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

Percentage Accurate: 100.0% → 100.0%

Time: 6.9s

Alternatives: 8

Speedup: 1.2×

• 42.9% 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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(y * y + 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. Step-by-step derivation

   1. +-commutative 100.0%

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

   2. fma-def 100.0%

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

   3. +-commutative 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 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 3: 79.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot y + x \cdot x\\ \mathbf{if}\;x \leq -1.2 \cdot 10^{-51}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.66 \cdot 10^{-85}:\\ \;\;\;\;x + x\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-168}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-81}:\\ \;\;\;\;x + x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (* y y) (* x x))))
   (if (<= x -1.2e-51)
     t_0
     (if (<= x -1.66e-85)
       (+ x x)
       (if (<= x 4.5e-168) (* y y) (if (<= x 2.9e-81) (+ x x) t_0))))))

C

double code(double x, double y) {
	double t_0 = (y * y) + (x * x);
	double tmp;
	if (x <= -1.2e-51) {
		tmp = t_0;
	} else if (x <= -1.66e-85) {
		tmp = x + x;
	} else if (x <= 4.5e-168) {
		tmp = y * y;
	} else if (x <= 2.9e-81) {
		tmp = x + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y * y) + (x * x)
    if (x <= (-1.2d-51)) then
        tmp = t_0
    else if (x <= (-1.66d-85)) then
        tmp = x + x
    else if (x <= 4.5d-168) then
        tmp = y * y
    else if (x <= 2.9d-81) then
        tmp = x + x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * y) + (x * x);
	double tmp;
	if (x <= -1.2e-51) {
		tmp = t_0;
	} else if (x <= -1.66e-85) {
		tmp = x + x;
	} else if (x <= 4.5e-168) {
		tmp = y * y;
	} else if (x <= 2.9e-81) {
		tmp = x + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * y) + (x * x)
	tmp = 0
	if x <= -1.2e-51:
		tmp = t_0
	elif x <= -1.66e-85:
		tmp = x + x
	elif x <= 4.5e-168:
		tmp = y * y
	elif x <= 2.9e-81:
		tmp = x + x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * y) + Float64(x * x))
	tmp = 0.0
	if (x <= -1.2e-51)
		tmp = t_0;
	elseif (x <= -1.66e-85)
		tmp = Float64(x + x);
	elseif (x <= 4.5e-168)
		tmp = Float64(y * y);
	elseif (x <= 2.9e-81)
		tmp = Float64(x + x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * y) + (x * x);
	tmp = 0.0;
	if (x <= -1.2e-51)
		tmp = t_0;
	elseif (x <= -1.66e-85)
		tmp = x + x;
	elseif (x <= 4.5e-168)
		tmp = y * y;
	elseif (x <= 2.9e-81)
		tmp = x + x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.2e-51], t$95$0, If[LessEqual[x, -1.66e-85], N[(x + x), $MachinePrecision], If[LessEqual[x, 4.5e-168], N[(y * y), $MachinePrecision], If[LessEqual[x, 2.9e-81], N[(x + x), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.2e-51 or 2.89999999999999989e-81 < 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 95.1%

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

      1. unpow2 95.1%

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

   6. Simplified 95.1%

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

   if -1.2e-51 < x < -1.66e-85 or 4.5000000000000001e-168 < x < 2.89999999999999989e-81

   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 83.0%

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

      1. count-2 83.0%

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

   9. Simplified 83.0%

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

   if -1.66e-85 < x < 4.5000000000000001e-168

   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 74.1%

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

      1. unpow2 74.1%

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

   6. Simplified 74.1%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-51}:\\ \;\;\;\;y \cdot y + x \cdot x\\ \mathbf{elif}\;x \leq -1.66 \cdot 10^{-85}:\\ \;\;\;\;x + x\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-168}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-81}:\\ \;\;\;\;x + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + x \cdot x\\ \end{array} \]

Alternative 4: 70.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.4 \cdot 10^{+58}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-167}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq 3.25 \cdot 10^{-79}:\\ \;\;\;\;x + x\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+68}:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6.4e+58)
   (* x x)
   (if (<= x 7.5e-167)
     (* y y)
     (if (<= x 3.25e-79) (+ x x) (if (<= x 2.2e+68) (* y y) (* x x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6.4e+58) {
		tmp = x * x;
	} else if (x <= 7.5e-167) {
		tmp = y * y;
	} else if (x <= 3.25e-79) {
		tmp = x + x;
	} else if (x <= 2.2e+68) {
		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 <= (-6.4d+58)) then
        tmp = x * x
    else if (x <= 7.5d-167) then
        tmp = y * y
    else if (x <= 3.25d-79) then
        tmp = x + x
    else if (x <= 2.2d+68) 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 <= -6.4e+58) {
		tmp = x * x;
	} else if (x <= 7.5e-167) {
		tmp = y * y;
	} else if (x <= 3.25e-79) {
		tmp = x + x;
	} else if (x <= 2.2e+68) {
		tmp = y * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6.4e+58:
		tmp = x * x
	elif x <= 7.5e-167:
		tmp = y * y
	elif x <= 3.25e-79:
		tmp = x + x
	elif x <= 2.2e+68:
		tmp = y * y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6.4e+58)
		tmp = Float64(x * x);
	elseif (x <= 7.5e-167)
		tmp = Float64(y * y);
	elseif (x <= 3.25e-79)
		tmp = Float64(x + x);
	elseif (x <= 2.2e+68)
		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 <= -6.4e+58)
		tmp = x * x;
	elseif (x <= 7.5e-167)
		tmp = y * y;
	elseif (x <= 3.25e-79)
		tmp = x + x;
	elseif (x <= 2.2e+68)
		tmp = y * y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6.4e+58], N[(x * x), $MachinePrecision], If[LessEqual[x, 7.5e-167], N[(y * y), $MachinePrecision], If[LessEqual[x, 3.25e-79], N[(x + x), $MachinePrecision], If[LessEqual[x, 2.2e+68], N[(y * y), $MachinePrecision], N[(x * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.40000000000000031e58 or 2.19999999999999987e68 < 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 90.0%

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

      1. unpow2 90.0%

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

   9. Simplified 90.0%

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

   if -6.40000000000000031e58 < x < 7.5000000000000007e-167 or 3.2500000000000001e-79 < x < 2.19999999999999987e68

   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 66.7%

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

      1. unpow2 66.7%

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

   6. Simplified 66.7%

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

   if 7.5000000000000007e-167 < x < 3.2500000000000001e-79

   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 81.8%

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

      1. count-2 81.8%

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

   9. Simplified 81.8%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.4 \cdot 10^{+58}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-167}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq 3.25 \cdot 10^{-79}:\\ \;\;\;\;x + x\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+68}:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 5: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \lor \neg \left(x \leq 4.5 \cdot 10^{-14}\right):\\ \;\;\;\;y \cdot y + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.0) (not (<= x 4.5e-14)))
   (+ (* y y) (* x x))
   (+ (* y y) (+ x x))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.0) || !(x <= 4.5e-14)) {
		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 <= (-2.0d0)) .or. (.not. (x <= 4.5d-14))) 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 <= -2.0) || !(x <= 4.5e-14)) {
		tmp = (y * y) + (x * x);
	} else {
		tmp = (y * y) + (x + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.0) or not (x <= 4.5e-14):
		tmp = (y * y) + (x * x)
	else:
		tmp = (y * y) + (x + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.0) || !(x <= 4.5e-14))
		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 <= -2.0) || ~((x <= 4.5e-14)))
		tmp = (y * y) + (x * x);
	else
		tmp = (y * y) + (x + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.0], N[Not[LessEqual[x, 4.5e-14]], $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 < -2 or 4.4999999999999998e-14 < 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 99.4%

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

      1. unpow2 99.4%

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

   6. Simplified 99.4%

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

   if -2 < x < 4.4999999999999998e-14

   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 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \lor \neg \left(x \leq 4.5 \cdot 10^{-14}\right):\\ \;\;\;\;y \cdot y + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + \left(x + x\right)\\ \end{array} \]

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: 72.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+43}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+68}:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.4e+43) (* x x) (if (<= x 1.2e+68) (* y y) (* x x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.4e+43) {
		tmp = x * x;
	} else if (x <= 1.2e+68) {
		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 <= (-1.4d+43)) then
        tmp = x * x
    else if (x <= 1.2d+68) 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 <= -1.4e+43) {
		tmp = x * x;
	} else if (x <= 1.2e+68) {
		tmp = y * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.4e+43:
		tmp = x * x
	elif x <= 1.2e+68:
		tmp = y * y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.4e+43)
		tmp = Float64(x * x);
	elseif (x <= 1.2e+68)
		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 <= -1.4e+43)
		tmp = x * x;
	elseif (x <= 1.2e+68)
		tmp = y * y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.4e+43], N[(x * x), $MachinePrecision], If[LessEqual[x, 1.2e+68], N[(y * y), $MachinePrecision], N[(x * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.40000000000000009e43 or 1.20000000000000004e68 < 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 90.0%

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

      1. unpow2 90.0%

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

   9. Simplified 90.0%

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

   if -1.40000000000000009e43 < x < 1.20000000000000004e68

   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 61.9%

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

      1. unpow2 61.9%

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

   6. Simplified 61.9%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+43}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+68}:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 8: 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 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 45.2%

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

   1. unpow2 45.2%

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

9. Simplified 45.2%

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

10. Final simplification 45.2%

    \[\leadsto x \cdot x \]

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 2023279 
(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)))