Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, A

Percentage Accurate: 100.0% → 100.0%

Time: 5.0s

Alternatives: 11

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ (- (* x (- y 1.0)) (* y 0.5)) 0.918938533204673))

C

double code(double x, double y) {
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * (y - 1.0d0)) - (y * 0.5d0)) + 0.918938533204673d0
end function

Java

public static double code(double x, double y) {
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
}

Python

def code(x, y):
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673

Julia

function code(x, y)
	return Float64(Float64(Float64(x * Float64(y - 1.0)) - Float64(y * 0.5)) + 0.918938533204673)
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
end

Wolfram

code[x_, y_] := N[(N[(N[(x * N[(y - 1.0), $MachinePrecision]), $MachinePrecision] - N[(y * 0.5), $MachinePrecision]), $MachinePrecision] + 0.918938533204673), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ (- (* x (- y 1.0)) (* y 0.5)) 0.918938533204673))

C

double code(double x, double y) {
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * (y - 1.0d0)) - (y * 0.5d0)) + 0.918938533204673d0
end function

Java

public static double code(double x, double y) {
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
}

Python

def code(x, y):
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673

Julia

function code(x, y)
	return Float64(Float64(Float64(x * Float64(y - 1.0)) - Float64(y * 0.5)) + 0.918938533204673)
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
end

Wolfram

code[x_, y_] := N[(N[(N[(x * N[(y - 1.0), $MachinePrecision]), $MachinePrecision] - N[(y * 0.5), $MachinePrecision]), $MachinePrecision] + 0.918938533204673), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.918938533204673 + \left(x \cdot \left(y + -1\right) - y \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ 0.918938533204673 (- (* x (+ y -1.0)) (* y 0.5))))

C

double code(double x, double y) {
	return 0.918938533204673 + ((x * (y + -1.0)) - (y * 0.5));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.918938533204673d0 + ((x * (y + (-1.0d0))) - (y * 0.5d0))
end function

Java

public static double code(double x, double y) {
	return 0.918938533204673 + ((x * (y + -1.0)) - (y * 0.5));
}

Python

def code(x, y):
	return 0.918938533204673 + ((x * (y + -1.0)) - (y * 0.5))

Julia

function code(x, y)
	return Float64(0.918938533204673 + Float64(Float64(x * Float64(y + -1.0)) - Float64(y * 0.5)))
end

MATLAB

function tmp = code(x, y)
	tmp = 0.918938533204673 + ((x * (y + -1.0)) - (y * 0.5));
end

Wolfram

code[x_, y_] := N[(0.918938533204673 + N[(N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision] - N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]

2. Final simplification 100.0%

   \[\leadsto 0.918938533204673 + \left(x \cdot \left(y + -1\right) - y \cdot 0.5\right) \]

Alternative 2: 98.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.26 \lor \neg \left(y \leq 1750000\right):\\ \;\;\;\;x \cdot \left(y + -1\right) - y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.918938533204673 - y \cdot 0.5\right) - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.26) (not (<= y 1750000.0)))
   (- (* x (+ y -1.0)) (* y 0.5))
   (- (- 0.918938533204673 (* y 0.5)) x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.26) || !(y <= 1750000.0)) {
		tmp = (x * (y + -1.0)) - (y * 0.5);
	} else {
		tmp = (0.918938533204673 - (y * 0.5)) - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.26d0)) .or. (.not. (y <= 1750000.0d0))) then
        tmp = (x * (y + (-1.0d0))) - (y * 0.5d0)
    else
        tmp = (0.918938533204673d0 - (y * 0.5d0)) - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.26) || !(y <= 1750000.0)) {
		tmp = (x * (y + -1.0)) - (y * 0.5);
	} else {
		tmp = (0.918938533204673 - (y * 0.5)) - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.26) or not (y <= 1750000.0):
		tmp = (x * (y + -1.0)) - (y * 0.5)
	else:
		tmp = (0.918938533204673 - (y * 0.5)) - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.26) || !(y <= 1750000.0))
		tmp = Float64(Float64(x * Float64(y + -1.0)) - Float64(y * 0.5));
	else
		tmp = Float64(Float64(0.918938533204673 - Float64(y * 0.5)) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.26) || ~((y <= 1750000.0)))
		tmp = (x * (y + -1.0)) - (y * 0.5);
	else
		tmp = (0.918938533204673 - (y * 0.5)) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.26], N[Not[LessEqual[y, 1750000.0]], $MachinePrecision]], N[(N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision] - N[(y * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(0.918938533204673 - N[(y * 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.26000000000000001 or 1.75e6 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   4. Taylor expanded in y around inf 100.0%

      \[\leadsto x \cdot \left(y + -1\right) - \color{blue}{0.5 \cdot y} \]
   5. Step-by-step derivation

      1. *-commutative 54.0%

         \[\leadsto \left(-x\right) - \color{blue}{y \cdot 0.5} \]

   6. Simplified 100.0%

      \[\leadsto x \cdot \left(y + -1\right) - \color{blue}{y \cdot 0.5} \]

   if -1.26000000000000001 < y < 1.75e6

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   4. Taylor expanded in y around 0 99.8%

      \[\leadsto \color{blue}{-1 \cdot x} - \left(y \cdot 0.5 - 0.918938533204673\right) \]
   5. Step-by-step derivation

      1. neg-mul-1 99.8%

         \[\leadsto \color{blue}{\left(-x\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   6. Simplified 99.8%

      \[\leadsto \color{blue}{\left(-x\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.26 \lor \neg \left(y \leq 1750000\right):\\ \;\;\;\;x \cdot \left(y + -1\right) - y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.918938533204673 - y \cdot 0.5\right) - x\\ \end{array} \]

Alternative 3: 73.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-x\right) - y \cdot 0.5\\ \mathbf{if}\;y \leq -3.2 \cdot 10^{+34}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -3.3 \cdot 10^{-6}:\\ \;\;\;\;x \cdot y - x\\ \mathbf{elif}\;y \leq 1.8:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (- x) (* y 0.5))))
   (if (<= y -3.2e+34)
     t_0
     (if (<= y -3.3e-6)
       (- (* x y) x)
       (if (<= y 1.8) (- 0.918938533204673 x) t_0)))))

C

double code(double x, double y) {
	double t_0 = -x - (y * 0.5);
	double tmp;
	if (y <= -3.2e+34) {
		tmp = t_0;
	} else if (y <= -3.3e-6) {
		tmp = (x * y) - x;
	} else if (y <= 1.8) {
		tmp = 0.918938533204673 - 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 = -x - (y * 0.5d0)
    if (y <= (-3.2d+34)) then
        tmp = t_0
    else if (y <= (-3.3d-6)) then
        tmp = (x * y) - x
    else if (y <= 1.8d0) then
        tmp = 0.918938533204673d0 - x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = -x - (y * 0.5);
	double tmp;
	if (y <= -3.2e+34) {
		tmp = t_0;
	} else if (y <= -3.3e-6) {
		tmp = (x * y) - x;
	} else if (y <= 1.8) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = -x - (y * 0.5)
	tmp = 0
	if y <= -3.2e+34:
		tmp = t_0
	elif y <= -3.3e-6:
		tmp = (x * y) - x
	elif y <= 1.8:
		tmp = 0.918938533204673 - x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(-x) - Float64(y * 0.5))
	tmp = 0.0
	if (y <= -3.2e+34)
		tmp = t_0;
	elseif (y <= -3.3e-6)
		tmp = Float64(Float64(x * y) - x);
	elseif (y <= 1.8)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = -x - (y * 0.5);
	tmp = 0.0;
	if (y <= -3.2e+34)
		tmp = t_0;
	elseif (y <= -3.3e-6)
		tmp = (x * y) - x;
	elseif (y <= 1.8)
		tmp = 0.918938533204673 - x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[((-x) - N[(y * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.2e+34], t$95$0, If[LessEqual[y, -3.3e-6], N[(N[(x * y), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[y, 1.8], N[(0.918938533204673 - x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.1999999999999998e34 or 1.80000000000000004 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   4. Taylor expanded in y around 0 56.6%

      \[\leadsto \color{blue}{-1 \cdot x} - \left(y \cdot 0.5 - 0.918938533204673\right) \]
   5. Step-by-step derivation

      1. neg-mul-1 56.6%

         \[\leadsto \color{blue}{\left(-x\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   6. Simplified 56.6%

      \[\leadsto \color{blue}{\left(-x\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   7. Taylor expanded in y around inf 56.0%

      \[\leadsto \left(-x\right) - \color{blue}{0.5 \cdot y} \]
   8. Step-by-step derivation

      1. *-commutative 56.0%

         \[\leadsto \left(-x\right) - \color{blue}{y \cdot 0.5} \]

   9. Simplified 56.0%

      \[\leadsto \left(-x\right) - \color{blue}{y \cdot 0.5} \]

   if -3.1999999999999998e34 < y < -3.30000000000000017e-6

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   4. Taylor expanded in y around 0 100.0%

      \[\leadsto x \cdot \left(y + -1\right) - \color{blue}{-0.918938533204673} \]

   5. Taylor expanded in x around inf 100.0%

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

      1. sub-neg 100.0%

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

      2. metadata-eval 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. mul-1-neg 100.0%

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

      6. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   if -3.30000000000000017e-6 < y < 1.80000000000000004

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   4. Taylor expanded in y around 0 98.4%

      \[\leadsto x \cdot \left(y + -1\right) - \color{blue}{-0.918938533204673} \]

   5. Taylor expanded in y around 0 98.3%

      \[\leadsto \color{blue}{0.918938533204673 + -1 \cdot x} \]
   6. Step-by-step derivation

      1. mul-1-neg 98.3%

         \[\leadsto 0.918938533204673 + \color{blue}{\left(-x\right)} \]

      2. unsub-neg 98.3%

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   7. Simplified 98.3%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+34}:\\ \;\;\;\;\left(-x\right) - y \cdot 0.5\\ \mathbf{elif}\;y \leq -3.3 \cdot 10^{-6}:\\ \;\;\;\;x \cdot y - x\\ \mathbf{elif}\;y \leq 1.8:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) - y \cdot 0.5\\ \end{array} \]

Alternative 4: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.5 \lor \neg \left(y \leq 1.75\right):\\ \;\;\;\;x \cdot y - y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.5) (not (<= y 1.75)))
   (- (* x y) (* y 0.5))
   (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.5) || !(y <= 1.75)) {
		tmp = (x * y) - (y * 0.5);
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.5d0)) .or. (.not. (y <= 1.75d0))) then
        tmp = (x * y) - (y * 0.5d0)
    else
        tmp = 0.918938533204673d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.5) || !(y <= 1.75)) {
		tmp = (x * y) - (y * 0.5);
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.5) or not (y <= 1.75):
		tmp = (x * y) - (y * 0.5)
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.5) || !(y <= 1.75))
		tmp = Float64(Float64(x * y) - Float64(y * 0.5));
	else
		tmp = Float64(0.918938533204673 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.5) || ~((y <= 1.75)))
		tmp = (x * y) - (y * 0.5);
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.5], N[Not[LessEqual[y, 1.75]], $MachinePrecision]], N[(N[(x * y), $MachinePrecision] - N[(y * 0.5), $MachinePrecision]), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.5 or 1.75 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   4. Taylor expanded in y around inf 99.4%

      \[\leadsto x \cdot \left(y + -1\right) - \color{blue}{0.5 \cdot y} \]
   5. Step-by-step derivation

      1. *-commutative 53.8%

         \[\leadsto \left(-x\right) - \color{blue}{y \cdot 0.5} \]

   6. Simplified 99.4%

      \[\leadsto x \cdot \left(y + -1\right) - \color{blue}{y \cdot 0.5} \]

   7. Taylor expanded in y around inf 98.1%

      \[\leadsto \color{blue}{x \cdot y} - y \cdot 0.5 \]

   if -1.5 < y < 1.75

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   4. Taylor expanded in y around 0 98.4%

      \[\leadsto x \cdot \left(y + -1\right) - \color{blue}{-0.918938533204673} \]

   5. Taylor expanded in y around 0 98.3%

      \[\leadsto \color{blue}{0.918938533204673 + -1 \cdot x} \]
   6. Step-by-step derivation

      1. mul-1-neg 98.3%

         \[\leadsto 0.918938533204673 + \color{blue}{\left(-x\right)} \]

      2. unsub-neg 98.3%

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   7. Simplified 98.3%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \lor \neg \left(y \leq 1.75\right):\\ \;\;\;\;x \cdot y - y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]

Alternative 5: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+34} \lor \neg \left(y \leq 360000000\right):\\ \;\;\;\;x \cdot y - y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + -1\right) - -0.918938533204673\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3e+34) (not (<= y 360000000.0)))
   (- (* x y) (* y 0.5))
   (- (* x (+ y -1.0)) -0.918938533204673)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3e+34) || !(y <= 360000000.0)) {
		tmp = (x * y) - (y * 0.5);
	} else {
		tmp = (x * (y + -1.0)) - -0.918938533204673;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-3d+34)) .or. (.not. (y <= 360000000.0d0))) then
        tmp = (x * y) - (y * 0.5d0)
    else
        tmp = (x * (y + (-1.0d0))) - (-0.918938533204673d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -3e+34) || !(y <= 360000000.0)) {
		tmp = (x * y) - (y * 0.5);
	} else {
		tmp = (x * (y + -1.0)) - -0.918938533204673;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3e+34) or not (y <= 360000000.0):
		tmp = (x * y) - (y * 0.5)
	else:
		tmp = (x * (y + -1.0)) - -0.918938533204673
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3e+34) || !(y <= 360000000.0))
		tmp = Float64(Float64(x * y) - Float64(y * 0.5));
	else
		tmp = Float64(Float64(x * Float64(y + -1.0)) - -0.918938533204673);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -3e+34) || ~((y <= 360000000.0)))
		tmp = (x * y) - (y * 0.5);
	else
		tmp = (x * (y + -1.0)) - -0.918938533204673;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3e+34], N[Not[LessEqual[y, 360000000.0]], $MachinePrecision]], N[(N[(x * y), $MachinePrecision] - N[(y * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision] - -0.918938533204673), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.00000000000000018e34 or 3.6e8 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   4. Taylor expanded in y around inf 100.0%

      \[\leadsto x \cdot \left(y + -1\right) - \color{blue}{0.5 \cdot y} \]
   5. Step-by-step derivation

      1. *-commutative 56.7%

         \[\leadsto \left(-x\right) - \color{blue}{y \cdot 0.5} \]

   6. Simplified 100.0%

      \[\leadsto x \cdot \left(y + -1\right) - \color{blue}{y \cdot 0.5} \]

   7. Taylor expanded in y around inf 100.0%

      \[\leadsto \color{blue}{x \cdot y} - y \cdot 0.5 \]

   if -3.00000000000000018e34 < y < 3.6e8

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   4. Taylor expanded in y around 0 97.8%

      \[\leadsto x \cdot \left(y + -1\right) - \color{blue}{-0.918938533204673} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+34} \lor \neg \left(y \leq 360000000\right):\\ \;\;\;\;x \cdot y - y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + -1\right) - -0.918938533204673\\ \end{array} \]

Alternative 6: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -102000000:\\ \;\;\;\;x \cdot y - x\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\left(0.918938533204673 - y \cdot 0.5\right) - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + -1\right) - -0.918938533204673\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -102000000.0)
   (- (* x y) x)
   (if (<= x 0.5)
     (- (- 0.918938533204673 (* y 0.5)) x)
     (- (* x (+ y -1.0)) -0.918938533204673))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -102000000.0) {
		tmp = (x * y) - x;
	} else if (x <= 0.5) {
		tmp = (0.918938533204673 - (y * 0.5)) - x;
	} else {
		tmp = (x * (y + -1.0)) - -0.918938533204673;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-102000000.0d0)) then
        tmp = (x * y) - x
    else if (x <= 0.5d0) then
        tmp = (0.918938533204673d0 - (y * 0.5d0)) - x
    else
        tmp = (x * (y + (-1.0d0))) - (-0.918938533204673d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -102000000.0) {
		tmp = (x * y) - x;
	} else if (x <= 0.5) {
		tmp = (0.918938533204673 - (y * 0.5)) - x;
	} else {
		tmp = (x * (y + -1.0)) - -0.918938533204673;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -102000000.0:
		tmp = (x * y) - x
	elif x <= 0.5:
		tmp = (0.918938533204673 - (y * 0.5)) - x
	else:
		tmp = (x * (y + -1.0)) - -0.918938533204673
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -102000000.0)
		tmp = Float64(Float64(x * y) - x);
	elseif (x <= 0.5)
		tmp = Float64(Float64(0.918938533204673 - Float64(y * 0.5)) - x);
	else
		tmp = Float64(Float64(x * Float64(y + -1.0)) - -0.918938533204673);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -102000000.0)
		tmp = (x * y) - x;
	elseif (x <= 0.5)
		tmp = (0.918938533204673 - (y * 0.5)) - x;
	else
		tmp = (x * (y + -1.0)) - -0.918938533204673;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -102000000.0], N[(N[(x * y), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[x, 0.5], N[(N[(0.918938533204673 - N[(y * 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], N[(N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision] - -0.918938533204673), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.02e8

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   4. Taylor expanded in y around 0 99.4%

      \[\leadsto x \cdot \left(y + -1\right) - \color{blue}{-0.918938533204673} \]

   5. Taylor expanded in x around inf 99.4%

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

      1. sub-neg 99.4%

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

      2. metadata-eval 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. *-commutative 99.4%

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

      5. mul-1-neg 99.4%

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

      6. unsub-neg 99.4%

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

   7. Simplified 99.4%

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

   if -1.02e8 < x < 0.5

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   4. Taylor expanded in y around 0 99.6%

      \[\leadsto \color{blue}{-1 \cdot x} - \left(y \cdot 0.5 - 0.918938533204673\right) \]
   5. Step-by-step derivation

      1. neg-mul-1 99.6%

         \[\leadsto \color{blue}{\left(-x\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   6. Simplified 99.6%

      \[\leadsto \color{blue}{\left(-x\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   if 0.5 < x

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   4. Taylor expanded in y around 0 100.0%

      \[\leadsto x \cdot \left(y + -1\right) - \color{blue}{-0.918938533204673} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -102000000:\\ \;\;\;\;x \cdot y - x\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\left(0.918938533204673 - y \cdot 0.5\right) - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + -1\right) - -0.918938533204673\\ \end{array} \]

Alternative 7: 74.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{-5} \lor \neg \left(y \leq 0.000106\right):\\ \;\;\;\;x \cdot y - x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -6.2e-5) (not (<= y 0.000106)))
   (- (* x y) x)
   (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -6.2e-5) || !(y <= 0.000106)) {
		tmp = (x * y) - x;
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-6.2d-5)) .or. (.not. (y <= 0.000106d0))) then
        tmp = (x * y) - x
    else
        tmp = 0.918938533204673d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -6.2e-5) || !(y <= 0.000106)) {
		tmp = (x * y) - x;
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -6.2e-5) or not (y <= 0.000106):
		tmp = (x * y) - x
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -6.2e-5) || !(y <= 0.000106))
		tmp = Float64(Float64(x * y) - x);
	else
		tmp = Float64(0.918938533204673 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -6.2e-5) || ~((y <= 0.000106)))
		tmp = (x * y) - x;
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -6.2e-5], N[Not[LessEqual[y, 0.000106]], $MachinePrecision]], N[(N[(x * y), $MachinePrecision] - x), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.20000000000000027e-5 or 1.06e-4 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   4. Taylor expanded in y around 0 48.7%

      \[\leadsto x \cdot \left(y + -1\right) - \color{blue}{-0.918938533204673} \]

   5. Taylor expanded in x around inf 48.6%

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

      1. sub-neg 48.6%

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

      2. metadata-eval 48.6%

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

      3. distribute-lft-in 48.6%

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

      4. *-commutative 48.6%

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

      5. mul-1-neg 48.6%

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

      6. unsub-neg 48.6%

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

   7. Simplified 48.6%

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

   if -6.20000000000000027e-5 < y < 1.06e-4

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   4. Taylor expanded in y around 0 98.4%

      \[\leadsto x \cdot \left(y + -1\right) - \color{blue}{-0.918938533204673} \]

   5. Taylor expanded in y around 0 98.3%

      \[\leadsto \color{blue}{0.918938533204673 + -1 \cdot x} \]
   6. Step-by-step derivation

      1. mul-1-neg 98.3%

         \[\leadsto 0.918938533204673 + \color{blue}{\left(-x\right)} \]

      2. unsub-neg 98.3%

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   7. Simplified 98.3%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{-5} \lor \neg \left(y \leq 0.000106\right):\\ \;\;\;\;x \cdot y - x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]

Alternative 8: 49.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.4:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.3:\\ \;\;\;\;0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.4) (* x y) (if (<= y 1.3) 0.918938533204673 (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.4) {
		tmp = x * y;
	} else if (y <= 1.3) {
		tmp = 0.918938533204673;
	} else {
		tmp = 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 (y <= (-4.4d0)) then
        tmp = x * y
    else if (y <= 1.3d0) then
        tmp = 0.918938533204673d0
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.4) {
		tmp = x * y;
	} else if (y <= 1.3) {
		tmp = 0.918938533204673;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.4:
		tmp = x * y
	elif y <= 1.3:
		tmp = 0.918938533204673
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.4)
		tmp = Float64(x * y);
	elseif (y <= 1.3)
		tmp = 0.918938533204673;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.4)
		tmp = x * y;
	elseif (y <= 1.3)
		tmp = 0.918938533204673;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.4], N[(x * y), $MachinePrecision], If[LessEqual[y, 1.3], 0.918938533204673, N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.4000000000000004 or 1.30000000000000004 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   4. Taylor expanded in y around 0 48.7%

      \[\leadsto x \cdot \left(y + -1\right) - \color{blue}{-0.918938533204673} \]

   5. Taylor expanded in y around inf 47.3%

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

   if -4.4000000000000004 < y < 1.30000000000000004

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   4. Taylor expanded in y around 0 98.4%

      \[\leadsto x \cdot \left(y + -1\right) - \color{blue}{-0.918938533204673} \]

   5. Taylor expanded in x around 0 51.9%

      \[\leadsto \color{blue}{0.918938533204673} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.4:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.3:\\ \;\;\;\;0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 9: 73.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -185:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.7:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -185.0) (* x y) (if (<= y 1.7) (- 0.918938533204673 x) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -185.0) {
		tmp = x * y;
	} else if (y <= 1.7) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = 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 (y <= (-185.0d0)) then
        tmp = x * y
    else if (y <= 1.7d0) then
        tmp = 0.918938533204673d0 - x
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -185.0) {
		tmp = x * y;
	} else if (y <= 1.7) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -185.0:
		tmp = x * y
	elif y <= 1.7:
		tmp = 0.918938533204673 - x
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -185.0)
		tmp = Float64(x * y);
	elseif (y <= 1.7)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -185.0)
		tmp = x * y;
	elseif (y <= 1.7)
		tmp = 0.918938533204673 - x;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -185.0], N[(x * y), $MachinePrecision], If[LessEqual[y, 1.7], N[(0.918938533204673 - x), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -185 or 1.69999999999999996 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   4. Taylor expanded in y around 0 48.7%

      \[\leadsto x \cdot \left(y + -1\right) - \color{blue}{-0.918938533204673} \]

   5. Taylor expanded in y around inf 47.3%

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

   if -185 < y < 1.69999999999999996

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   4. Taylor expanded in y around 0 98.4%

      \[\leadsto x \cdot \left(y + -1\right) - \color{blue}{-0.918938533204673} \]

   5. Taylor expanded in y around 0 98.3%

      \[\leadsto \color{blue}{0.918938533204673 + -1 \cdot x} \]
   6. Step-by-step derivation

      1. mul-1-neg 98.3%

         \[\leadsto 0.918938533204673 + \color{blue}{\left(-x\right)} \]

      2. unsub-neg 98.3%

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   7. Simplified 98.3%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -185:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.7:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 10: 74.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot y - -0.918938533204673\\ \mathbf{elif}\;y \leq 1.5:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   (- (* x y) -0.918938533204673)
   (if (<= y 1.5) (- 0.918938533204673 x) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = (x * y) - -0.918938533204673;
	} else if (y <= 1.5) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = 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 (y <= (-1.0d0)) then
        tmp = (x * y) - (-0.918938533204673d0)
    else if (y <= 1.5d0) then
        tmp = 0.918938533204673d0 - x
    else
        tmp = x * 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) - -0.918938533204673;
	} else if (y <= 1.5) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = (x * y) - -0.918938533204673
	elif y <= 1.5:
		tmp = 0.918938533204673 - x
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(Float64(x * y) - -0.918938533204673);
	elseif (y <= 1.5)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = (x * y) - -0.918938533204673;
	elseif (y <= 1.5)
		tmp = 0.918938533204673 - x;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], N[(N[(x * y), $MachinePrecision] - -0.918938533204673), $MachinePrecision], If[LessEqual[y, 1.5], N[(0.918938533204673 - x), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   4. Taylor expanded in y around 0 49.5%

      \[\leadsto x \cdot \left(y + -1\right) - \color{blue}{-0.918938533204673} \]

   5. Taylor expanded in y around inf 47.4%

      \[\leadsto \color{blue}{x \cdot y} - -0.918938533204673 \]

   if -1 < y < 1.5

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   4. Taylor expanded in y around 0 98.4%

      \[\leadsto x \cdot \left(y + -1\right) - \color{blue}{-0.918938533204673} \]

   5. Taylor expanded in y around 0 98.3%

      \[\leadsto \color{blue}{0.918938533204673 + -1 \cdot x} \]
   6. Step-by-step derivation

      1. mul-1-neg 98.3%

         \[\leadsto 0.918938533204673 + \color{blue}{\left(-x\right)} \]

      2. unsub-neg 98.3%

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   7. Simplified 98.3%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]

   if 1.5 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   4. Taylor expanded in y around 0 47.9%

      \[\leadsto x \cdot \left(y + -1\right) - \color{blue}{-0.918938533204673} \]

   5. Taylor expanded in y around inf 47.7%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot y - -0.918938533204673\\ \mathbf{elif}\;y \leq 1.5:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 11: 25.8% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ 0.918938533204673 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.918938533204673)

C

double code(double x, double y) {
	return 0.918938533204673;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.918938533204673d0
end function

Java

public static double code(double x, double y) {
	return 0.918938533204673;
}

Python

def code(x, y):
	return 0.918938533204673

Julia

function code(x, y)
	return 0.918938533204673
end

MATLAB

function tmp = code(x, y)
	tmp = 0.918938533204673;
end

Wolfram

code[x_, y_] := 0.918938533204673

Derivation

1. Initial program 100.0%

   \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
2. Step-by-step derivation

   1. associate-+l- 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   2. sub-neg 100.0%

      \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. metadata-eval 100.0%

      \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

4. Taylor expanded in y around 0 74.1%

   \[\leadsto x \cdot \left(y + -1\right) - \color{blue}{-0.918938533204673} \]

5. Taylor expanded in x around 0 27.8%

   \[\leadsto \color{blue}{0.918938533204673} \]

6. Final simplification 27.8%

   \[\leadsto 0.918938533204673 \]

Reproduce

herbie shell --seed 2023290 
(FPCore (x y)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (+ (- (* x (- y 1.0)) (* y 0.5)) 0.918938533204673))