0d0) 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 <= -9.8e+17) {
		tmp = x * x;
	} else if (x <= 2.2e-81) {
		tmp = y * y;
	} else if (x <= 6.6e-10) {
		tmp = x + x;
	} else if (x <= 3700000000000.0) {
		tmp = y * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -9.8e+17:
		tmp = x * x
	elif x <= 2.2e-81:
		tmp = y * y
	elif x <= 6.6e-10:
		tmp = x + x
	elif x <= 3700000000000.0:
		tmp = y * y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9.8e+17)
		tmp = Float64(x * x);
	elseif (x <= 2.2e-81)
		tmp = Float64(y * y);
	elseif (x <= 6.6e-10)
		tmp = Float64(x + x);
	elseif (x <= 3700000000000.0)
		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 <= -9.8e+17)
		tmp = x * x;
	elseif (x <= 2.2e-81)
		tmp = y * y;
	elseif (x <= 6.6e-10)
		tmp = x + x;
	elseif (x <= 3700000000000.0)
		tmp = y * y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9.8e+17], N[(x * x), $MachinePrecision], If[LessEqual[x, 2.2e-81], N[(y * y), $MachinePrecision], If[LessEqual[x, 6.6e-10], N[(x + x), $MachinePrecision], If[LessEqual[x, 3700000000000.0], N[(y * y), $MachinePrecision], N[(x * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.8e17 or 3.7e12 < 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 y around 0 85.4%

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

   5. Taylor expanded in x around inf 85.3%

      \[\leadsto \color{blue}{{x}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 85.3%

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

   7. Simplified 85.3%

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

   if -9.8e17 < x < 2.1999999999999999e-81 or 6.6e-10 < x < 3.7e12

   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 66.2%

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

      1. unpow2 66.2%

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

   6. Simplified 66.2%

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

   if 2.1999999999999999e-81 < x < 6.6e-10

   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 y around 0 83.9%

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

   5. Taylor expanded in x around 0 78.9%

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

      1. count-2 78.9%

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

   7. Simplified 78.9%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{+17}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-81}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-10}:\\ \;\;\;\;x + x\\ \mathbf{elif}\;x \leq 3700000000000:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 4: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -420000000000 \lor \neg \left(x \leq 1.7 \cdot 10^{-8}\right):\\ \;\;\;\;y \cdot y + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + x \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -420000000000.0) (not (<= x 1.7e-8)))
   (+ (* y y) (* x x))
   (+ (* y y) (* x 2.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -420000000000.0) || !(x <= 1.7e-8)) {
		tmp = (y * y) + (x * x);
	} else {
		tmp = (y * y) + (x * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-420000000000.0d0)) .or. (.not. (x <= 1.7d-8))) then
        tmp = (y * y) + (x * x)
    else
        tmp = (y * y) + (x * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -420000000000.0) || !(x <= 1.7e-8)) {
		tmp = (y * y) + (x * x);
	} else {
		tmp = (y * y) + (x * 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -420000000000.0) or not (x <= 1.7e-8):
		tmp = (y * y) + (x * x)
	else:
		tmp = (y * y) + (x * 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -420000000000.0) || !(x <= 1.7e-8))
		tmp = Float64(Float64(y * y) + Float64(x * x));
	else
		tmp = Float64(Float64(y * y) + Float64(x * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -420000000000.0) || ~((x <= 1.7e-8)))
		tmp = (y * y) + (x * x);
	else
		tmp = (y * y) + (x * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -420000000000.0], N[Not[LessEqual[x, 1.7e-8]], $MachinePrecision]], N[(N[(y * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.2e11 or 1.7e-8 < 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 99.8%

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

      1. unpow2 99.8%

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

   6. Simplified 99.8%

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

   if -4.2e11 < x < 1.7e-8

   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 99.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -420000000000 \lor \neg \left(x \leq 1.7 \cdot 10^{-8}\right):\\ \;\;\;\;y \cdot y + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + x \cdot 2\\ \end{array} \]

Alternative 5: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -34000000000000:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 750000000000:\\ \;\;\;\;y \cdot y + x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -34000000000000.0)
   (* x x)
   (if (<= x 750000000000.0) (+ (* y y) (* x 2.0)) (* x (+ x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -34000000000000.0) {
		tmp = x * x;
	} else if (x <= 750000000000.0) {
		tmp = (y * y) + (x * 2.0);
	} else {
		tmp = x * (x + 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-34000000000000.0d0)) then
        tmp = x * x
    else if (x <= 750000000000.0d0) then
        tmp = (y * y) + (x * 2.0d0)
    else
        tmp = x * (x + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -34000000000000.0) {
		tmp = x * x;
	} else if (x <= 750000000000.0) {
		tmp = (y * y) + (x * 2.0);
	} else {
		tmp = x * (x + 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -34000000000000.0:
		tmp = x * x
	elif x <= 750000000000.0:
		tmp = (y * y) + (x * 2.0)
	else:
		tmp = x * (x + 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -34000000000000.0)
		tmp = Float64(x * x);
	elseif (x <= 750000000000.0)
		tmp = Float64(Float64(y * y) + Float64(x * 2.0));
	else
		tmp = Float64(x * Float64(x + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -34000000000000.0)
		tmp = x * x;
	elseif (x <= 750000000000.0)
		tmp = (y * y) + (x * 2.0);
	else
		tmp = x * (x + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -34000000000000.0], N[(x * x), $MachinePrecision], If[LessEqual[x, 750000000000.0], N[(N[(y * y), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.4e13

   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 y around 0 84.2%

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

   5. Taylor expanded in x around inf 84.2%

      \[\leadsto \color{blue}{{x}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 84.2%

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

   7. Simplified 84.2%

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

   if -3.4e13 < x < 7.5e11

   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 99.0%

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

   if 7.5e11 < 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 y around 0 86.8%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -34000000000000:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 750000000000:\\ \;\;\;\;y \cdot y + x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \end{array} \]

Alternative 6: 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 7: 72.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+15}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 7000000000000:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7.2e+15) (* x x) (if (<= x 7000000000000.0) (* y y) (* x x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.2e+15) {
		tmp = x * x;
	} else if (x <= 7000000000000.0) {
		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 <= (-7.2d+15)) then
        tmp = x * x
    else if (x <= 7000000000000.0d0) 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 <= -7.2e+15) {
		tmp = x * x;
	} else if (x <= 7000000000000.0) {
		tmp = y * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7.2e+15:
		tmp = x * x
	elif x <= 7000000000000.0:
		tmp = y * y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7.2e+15)
		tmp = Float64(x * x);
	elseif (x <= 7000000000000.0)
		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 <= -7.2e+15)
		tmp = x * x;
	elseif (x <= 7000000000000.0)
		tmp = y * y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7.2e+15], N[(x * x), $MachinePrecision], If[LessEqual[x, 7000000000000.0], N[(y * y), $MachinePrecision], N[(x * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.2e15 or 7e12 < 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 y around 0 85.4%

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

   5. Taylor expanded in x around inf 85.3%

      \[\leadsto \color{blue}{{x}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 85.3%

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

   7. Simplified 85.3%

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

   if -7.2e15 < x < 7e12

   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 61.8%

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

      1. unpow2 61.8%

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

   6. Simplified 61.8%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+15}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 7000000000000:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 8: 42.1% 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 y around 0 64.4%

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

5. Taylor expanded in x around inf 46.1%

   \[\leadsto \color{blue}{{x}^{2}} \]
6. Step-by-step derivation

   1. unpow2 46.1%

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

7. Simplified 46.1%

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

8. Final simplification 46.1%

   \[\leadsto x \cdot x \]

Developer target: 99.9% 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 2023252 
(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)))