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

Percentage Accurate: 100.0% → 100.0%

Time: 5.6s

Alternatives: 10

Speedup: 1.2×

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, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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. +-commutative 100.0%

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

   2. cancel-sign-sub-inv 100.0%

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

   3. +-commutative 100.0%

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

   4. associate-+r+ 100.0%

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

   5. cancel-sign-sub-inv 100.0%

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

   6. associate-+l- 100.0%

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

   7. sub-neg 100.0%

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

   8. distribute-rgt-in 100.0%

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

   9. metadata-eval 100.0%

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

   10. neg-mul-1 100.0%

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

   11. associate--r+ 100.0%

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

   12. distribute-lft-out-- 100.0%

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

   13. unsub-neg 100.0%

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

   14. fma-neg 100.0%

       \[\leadsto 0.918938533204673 - \color{blue}{\mathsf{fma}\left(y, 0.5 + \left(-x\right), -\left(-x\right)\right)} \]

   15. unsub-neg 100.0%

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

   16. remove-double-neg 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{0.918938533204673 - \mathsf{fma}\left(y, 0.5 - x, x\right)} \]
4. Step-by-step derivation

   1. fma-udef 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 98.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -36000:\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;0.918938533204673 - \left(x + y \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 + \left(y \cdot x - y \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -36000.0)
   (* y (- x 0.5))
   (if (<= y 1.0)
     (- 0.918938533204673 (+ x (* y 0.5)))
     (+ 0.918938533204673 (- (* y x) (* y 0.5))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -36000.0) {
		tmp = y * (x - 0.5);
	} else if (y <= 1.0) {
		tmp = 0.918938533204673 - (x + (y * 0.5));
	} else {
		tmp = 0.918938533204673 + ((y * x) - (y * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-36000.0d0)) then
        tmp = y * (x - 0.5d0)
    else if (y <= 1.0d0) then
        tmp = 0.918938533204673d0 - (x + (y * 0.5d0))
    else
        tmp = 0.918938533204673d0 + ((y * x) - (y * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -36000.0) {
		tmp = y * (x - 0.5);
	} else if (y <= 1.0) {
		tmp = 0.918938533204673 - (x + (y * 0.5));
	} else {
		tmp = 0.918938533204673 + ((y * x) - (y * 0.5));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -36000.0:
		tmp = y * (x - 0.5)
	elif y <= 1.0:
		tmp = 0.918938533204673 - (x + (y * 0.5))
	else:
		tmp = 0.918938533204673 + ((y * x) - (y * 0.5))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -36000.0)
		tmp = Float64(y * Float64(x - 0.5));
	elseif (y <= 1.0)
		tmp = Float64(0.918938533204673 - Float64(x + Float64(y * 0.5)));
	else
		tmp = Float64(0.918938533204673 + Float64(Float64(y * x) - Float64(y * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -36000.0)
		tmp = y * (x - 0.5);
	elseif (y <= 1.0)
		tmp = 0.918938533204673 - (x + (y * 0.5));
	else
		tmp = 0.918938533204673 + ((y * x) - (y * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -36000.0], N[(y * N[(x - 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(0.918938533204673 - N[(x + N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.918938533204673 + N[(N[(y * x), $MachinePrecision] - N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -36000

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 99.0%

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

   if -36000 < 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. +-commutative 100.0%

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

      2. cancel-sign-sub-inv 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-+r+ 100.0%

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

      5. cancel-sign-sub-inv 100.0%

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

      6. associate-+l- 100.0%

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

      7. sub-neg 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. metadata-eval 100.0%

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

      10. neg-mul-1 100.0%

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

      11. associate--r+ 100.0%

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

      12. distribute-lft-out-- 100.0%

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

      13. unsub-neg 100.0%

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

      14. fma-neg 100.0%

          \[\leadsto 0.918938533204673 - \color{blue}{\mathsf{fma}\left(y, 0.5 + \left(-x\right), -\left(-x\right)\right)} \]

      15. unsub-neg 100.0%

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

      16. remove-double-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.918938533204673 - \mathsf{fma}\left(y, 0.5 - x, x\right)} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 99.6%

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

   if 1 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 98.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -36000:\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;0.918938533204673 - \left(x + y \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 + \left(y \cdot x - y \cdot 0.5\right)\\ \end{array} \]

Alternative 3: 49.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-7}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq 6.3 \cdot 10^{+175} \lor \neg \left(y \leq 2.1 \cdot 10^{+293}\right):\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   (* y x)
   (if (<= y 4e-7)
     (- x)
     (if (or (<= y 6.3e+175) (not (<= y 2.1e+293))) (* y -0.5) (* y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = y * x;
	} else if (y <= 4e-7) {
		tmp = -x;
	} else if ((y <= 6.3e+175) || !(y <= 2.1e+293)) {
		tmp = y * -0.5;
	} else {
		tmp = y * 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 <= (-1.0d0)) then
        tmp = y * x
    else if (y <= 4d-7) then
        tmp = -x
    else if ((y <= 6.3d+175) .or. (.not. (y <= 2.1d+293))) then
        tmp = y * (-0.5d0)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = y * x;
	} else if (y <= 4e-7) {
		tmp = -x;
	} else if ((y <= 6.3e+175) || !(y <= 2.1e+293)) {
		tmp = y * -0.5;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = y * x
	elif y <= 4e-7:
		tmp = -x
	elif (y <= 6.3e+175) or not (y <= 2.1e+293):
		tmp = y * -0.5
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(y * x);
	elseif (y <= 4e-7)
		tmp = Float64(-x);
	elseif ((y <= 6.3e+175) || !(y <= 2.1e+293))
		tmp = Float64(y * -0.5);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = y * x;
	elseif (y <= 4e-7)
		tmp = -x;
	elseif ((y <= 6.3e+175) || ~((y <= 2.1e+293)))
		tmp = y * -0.5;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], N[(y * x), $MachinePrecision], If[LessEqual[y, 4e-7], (-x), If[Or[LessEqual[y, 6.3e+175], N[Not[LessEqual[y, 2.1e+293]], $MachinePrecision]], N[(y * -0.5), $MachinePrecision], N[(y * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 6.2999999999999997e175 < y < 2.1e293

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 98.5%

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

   3. Taylor expanded in x around inf 62.7%

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

      1. *-commutative 62.7%

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

   5. Simplified 62.7%

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

   if -1 < y < 3.9999999999999998e-7

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 50.0%

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

   3. Taylor expanded in y around 0 49.8%

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

      1. neg-mul-1 49.8%

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

   5. Simplified 49.8%

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

   if 3.9999999999999998e-7 < y < 6.2999999999999997e175 or 2.1e293 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 93.3%

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

   3. Taylor expanded in x around 0 57.6%

      \[\leadsto \color{blue}{-0.5 \cdot y} \]
   4. Step-by-step derivation

      1. *-commutative 57.6%

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

   5. Simplified 57.6%

      \[\leadsto \color{blue}{y \cdot -0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-7}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq 6.3 \cdot 10^{+175} \lor \neg \left(y \leq 2.1 \cdot 10^{+293}\right):\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 4: 74.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -15200:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+176} \lor \neg \left(y \leq 1.3 \cdot 10^{+295}\right):\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -15200.0)
   (* y x)
   (if (<= y 1.85)
     (- 0.918938533204673 x)
     (if (or (<= y 9e+176) (not (<= y 1.3e+295))) (* y -0.5) (* y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -15200.0) {
		tmp = y * x;
	} else if (y <= 1.85) {
		tmp = 0.918938533204673 - x;
	} else if ((y <= 9e+176) || !(y <= 1.3e+295)) {
		tmp = y * -0.5;
	} else {
		tmp = y * 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 <= (-15200.0d0)) then
        tmp = y * x
    else if (y <= 1.85d0) then
        tmp = 0.918938533204673d0 - x
    else if ((y <= 9d+176) .or. (.not. (y <= 1.3d+295))) then
        tmp = y * (-0.5d0)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -15200.0) {
		tmp = y * x;
	} else if (y <= 1.85) {
		tmp = 0.918938533204673 - x;
	} else if ((y <= 9e+176) || !(y <= 1.3e+295)) {
		tmp = y * -0.5;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -15200.0:
		tmp = y * x
	elif y <= 1.85:
		tmp = 0.918938533204673 - x
	elif (y <= 9e+176) or not (y <= 1.3e+295):
		tmp = y * -0.5
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -15200.0)
		tmp = Float64(y * x);
	elseif (y <= 1.85)
		tmp = Float64(0.918938533204673 - x);
	elseif ((y <= 9e+176) || !(y <= 1.3e+295))
		tmp = Float64(y * -0.5);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -15200.0)
		tmp = y * x;
	elseif (y <= 1.85)
		tmp = 0.918938533204673 - x;
	elseif ((y <= 9e+176) || ~((y <= 1.3e+295)))
		tmp = y * -0.5;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -15200.0], N[(y * x), $MachinePrecision], If[LessEqual[y, 1.85], N[(0.918938533204673 - x), $MachinePrecision], If[Or[LessEqual[y, 9e+176], N[Not[LessEqual[y, 1.3e+295]], $MachinePrecision]], N[(y * -0.5), $MachinePrecision], N[(y * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -15200 or 9.00000000000000007e176 < y < 1.29999999999999993e295

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 99.3%

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

   3. Taylor expanded in x around inf 63.4%

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

      1. *-commutative 63.4%

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

   5. Simplified 63.4%

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

   if -15200 < y < 1.8500000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 98.6%

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

      1. neg-mul-1 98.6%

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

      2. sub-neg 98.6%

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

   4. Simplified 98.6%

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

   if 1.8500000000000001 < y < 9.00000000000000007e176 or 1.29999999999999993e295 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 95.2%

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

   3. Taylor expanded in x around 0 58.7%

      \[\leadsto \color{blue}{-0.5 \cdot y} \]
   4. Step-by-step derivation

      1. *-commutative 58.7%

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

   5. Simplified 58.7%

      \[\leadsto \color{blue}{y \cdot -0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -15200:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+176} \lor \neg \left(y \leq 1.3 \cdot 10^{+295}\right):\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 5: 74.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+176} \lor \neg \left(y \leq 5.5 \cdot 10^{+294}\right):\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6.4e-12)
   (* x (+ y -1.0))
   (if (<= y 1.85)
     (- 0.918938533204673 x)
     (if (or (<= y 4.2e+176) (not (<= y 5.5e+294))) (* y -0.5) (* y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6.4e-12) {
		tmp = x * (y + -1.0);
	} else if (y <= 1.85) {
		tmp = 0.918938533204673 - x;
	} else if ((y <= 4.2e+176) || !(y <= 5.5e+294)) {
		tmp = y * -0.5;
	} else {
		tmp = y * 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 <= (-6.4d-12)) then
        tmp = x * (y + (-1.0d0))
    else if (y <= 1.85d0) then
        tmp = 0.918938533204673d0 - x
    else if ((y <= 4.2d+176) .or. (.not. (y <= 5.5d+294))) then
        tmp = y * (-0.5d0)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -6.4e-12) {
		tmp = x * (y + -1.0);
	} else if (y <= 1.85) {
		tmp = 0.918938533204673 - x;
	} else if ((y <= 4.2e+176) || !(y <= 5.5e+294)) {
		tmp = y * -0.5;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -6.4e-12:
		tmp = x * (y + -1.0)
	elif y <= 1.85:
		tmp = 0.918938533204673 - x
	elif (y <= 4.2e+176) or not (y <= 5.5e+294):
		tmp = y * -0.5
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6.4e-12)
		tmp = Float64(x * Float64(y + -1.0));
	elseif (y <= 1.85)
		tmp = Float64(0.918938533204673 - x);
	elseif ((y <= 4.2e+176) || !(y <= 5.5e+294))
		tmp = Float64(y * -0.5);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -6.4e-12)
		tmp = x * (y + -1.0);
	elseif (y <= 1.85)
		tmp = 0.918938533204673 - x;
	elseif ((y <= 4.2e+176) || ~((y <= 5.5e+294)))
		tmp = y * -0.5;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6.4e-12], N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.85], N[(0.918938533204673 - x), $MachinePrecision], If[Or[LessEqual[y, 4.2e+176], N[Not[LessEqual[y, 5.5e+294]], $MachinePrecision]], N[(y * -0.5), $MachinePrecision], N[(y * x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -6.4000000000000002e-12

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 62.8%

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

   if -6.4000000000000002e-12 < y < 1.8500000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.4%

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

      1. neg-mul-1 99.4%

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

      2. sub-neg 99.4%

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

   4. Simplified 99.4%

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

   if 1.8500000000000001 < y < 4.1999999999999998e176 or 5.4999999999999994e294 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 95.2%

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

   3. Taylor expanded in x around 0 58.7%

      \[\leadsto \color{blue}{-0.5 \cdot y} \]
   4. Step-by-step derivation

      1. *-commutative 58.7%

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

   5. Simplified 58.7%

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

   if 4.1999999999999998e176 < y < 5.4999999999999994e294

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 100.0%

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

   3. Taylor expanded in x around inf 66.2%

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

      1. *-commutative 66.2%

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

   5. Simplified 66.2%

      \[\leadsto \color{blue}{y \cdot x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+176} \lor \neg \left(y \leq 5.5 \cdot 10^{+294}\right):\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 6: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.00032 \lor \neg \left(y \leq 2.2 \cdot 10^{-15}\right):\\ \;\;\;\;0.918938533204673 + y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -0.00032) (not (<= y 2.2e-15)))
   (+ 0.918938533204673 (* y (- x 0.5)))
   (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -0.00032) || !(y <= 2.2e-15)) {
		tmp = 0.918938533204673 + (y * (x - 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 <= (-0.00032d0)) .or. (.not. (y <= 2.2d-15))) then
        tmp = 0.918938533204673d0 + (y * (x - 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 <= -0.00032) || !(y <= 2.2e-15)) {
		tmp = 0.918938533204673 + (y * (x - 0.5));
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -0.00032) or not (y <= 2.2e-15):
		tmp = 0.918938533204673 + (y * (x - 0.5))
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -0.00032) || !(y <= 2.2e-15))
		tmp = Float64(0.918938533204673 + Float64(y * Float64(x - 0.5)));
	else
		tmp = Float64(0.918938533204673 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -0.00032) || ~((y <= 2.2e-15)))
		tmp = 0.918938533204673 + (y * (x - 0.5));
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -0.00032], N[Not[LessEqual[y, 2.2e-15]], $MachinePrecision]], N[(0.918938533204673 + N[(y * N[(x - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.20000000000000026e-4 or 2.19999999999999986e-15 < 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. +-commutative 100.0%

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

      2. cancel-sign-sub-inv 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-+r+ 100.0%

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

      5. cancel-sign-sub-inv 100.0%

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

      6. associate-+l- 100.0%

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

      7. sub-neg 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. metadata-eval 100.0%

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

      10. neg-mul-1 100.0%

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

      11. associate--r+ 100.0%

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

      12. distribute-lft-out-- 100.0%

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

      13. unsub-neg 100.0%

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

      14. fma-neg 100.0%

          \[\leadsto 0.918938533204673 - \color{blue}{\mathsf{fma}\left(y, 0.5 + \left(-x\right), -\left(-x\right)\right)} \]

      15. unsub-neg 100.0%

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

      16. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 98.4%

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

   if -3.20000000000000026e-4 < y < 2.19999999999999986e-15

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.7%

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

      1. neg-mul-1 99.7%

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

      2. sub-neg 99.7%

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

   4. Simplified 99.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00032 \lor \neg \left(y \leq 2.2 \cdot 10^{-15}\right):\\ \;\;\;\;0.918938533204673 + y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]

Alternative 7: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x - 0.5\right)\\ \mathbf{if}\;y \leq -510000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;0.918938533204673 - \left(x + y \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 + t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (- x 0.5))))
   (if (<= y -510000.0)
     t_0
     (if (<= y 1.0)
       (- 0.918938533204673 (+ x (* y 0.5)))
       (+ 0.918938533204673 t_0)))))

C

double code(double x, double y) {
	double t_0 = y * (x - 0.5);
	double tmp;
	if (y <= -510000.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = 0.918938533204673 - (x + (y * 0.5));
	} else {
		tmp = 0.918938533204673 + 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 * (x - 0.5d0)
    if (y <= (-510000.0d0)) then
        tmp = t_0
    else if (y <= 1.0d0) then
        tmp = 0.918938533204673d0 - (x + (y * 0.5d0))
    else
        tmp = 0.918938533204673d0 + t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (x - 0.5);
	double tmp;
	if (y <= -510000.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = 0.918938533204673 - (x + (y * 0.5));
	} else {
		tmp = 0.918938533204673 + t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (x - 0.5)
	tmp = 0
	if y <= -510000.0:
		tmp = t_0
	elif y <= 1.0:
		tmp = 0.918938533204673 - (x + (y * 0.5))
	else:
		tmp = 0.918938533204673 + t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x - 0.5))
	tmp = 0.0
	if (y <= -510000.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = Float64(0.918938533204673 - Float64(x + Float64(y * 0.5)));
	else
		tmp = Float64(0.918938533204673 + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (x - 0.5);
	tmp = 0.0;
	if (y <= -510000.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = 0.918938533204673 - (x + (y * 0.5));
	else
		tmp = 0.918938533204673 + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x - 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -510000.0], t$95$0, If[LessEqual[y, 1.0], N[(0.918938533204673 - N[(x + N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.918938533204673 + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.1e5

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 99.0%

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

   if -5.1e5 < 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. +-commutative 100.0%

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

      2. cancel-sign-sub-inv 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-+r+ 100.0%

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

      5. cancel-sign-sub-inv 100.0%

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

      6. associate-+l- 100.0%

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

      7. sub-neg 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. metadata-eval 100.0%

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

      10. neg-mul-1 100.0%

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

      11. associate--r+ 100.0%

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

      12. distribute-lft-out-- 100.0%

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

      13. unsub-neg 100.0%

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

      14. fma-neg 100.0%

          \[\leadsto 0.918938533204673 - \color{blue}{\mathsf{fma}\left(y, 0.5 + \left(-x\right), -\left(-x\right)\right)} \]

      15. unsub-neg 100.0%

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

      16. remove-double-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.918938533204673 - \mathsf{fma}\left(y, 0.5 - x, x\right)} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 99.6%

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

   if 1 < 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. +-commutative 100.0%

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

      2. cancel-sign-sub-inv 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-+r+ 100.0%

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

      5. cancel-sign-sub-inv 100.0%

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

      6. associate-+l- 100.0%

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

      7. sub-neg 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. metadata-eval 100.0%

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

      10. neg-mul-1 100.0%

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

      11. associate--r+ 100.0%

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

      12. distribute-lft-out-- 100.0%

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

      13. unsub-neg 100.0%

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

      14. fma-neg 100.0%

          \[\leadsto 0.918938533204673 - \color{blue}{\mathsf{fma}\left(y, 0.5 + \left(-x\right), -\left(-x\right)\right)} \]

      15. unsub-neg 100.0%

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

      16. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 98.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -510000:\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;0.918938533204673 - \left(x + y \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 + y \cdot \left(x - 0.5\right)\\ \end{array} \]

Alternative 8: 97.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.25 \lor \neg \left(y \leq 1.85\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.25) (not (<= y 1.85)))
   (* y (- x 0.5))
   (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.25) || !(y <= 1.85)) {
		tmp = y * (x - 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 <= (-1.25d0)) .or. (.not. (y <= 1.85d0))) then
        tmp = y * (x - 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 <= -1.25) || !(y <= 1.85)) {
		tmp = y * (x - 0.5);
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.25) or not (y <= 1.85):
		tmp = y * (x - 0.5)
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.25) || !(y <= 1.85))
		tmp = Float64(y * Float64(x - 0.5));
	else
		tmp = Float64(0.918938533204673 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.25) || ~((y <= 1.85)))
		tmp = y * (x - 0.5);
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.25], N[Not[LessEqual[y, 1.85]], $MachinePrecision]], N[(y * N[(x - 0.5), $MachinePrecision]), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.25 or 1.8500000000000001 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 97.2%

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

   if -1.25 < y < 1.8500000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.2%

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

      1. neg-mul-1 99.2%

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

      2. sub-neg 99.2%

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

   4. Simplified 99.2%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \lor \neg \left(y \leq 1.85\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]

Alternative 9: 48.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-9} \lor \neg \left(x \leq 0.9\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.1e-9) (not (<= x 0.9))) (- x) (* y -0.5)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.1e-9) || !(x <= 0.9)) {
		tmp = -x;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-3.1d-9)) .or. (.not. (x <= 0.9d0))) then
        tmp = -x
    else
        tmp = y * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -3.1e-9) || !(x <= 0.9)) {
		tmp = -x;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -3.1e-9) or not (x <= 0.9):
		tmp = -x
	else:
		tmp = y * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3.1e-9) || !(x <= 0.9))
		tmp = Float64(-x);
	else
		tmp = Float64(y * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -3.1e-9) || ~((x <= 0.9)))
		tmp = -x;
	else
		tmp = y * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3.1e-9], N[Not[LessEqual[x, 0.9]], $MachinePrecision]], (-x), N[(y * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.10000000000000005e-9 or 0.900000000000000022 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 95.9%

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

   3. Taylor expanded in y around 0 47.4%

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

      1. neg-mul-1 47.4%

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

   5. Simplified 47.4%

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

   if -3.10000000000000005e-9 < x < 0.900000000000000022

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 49.1%

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

   3. Taylor expanded in x around 0 48.1%

      \[\leadsto \color{blue}{-0.5 \cdot y} \]
   4. Step-by-step derivation

      1. *-commutative 48.1%

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

   5. Simplified 48.1%

      \[\leadsto \color{blue}{y \cdot -0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-9} \lor \neg \left(x \leq 0.9\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]

Alternative 10: 26.2% accurate, 5.5× 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 Float64(-x)
end

MATLAB

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

Wolfram

code[x_, y_] := (-x)

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around inf 51.9%

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

3. Taylor expanded in y around 0 26.3%

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

   1. neg-mul-1 26.3%

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

5. Simplified 26.3%

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

6. Final simplification 26.3%

   \[\leadsto -x \]

Reproduce

herbie shell --seed 2023336 
(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))