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

Percentage Accurate: 100.0% → 100.0%

Time: 4.9s

Alternatives: 9

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, 1.0× 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]

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%

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

Alternative 2: 49.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+224}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -11.5:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+91} \lor \neg \left(y \leq 3.8 \cdot 10^{+228}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.7e+224)
   (* y -0.5)
   (if (<= y -11.5)
     (* x y)
     (if (<= y 1.85)
       0.918938533204673
       (if (or (<= y 2e+91) (not (<= y 3.8e+228))) (* x y) (* y -0.5))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.7e+224) {
		tmp = y * -0.5;
	} else if (y <= -11.5) {
		tmp = x * y;
	} else if (y <= 1.85) {
		tmp = 0.918938533204673;
	} else if ((y <= 2e+91) || !(y <= 3.8e+228)) {
		tmp = x * y;
	} 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 (y <= (-3.7d+224)) then
        tmp = y * (-0.5d0)
    else if (y <= (-11.5d0)) then
        tmp = x * y
    else if (y <= 1.85d0) then
        tmp = 0.918938533204673d0
    else if ((y <= 2d+91) .or. (.not. (y <= 3.8d+228))) then
        tmp = x * y
    else
        tmp = y * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.7e+224) {
		tmp = y * -0.5;
	} else if (y <= -11.5) {
		tmp = x * y;
	} else if (y <= 1.85) {
		tmp = 0.918938533204673;
	} else if ((y <= 2e+91) || !(y <= 3.8e+228)) {
		tmp = x * y;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.7e+224:
		tmp = y * -0.5
	elif y <= -11.5:
		tmp = x * y
	elif y <= 1.85:
		tmp = 0.918938533204673
	elif (y <= 2e+91) or not (y <= 3.8e+228):
		tmp = x * y
	else:
		tmp = y * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.7e+224)
		tmp = Float64(y * -0.5);
	elseif (y <= -11.5)
		tmp = Float64(x * y);
	elseif (y <= 1.85)
		tmp = 0.918938533204673;
	elseif ((y <= 2e+91) || !(y <= 3.8e+228))
		tmp = Float64(x * y);
	else
		tmp = Float64(y * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.7e+224)
		tmp = y * -0.5;
	elseif (y <= -11.5)
		tmp = x * y;
	elseif (y <= 1.85)
		tmp = 0.918938533204673;
	elseif ((y <= 2e+91) || ~((y <= 3.8e+228)))
		tmp = x * y;
	else
		tmp = y * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.7e+224], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, -11.5], N[(x * y), $MachinePrecision], If[LessEqual[y, 1.85], 0.918938533204673, If[Or[LessEqual[y, 2e+91], N[Not[LessEqual[y, 3.8e+228]], $MachinePrecision]], N[(x * y), $MachinePrecision], N[(y * -0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.70000000000000003e224 or 2.00000000000000016e91 < y < 3.8000000000000002e228

   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%

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

   5. Taylor expanded in x around 0 68.5%

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

      1. *-commutative 68.5%

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

   7. Simplified 68.5%

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

   if -3.70000000000000003e224 < y < -11.5 or 1.8500000000000001 < y < 2.00000000000000016e91 or 3.8000000000000002e228 < 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 97.7%

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

   5. Taylor expanded in x around inf 61.7%

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

      1. *-commutative 61.7%

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

   7. Simplified 61.7%

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

   if -11.5 < y < 1.8500000000000001

   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 x around 0 62.6%

      \[\leadsto 0.918938533204673 - \color{blue}{0.5 \cdot y} \]
   5. Step-by-step derivation

      1. *-commutative 62.6%

         \[\leadsto 0.918938533204673 - \color{blue}{y \cdot 0.5} \]

   6. Simplified 62.6%

      \[\leadsto 0.918938533204673 - \color{blue}{y \cdot 0.5} \]

   7. Taylor expanded in y around 0 60.1%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+224}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -11.5:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+91} \lor \neg \left(y \leq 3.8 \cdot 10^{+228}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]

Alternative 3: 74.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+226}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -30.5:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.55:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+85} \lor \neg \left(y \leq 1.22 \cdot 10^{+231}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.5e+226)
   (* y -0.5)
   (if (<= y -30.5)
     (* x y)
     (if (<= y 1.55)
       (- 0.918938533204673 x)
       (if (or (<= y 2.2e+85) (not (<= y 1.22e+231))) (* x y) (* y -0.5))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.5e+226) {
		tmp = y * -0.5;
	} else if (y <= -30.5) {
		tmp = x * y;
	} else if (y <= 1.55) {
		tmp = 0.918938533204673 - x;
	} else if ((y <= 2.2e+85) || !(y <= 1.22e+231)) {
		tmp = x * y;
	} 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 (y <= (-2.5d+226)) then
        tmp = y * (-0.5d0)
    else if (y <= (-30.5d0)) then
        tmp = x * y
    else if (y <= 1.55d0) then
        tmp = 0.918938533204673d0 - x
    else if ((y <= 2.2d+85) .or. (.not. (y <= 1.22d+231))) then
        tmp = x * y
    else
        tmp = y * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.5e+226) {
		tmp = y * -0.5;
	} else if (y <= -30.5) {
		tmp = x * y;
	} else if (y <= 1.55) {
		tmp = 0.918938533204673 - x;
	} else if ((y <= 2.2e+85) || !(y <= 1.22e+231)) {
		tmp = x * y;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.5e+226:
		tmp = y * -0.5
	elif y <= -30.5:
		tmp = x * y
	elif y <= 1.55:
		tmp = 0.918938533204673 - x
	elif (y <= 2.2e+85) or not (y <= 1.22e+231):
		tmp = x * y
	else:
		tmp = y * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.5e+226)
		tmp = Float64(y * -0.5);
	elseif (y <= -30.5)
		tmp = Float64(x * y);
	elseif (y <= 1.55)
		tmp = Float64(0.918938533204673 - x);
	elseif ((y <= 2.2e+85) || !(y <= 1.22e+231))
		tmp = Float64(x * y);
	else
		tmp = Float64(y * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.5e+226)
		tmp = y * -0.5;
	elseif (y <= -30.5)
		tmp = x * y;
	elseif (y <= 1.55)
		tmp = 0.918938533204673 - x;
	elseif ((y <= 2.2e+85) || ~((y <= 1.22e+231)))
		tmp = x * y;
	else
		tmp = y * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.5e+226], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, -30.5], N[(x * y), $MachinePrecision], If[LessEqual[y, 1.55], N[(0.918938533204673 - x), $MachinePrecision], If[Or[LessEqual[y, 2.2e+85], N[Not[LessEqual[y, 1.22e+231]], $MachinePrecision]], N[(x * y), $MachinePrecision], N[(y * -0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.5000000000000002e226 or 2.2000000000000002e85 < y < 1.21999999999999998e231

   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%

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

   5. Taylor expanded in x around 0 68.5%

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

      1. *-commutative 68.5%

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

   7. Simplified 68.5%

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

   if -2.5000000000000002e226 < y < -30.5 or 1.55000000000000004 < y < 2.2000000000000002e85 or 1.21999999999999998e231 < 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 97.7%

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

   5. Taylor expanded in x around inf 61.7%

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

      1. *-commutative 61.7%

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

   7. Simplified 61.7%

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

   if -30.5 < y < 1.55000000000000004

   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 0 97.0%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+226}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -30.5:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.55:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+85} \lor \neg \left(y \leq 1.22 \cdot 10^{+231}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]

Alternative 4: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-10} \lor \neg \left(x \leq 6.2 \cdot 10^{-12}\right):\\ \;\;\;\;x \cdot \left(y + -1\right) - -0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.2e-10) (not (<= x 6.2e-12)))
   (- (* x (+ y -1.0)) -0.918938533204673)
   (- 0.918938533204673 (* y 0.5))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.2e-10) || !(x <= 6.2e-12)) {
		tmp = (x * (y + -1.0)) - -0.918938533204673;
	} else {
		tmp = 0.918938533204673 - (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 <= (-3.2d-10)) .or. (.not. (x <= 6.2d-12))) then
        tmp = (x * (y + (-1.0d0))) - (-0.918938533204673d0)
    else
        tmp = 0.918938533204673d0 - (y * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -3.2e-10) || !(x <= 6.2e-12)) {
		tmp = (x * (y + -1.0)) - -0.918938533204673;
	} else {
		tmp = 0.918938533204673 - (y * 0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -3.2e-10) or not (x <= 6.2e-12):
		tmp = (x * (y + -1.0)) - -0.918938533204673
	else:
		tmp = 0.918938533204673 - (y * 0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3.2e-10) || !(x <= 6.2e-12))
		tmp = Float64(Float64(x * Float64(y + -1.0)) - -0.918938533204673);
	else
		tmp = Float64(0.918938533204673 - Float64(y * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -3.2e-10) || ~((x <= 6.2e-12)))
		tmp = (x * (y + -1.0)) - -0.918938533204673;
	else
		tmp = 0.918938533204673 - (y * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3.2e-10], N[Not[LessEqual[x, 6.2e-12]], $MachinePrecision]], N[(N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision] - -0.918938533204673), $MachinePrecision], N[(0.918938533204673 - N[(y * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.19999999999999981e-10 or 6.2000000000000002e-12 < 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 98.0%

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

   if -3.19999999999999981e-10 < x < 6.2000000000000002e-12

   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 x around 0 99.9%

      \[\leadsto 0.918938533204673 - \color{blue}{0.5 \cdot y} \]
   5. Step-by-step derivation

      1. *-commutative 99.9%

         \[\leadsto 0.918938533204673 - \color{blue}{y \cdot 0.5} \]

   6. Simplified 99.9%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-10} \lor \neg \left(x \leq 6.2 \cdot 10^{-12}\right):\\ \;\;\;\;x \cdot \left(y + -1\right) - -0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - y \cdot 0.5\\ \end{array} \]

Alternative 5: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4:\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.4)
   (* y (- x 0.5))
   (if (<= y 1.0) (- 0.918938533204673 x) (+ (* x y) (* y -0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.4) {
		tmp = y * (x - 0.5);
	} else if (y <= 1.0) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = (x * y) + (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 <= (-1.4d0)) then
        tmp = y * (x - 0.5d0)
    else if (y <= 1.0d0) then
        tmp = 0.918938533204673d0 - x
    else
        tmp = (x * y) + (y * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.4) {
		tmp = y * (x - 0.5);
	} else if (y <= 1.0) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = (x * y) + (y * -0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.4:
		tmp = y * (x - 0.5)
	elif y <= 1.0:
		tmp = 0.918938533204673 - x
	else:
		tmp = (x * y) + (y * -0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.4)
		tmp = Float64(y * Float64(x - 0.5));
	elseif (y <= 1.0)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = Float64(Float64(x * y) + Float64(y * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.4)
		tmp = y * (x - 0.5);
	elseif (y <= 1.0)
		tmp = 0.918938533204673 - x;
	else
		tmp = (x * y) + (y * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.4], N[(y * N[(x - 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(0.918938533204673 - x), $MachinePrecision], N[(N[(x * y), $MachinePrecision] + N[(y * -0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.3999999999999999

   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 98.6%

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

   if -1.3999999999999999 < 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. Taylor expanded in y around 0 97.6%

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

   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. 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 97.0%

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

      1. sub-neg 97.0%

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

      2. distribute-lft-in 97.1%

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

      3. metadata-eval 97.1%

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

   6. Applied egg-rr 97.1%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4:\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + y \cdot -0.5\\ \end{array} \]

Alternative 6: 98.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \lor \neg \left(y \leq 1.8\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.45) (not (<= y 1.8)))
   (* y (- x 0.5))
   (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.45) || !(y <= 1.8)) {
		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 <= (-1.45d0)) .or. (.not. (y <= 1.8d0))) 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 <= -1.45) || !(y <= 1.8)) {
		tmp = y * (x - 0.5);
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.45) or not (y <= 1.8):
		tmp = y * (x - 0.5)
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.45) || !(y <= 1.8))
		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 <= -1.45) || ~((y <= 1.8)))
		tmp = y * (x - 0.5);
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.45], N[Not[LessEqual[y, 1.8]], $MachinePrecision]], N[(y * N[(x - 0.5), $MachinePrecision]), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.44999999999999996 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 inf 97.8%

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

   if -1.44999999999999996 < 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. +-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 0 97.6%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \lor \neg \left(y \leq 1.8\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]

Alternative 7: 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 8: 49.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.85) || !(y <= 1.85)) {
		tmp = y * -0.5;
	} else {
		tmp = 0.918938533204673;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1.8500000000000001 or 1.8500000000000001 < 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 97.8%

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

   5. Taylor expanded in x around 0 48.0%

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

      1. *-commutative 48.0%

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

   7. Simplified 48.0%

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

   if -1.8500000000000001 < y < 1.8500000000000001

   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 x around 0 62.3%

      \[\leadsto 0.918938533204673 - \color{blue}{0.5 \cdot y} \]
   5. Step-by-step derivation

      1. *-commutative 62.3%

         \[\leadsto 0.918938533204673 - \color{blue}{y \cdot 0.5} \]

   6. Simplified 62.3%

      \[\leadsto 0.918938533204673 - \color{blue}{y \cdot 0.5} \]

   7. Taylor expanded in y around 0 60.5%

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

4. Final simplification 54.4%

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

Alternative 9: 26.6% 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. +-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 x around 0 56.0%

   \[\leadsto 0.918938533204673 - \color{blue}{0.5 \cdot y} \]
5. Step-by-step derivation

   1. *-commutative 56.0%

      \[\leadsto 0.918938533204673 - \color{blue}{y \cdot 0.5} \]

6. Simplified 56.0%

   \[\leadsto 0.918938533204673 - \color{blue}{y \cdot 0.5} \]

7. Taylor expanded in y around 0 32.5%

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

8. Final simplification 32.5%

   \[\leadsto 0.918938533204673 \]

Reproduce

herbie shell --seed 2023333 
(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))