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

Percentage Accurate: 100.0% → 100.0%

Time: 3.8s

Alternatives: 8

Speedup: 1.2×

• 43.0% 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

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

FPCore

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

C

double code(double x, double y) {
	return ((x * 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) + (x * 2.0d0)) + (y * y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 2: 82.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{-123} \lor \neg \left(y \cdot y \leq 1.9 \cdot 10^{-78}\right) \land y \cdot y \leq 2 \cdot 10^{+96}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* y y) 1e-123)
         (and (not (<= (* y y) 1.9e-78)) (<= (* y y) 2e+96)))
   (* x (+ x 2.0))
   (* y y)))

C

double code(double x, double y) {
	double tmp;
	if (((y * y) <= 1e-123) || (!((y * y) <= 1.9e-78) && ((y * y) <= 2e+96))) {
		tmp = x * (x + 2.0);
	} 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) <= 1d-123) .or. (.not. ((y * y) <= 1.9d-78)) .and. ((y * y) <= 2d+96)) then
        tmp = x * (x + 2.0d0)
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((y * y) <= 1e-123) || (!((y * y) <= 1.9e-78) && ((y * y) <= 2e+96))) {
		tmp = x * (x + 2.0);
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((y * y) <= 1e-123) or (not ((y * y) <= 1.9e-78) and ((y * y) <= 2e+96)):
		tmp = x * (x + 2.0)
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(y * y) <= 1e-123) || (!(Float64(y * y) <= 1.9e-78) && (Float64(y * y) <= 2e+96)))
		tmp = Float64(x * Float64(x + 2.0));
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((y * y) <= 1e-123) || (~(((y * y) <= 1.9e-78)) && ((y * y) <= 2e+96)))
		tmp = x * (x + 2.0);
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(y * y), $MachinePrecision], 1e-123], And[N[Not[LessEqual[N[(y * y), $MachinePrecision], 1.9e-78]], $MachinePrecision], LessEqual[N[(y * y), $MachinePrecision], 2e+96]]], N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1.0000000000000001e-123 or 1.8999999999999999e-78 < (*.f64 y y) < 2.0000000000000001e96

   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 88.6%

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

   if 1.0000000000000001e-123 < (*.f64 y y) < 1.8999999999999999e-78 or 2.0000000000000001e96 < (*.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 84.8%

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

      1. unpow2 84.8%

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

   6. Simplified 84.8%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{-123} \lor \neg \left(y \cdot y \leq 1.9 \cdot 10^{-78}\right) \land y \cdot y \leq 2 \cdot 10^{+96}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 3: 82.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5.4 \cdot 10^{-123}:\\ \;\;\;\;x \cdot x + x \cdot 2\\ \mathbf{elif}\;y \cdot y \leq 1.7 \cdot 10^{-76}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;y \cdot y \leq 2 \cdot 10^{+96}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 5.4e-123)
   (+ (* x x) (* x 2.0))
   (if (<= (* y y) 1.7e-76)
     (* y y)
     (if (<= (* y y) 2e+96) (* x (+ x 2.0)) (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5.4e-123) {
		tmp = (x * x) + (x * 2.0);
	} else if ((y * y) <= 1.7e-76) {
		tmp = y * y;
	} else if ((y * y) <= 2e+96) {
		tmp = x * (x + 2.0);
	} 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) <= 5.4d-123) then
        tmp = (x * x) + (x * 2.0d0)
    else if ((y * y) <= 1.7d-76) then
        tmp = y * y
    else if ((y * y) <= 2d+96) then
        tmp = x * (x + 2.0d0)
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5.4e-123) {
		tmp = (x * x) + (x * 2.0);
	} else if ((y * y) <= 1.7e-76) {
		tmp = y * y;
	} else if ((y * y) <= 2e+96) {
		tmp = x * (x + 2.0);
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 5.4e-123:
		tmp = (x * x) + (x * 2.0)
	elif (y * y) <= 1.7e-76:
		tmp = y * y
	elif (y * y) <= 2e+96:
		tmp = x * (x + 2.0)
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 5.4e-123)
		tmp = Float64(Float64(x * x) + Float64(x * 2.0));
	elseif (Float64(y * y) <= 1.7e-76)
		tmp = Float64(y * y);
	elseif (Float64(y * y) <= 2e+96)
		tmp = Float64(x * Float64(x + 2.0));
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 5.4e-123)
		tmp = (x * x) + (x * 2.0);
	elseif ((y * y) <= 1.7e-76)
		tmp = y * y;
	elseif ((y * y) <= 2e+96)
		tmp = x * (x + 2.0);
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 5.4e-123], N[(N[(x * x), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y * y), $MachinePrecision], 1.7e-76], N[(y * y), $MachinePrecision], If[LessEqual[N[(y * y), $MachinePrecision], 2e+96], N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision], N[(y * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y y) < 5.4000000000000002e-123

   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 94.6%

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

      1. *-commutative 94.6%

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

      2. +-commutative 94.6%

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

      3. distribute-rgt-in 94.6%

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

   6. Applied egg-rr 94.6%

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

   if 5.4000000000000002e-123 < (*.f64 y y) < 1.7e-76 or 2.0000000000000001e96 < (*.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 84.8%

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

      1. unpow2 84.8%

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

   6. Simplified 84.8%

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

   if 1.7e-76 < (*.f64 y y) < 2.0000000000000001e96

   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 y around 0 68.0%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5.4 \cdot 10^{-123}:\\ \;\;\;\;x \cdot x + x \cdot 2\\ \mathbf{elif}\;y \cdot y \leq 1.7 \cdot 10^{-76}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;y \cdot y \leq 2 \cdot 10^{+96}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 4: 72.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1250000000:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{-128}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-144}:\\ \;\;\;\;x \cdot 2\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{-64}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-23}:\\ \;\;\;\;x \cdot 2\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+16}:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1250000000.0)
   (* x x)
   (if (<= x -3.9e-128)
     (* y y)
     (if (<= x -9e-144)
       (* x 2.0)
       (if (<= x 9.6e-64)
         (* y y)
         (if (<= x 6.5e-23) (* x 2.0) (if (<= x 2.1e+16) (* y y) (* x x))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1250000000.0) {
		tmp = x * x;
	} else if (x <= -3.9e-128) {
		tmp = y * y;
	} else if (x <= -9e-144) {
		tmp = x * 2.0;
	} else if (x <= 9.6e-64) {
		tmp = y * y;
	} else if (x <= 6.5e-23) {
		tmp = x * 2.0;
	} else if (x <= 2.1e+16) {
		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 <= (-1250000000.0d0)) then
        tmp = x * x
    else if (x <= (-3.9d-128)) then
        tmp = y * y
    else if (x <= (-9d-144)) then
        tmp = x * 2.0d0
    else if (x <= 9.6d-64) then
        tmp = y * y
    else if (x <= 6.5d-23) then
        tmp = x * 2.0d0
    else if (x <= 2.1d+16) 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 <= -1250000000.0) {
		tmp = x * x;
	} else if (x <= -3.9e-128) {
		tmp = y * y;
	} else if (x <= -9e-144) {
		tmp = x * 2.0;
	} else if (x <= 9.6e-64) {
		tmp = y * y;
	} else if (x <= 6.5e-23) {
		tmp = x * 2.0;
	} else if (x <= 2.1e+16) {
		tmp = y * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1250000000.0:
		tmp = x * x
	elif x <= -3.9e-128:
		tmp = y * y
	elif x <= -9e-144:
		tmp = x * 2.0
	elif x <= 9.6e-64:
		tmp = y * y
	elif x <= 6.5e-23:
		tmp = x * 2.0
	elif x <= 2.1e+16:
		tmp = y * y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1250000000.0)
		tmp = Float64(x * x);
	elseif (x <= -3.9e-128)
		tmp = Float64(y * y);
	elseif (x <= -9e-144)
		tmp = Float64(x * 2.0);
	elseif (x <= 9.6e-64)
		tmp = Float64(y * y);
	elseif (x <= 6.5e-23)
		tmp = Float64(x * 2.0);
	elseif (x <= 2.1e+16)
		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 <= -1250000000.0)
		tmp = x * x;
	elseif (x <= -3.9e-128)
		tmp = y * y;
	elseif (x <= -9e-144)
		tmp = x * 2.0;
	elseif (x <= 9.6e-64)
		tmp = y * y;
	elseif (x <= 6.5e-23)
		tmp = x * 2.0;
	elseif (x <= 2.1e+16)
		tmp = y * y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1250000000.0], N[(x * x), $MachinePrecision], If[LessEqual[x, -3.9e-128], N[(y * y), $MachinePrecision], If[LessEqual[x, -9e-144], N[(x * 2.0), $MachinePrecision], If[LessEqual[x, 9.6e-64], N[(y * y), $MachinePrecision], If[LessEqual[x, 6.5e-23], N[(x * 2.0), $MachinePrecision], If[LessEqual[x, 2.1e+16], N[(y * y), $MachinePrecision], N[(x * x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25e9 or 2.1e16 < 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 86.5%

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

   5. Taylor expanded in x around inf 85.6%

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

      1. unpow2 85.6%

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

   7. Simplified 85.6%

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

   if -1.25e9 < x < -3.89999999999999997e-128 or -8.9999999999999996e-144 < x < 9.59999999999999994e-64 or 6.5e-23 < x < 2.1e16

   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 68.6%

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

      1. unpow2 68.6%

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

   6. Simplified 68.6%

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

   if -3.89999999999999997e-128 < x < -8.9999999999999996e-144 or 9.59999999999999994e-64 < x < 6.5e-23

   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 88.6%

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

   5. Taylor expanded in x around 0 88.6%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1250000000:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{-128}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-144}:\\ \;\;\;\;x \cdot 2\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{-64}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-23}:\\ \;\;\;\;x \cdot 2\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+16}:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 5: 90.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -260000000:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{+16}:\\ \;\;\;\;y \cdot y + \left(x + x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -260000000.0)
   (* x (+ x 2.0))
   (if (<= x 2.55e+16) (+ (* y y) (+ x x)) (* x x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -260000000.0) {
		tmp = x * (x + 2.0);
	} else if (x <= 2.55e+16) {
		tmp = (y * y) + (x + x);
	} 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 <= (-260000000.0d0)) then
        tmp = x * (x + 2.0d0)
    else if (x <= 2.55d+16) then
        tmp = (y * y) + (x + x)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -260000000.0) {
		tmp = x * (x + 2.0);
	} else if (x <= 2.55e+16) {
		tmp = (y * y) + (x + x);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -260000000.0:
		tmp = x * (x + 2.0)
	elif x <= 2.55e+16:
		tmp = (y * y) + (x + x)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -260000000.0)
		tmp = Float64(x * Float64(x + 2.0));
	elseif (x <= 2.55e+16)
		tmp = Float64(Float64(y * y) + Float64(x + x));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -260000000.0)
		tmp = x * (x + 2.0);
	elseif (x <= 2.55e+16)
		tmp = (y * y) + (x + x);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -260000000.0], N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.55e+16], N[(N[(y * y), $MachinePrecision] + N[(x + x), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.6e8

   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 88.4%

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

   if -2.6e8 < x < 2.55e16

   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 99.1%

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

      1. count-2 99.1%

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

   6. Simplified 99.1%

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

   if 2.55e16 < 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 84.5%

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

   5. Taylor expanded in x around inf 84.5%

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

      1. unpow2 84.5%

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

   7. Simplified 84.5%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -260000000:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{+16}:\\ \;\;\;\;y \cdot y + \left(x + x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

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: 59.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.0) {
		tmp = x * x;
	} else if (x <= 2.0) {
		tmp = x * 2.0;
	} 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 <= (-2.0d0)) then
        tmp = x * x
    else if (x <= 2.0d0) then
        tmp = x * 2.0d0
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -2 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 y around 0 83.6%

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

   5. Taylor expanded in x around inf 82.8%

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

      1. unpow2 82.8%

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

   7. Simplified 82.8%

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

   if -2 < 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 y around 0 41.5%

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

   5. Taylor expanded in x around 0 40.7%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 8: 20.6% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(x * 2.0), $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 60.6%

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

5. Taylor expanded in x around 0 23.8%

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

6. Final simplification 23.8%

   \[\leadsto x \cdot 2 \]

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 2023221 
(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)))