Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B

Percentage Accurate: 100.0% → 100.0%

Time: 4.1s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

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

   2. associate-+r+ 100.0%

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

   3. fma-def 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 60.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-102}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+215}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+250}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   (* x y)
   (if (<= y 2.9e-102)
     x
     (if (<= y 1.35e+215) y (if (<= y 1.7e+250) (* x y) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x * y;
	} else if (y <= 2.9e-102) {
		tmp = x;
	} else if (y <= 1.35e+215) {
		tmp = y;
	} else if (y <= 1.7e+250) {
		tmp = x * y;
	} else {
		tmp = 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 <= (-1.0d0)) then
        tmp = x * y
    else if (y <= 2.9d-102) then
        tmp = x
    else if (y <= 1.35d+215) then
        tmp = y
    else if (y <= 1.7d+250) then
        tmp = x * y
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x * y;
	} else if (y <= 2.9e-102) {
		tmp = x;
	} else if (y <= 1.35e+215) {
		tmp = y;
	} else if (y <= 1.7e+250) {
		tmp = x * y;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x * y
	elif y <= 2.9e-102:
		tmp = x
	elif y <= 1.35e+215:
		tmp = y
	elif y <= 1.7e+250:
		tmp = x * y
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(x * y);
	elseif (y <= 2.9e-102)
		tmp = x;
	elseif (y <= 1.35e+215)
		tmp = y;
	elseif (y <= 1.7e+250)
		tmp = Float64(x * y);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x * y;
	elseif (y <= 2.9e-102)
		tmp = x;
	elseif (y <= 1.35e+215)
		tmp = y;
	elseif (y <= 1.7e+250)
		tmp = x * y;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], N[(x * y), $MachinePrecision], If[LessEqual[y, 2.9e-102], x, If[LessEqual[y, 1.35e+215], y, If[LessEqual[y, 1.7e+250], N[(x * y), $MachinePrecision], y]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 1.35e215 < y < 1.69999999999999987e250

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+r+ 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 57.6%

      \[\leadsto \color{blue}{x \cdot \left(1 + y\right)} \]
   5. Step-by-step derivation

      1. +-commutative 57.6%

         \[\leadsto x \cdot \color{blue}{\left(y + 1\right)} \]

   6. Simplified 57.6%

      \[\leadsto \color{blue}{x \cdot \left(y + 1\right)} \]

   7. Taylor expanded in y around inf 55.7%

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

   if -1 < y < 2.89999999999999986e-102

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+r+ 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 77.9%

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

   if 2.89999999999999986e-102 < y < 1.35e215 or 1.69999999999999987e250 < y

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+r+ 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 50.1%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-102}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+215}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+250}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 3: 85.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-108} \lor \neg \left(x \leq 0.00105\right):\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.25e-108) (not (<= x 0.00105))) (* x (+ y 1.0)) y))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.25e-108) || !(x <= 0.00105)) {
		tmp = x * (y + 1.0);
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.25d-108)) .or. (.not. (x <= 0.00105d0))) then
        tmp = x * (y + 1.0d0)
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.25e-108) || !(x <= 0.00105)) {
		tmp = x * (y + 1.0);
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.25e-108) or not (x <= 0.00105):
		tmp = x * (y + 1.0)
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.25e-108) || !(x <= 0.00105))
		tmp = Float64(x * Float64(y + 1.0));
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.25e-108) || ~((x <= 0.00105)))
		tmp = x * (y + 1.0);
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.25e-108], N[Not[LessEqual[x, 0.00105]], $MachinePrecision]], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision], y]

Derivation

1. Split input into 2 regimes

2. if x < -1.25e-108 or 0.00104999999999999994 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+r+ 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 93.1%

      \[\leadsto \color{blue}{x \cdot \left(1 + y\right)} \]
   5. Step-by-step derivation

      1. +-commutative 93.1%

         \[\leadsto x \cdot \color{blue}{\left(y + 1\right)} \]

   6. Simplified 93.1%

      \[\leadsto \color{blue}{x \cdot \left(y + 1\right)} \]

   if -1.25e-108 < x < 0.00104999999999999994

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+r+ 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 84.1%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-108} \lor \neg \left(x \leq 0.00105\right):\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 4: 72.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{-109}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7.4e-109) (* x (+ y 1.0)) (* y (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.4e-109) {
		tmp = x * (y + 1.0);
	} else {
		tmp = y * (x + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-7.4d-109)) then
        tmp = x * (y + 1.0d0)
    else
        tmp = y * (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -7.4e-109) {
		tmp = x * (y + 1.0);
	} else {
		tmp = y * (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7.4e-109:
		tmp = x * (y + 1.0)
	else:
		tmp = y * (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7.4e-109)
		tmp = Float64(x * Float64(y + 1.0));
	else
		tmp = Float64(y * Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7.4e-109)
		tmp = x * (y + 1.0);
	else
		tmp = y * (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7.4e-109], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.39999999999999961e-109

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+r+ 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 88.3%

      \[\leadsto \color{blue}{x \cdot \left(1 + y\right)} \]
   5. Step-by-step derivation

      1. +-commutative 88.3%

         \[\leadsto x \cdot \color{blue}{\left(y + 1\right)} \]

   6. Simplified 88.3%

      \[\leadsto \color{blue}{x \cdot \left(y + 1\right)} \]

   if -7.39999999999999961e-109 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+r+ 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 68.9%

      \[\leadsto \color{blue}{y \cdot \left(1 + x\right)} \]
   5. Step-by-step derivation

      1. +-commutative 68.9%

         \[\leadsto y \cdot \color{blue}{\left(x + 1\right)} \]

   6. Simplified 68.9%

      \[\leadsto \color{blue}{y \cdot \left(x + 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{-109}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \]

Alternative 5: 72.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-108}:\\ \;\;\;\;x + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.25e-108) (+ x (* x y)) (* y (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.25e-108) {
		tmp = x + (x * y);
	} else {
		tmp = y * (x + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.25d-108)) then
        tmp = x + (x * y)
    else
        tmp = y * (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.25e-108) {
		tmp = x + (x * y);
	} else {
		tmp = y * (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.25e-108:
		tmp = x + (x * y)
	else:
		tmp = y * (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.25e-108)
		tmp = Float64(x + Float64(x * y));
	else
		tmp = Float64(y * Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.25e-108)
		tmp = x + (x * y);
	else
		tmp = y * (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.25e-108], N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(y * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.25e-108

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+r+ 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 88.3%

      \[\leadsto \color{blue}{x \cdot \left(1 + y\right)} \]
   5. Step-by-step derivation

      1. +-commutative 88.3%

         \[\leadsto x \cdot \color{blue}{\left(y + 1\right)} \]

   6. Simplified 88.3%

      \[\leadsto \color{blue}{x \cdot \left(y + 1\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in 88.4%

         \[\leadsto \color{blue}{x \cdot y + x \cdot 1} \]

      2. *-rgt-identity 88.4%

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

   8. Applied egg-rr 88.4%

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

   if -1.25e-108 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+r+ 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 68.9%

      \[\leadsto \color{blue}{y \cdot \left(1 + x\right)} \]
   5. Step-by-step derivation

      1. +-commutative 68.9%

         \[\leadsto y \cdot \color{blue}{\left(x + 1\right)} \]

   6. Simplified 68.9%

      \[\leadsto \color{blue}{y \cdot \left(x + 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-108}:\\ \;\;\;\;x + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \]

Alternative 6: 72.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-108}:\\ \;\;\;\;x + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.1e-108) (+ x (* x y)) (+ y (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.1e-108) {
		tmp = x + (x * y);
	} else {
		tmp = y + (x * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.1d-108)) then
        tmp = x + (x * y)
    else
        tmp = y + (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.1e-108) {
		tmp = x + (x * y);
	} else {
		tmp = y + (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.1e-108:
		tmp = x + (x * y)
	else:
		tmp = y + (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.1e-108)
		tmp = Float64(x + Float64(x * y));
	else
		tmp = Float64(y + Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.1e-108)
		tmp = x + (x * y);
	else
		tmp = y + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.1e-108], N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(y + N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1000000000000001e-108

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+r+ 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 88.3%

      \[\leadsto \color{blue}{x \cdot \left(1 + y\right)} \]
   5. Step-by-step derivation

      1. +-commutative 88.3%

         \[\leadsto x \cdot \color{blue}{\left(y + 1\right)} \]

   6. Simplified 88.3%

      \[\leadsto \color{blue}{x \cdot \left(y + 1\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in 88.4%

         \[\leadsto \color{blue}{x \cdot y + x \cdot 1} \]

      2. *-rgt-identity 88.4%

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

   8. Applied egg-rr 88.4%

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

   if -1.1000000000000001e-108 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 68.9%

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

      1. *-commutative 68.9%

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

   4. Simplified 68.9%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-108}:\\ \;\;\;\;x + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot y\\ \end{array} \]

Alternative 7: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 8: 50.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{-109}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -7.4e-109) x y))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.4e-109) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-7.4d-109)) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -7.4e-109) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7.4e-109:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7.4e-109)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7.4e-109)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7.4e-109], x, y]

Derivation

1. Split input into 2 regimes

2. if x < -7.39999999999999961e-109

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+r+ 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 50.3%

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

   if -7.39999999999999961e-109 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+r+ 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 50.7%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{-109}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 9: 39.2% accurate, 7.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

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

   2. associate-+r+ 100.0%

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

   3. fma-def 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in y around 0 38.8%

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

5. Final simplification 38.8%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023279 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))