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

Percentage Accurate: 100.0% → 100.0%

Time: 6.0s

Alternatives: 10

Speedup: 0.1×

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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(y * N[(x + -0.5), $MachinePrecision] + N[(0.918938533204673 - 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. cancel-sign-sub-inv 100.0%

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

   2. +-commutative 100.0%

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

   3. sub-neg 100.0%

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

   4. distribute-rgt-in 100.0%

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

   5. metadata-eval 100.0%

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

   6. neg-mul-1 100.0%

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

   7. associate-+r+ 100.0%

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

   8. unsub-neg 100.0%

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

   9. associate-+l- 100.0%

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

   10. distribute-lft-neg-out 100.0%

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

   11. distribute-rgt-neg-in 100.0%

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

   12. distribute-lft-out 100.0%

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

   13. fma-neg 100.0%

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

   14. +-commutative 100.0%

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

   15. metadata-eval 100.0%

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

   16. neg-sub0 100.0%

       \[\leadsto \mathsf{fma}\left(y, x + -0.5, \color{blue}{0 - \left(x - 0.918938533204673\right)}\right) \]

   17. associate-+l- 100.0%

       \[\leadsto \mathsf{fma}\left(y, x + -0.5, \color{blue}{\left(0 - x\right) + 0.918938533204673}\right) \]

   18. neg-sub0 100.0%

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

   19. +-commutative 100.0%

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

   20. unsub-neg 100.0%

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

3. Simplified 100.0%

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

Alternative 2: 74.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -140:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+75} \lor \neg \left(y \leq 1.65 \cdot 10^{+146}\right) \land y \leq 7.2 \cdot 10^{+229}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -140.0)
   (* y -0.5)
   (if (<= y 1.85)
     (- 0.918938533204673 x)
     (if (or (<= y 2.3e+75) (and (not (<= y 1.65e+146)) (<= y 7.2e+229)))
       (* y -0.5)
       (* y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -140.0) {
		tmp = y * -0.5;
	} else if (y <= 1.85) {
		tmp = 0.918938533204673 - x;
	} else if ((y <= 2.3e+75) || (!(y <= 1.65e+146) && (y <= 7.2e+229))) {
		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 <= (-140.0d0)) then
        tmp = y * (-0.5d0)
    else if (y <= 1.85d0) then
        tmp = 0.918938533204673d0 - x
    else if ((y <= 2.3d+75) .or. (.not. (y <= 1.65d+146)) .and. (y <= 7.2d+229)) 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 <= -140.0) {
		tmp = y * -0.5;
	} else if (y <= 1.85) {
		tmp = 0.918938533204673 - x;
	} else if ((y <= 2.3e+75) || (!(y <= 1.65e+146) && (y <= 7.2e+229))) {
		tmp = y * -0.5;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -140.0:
		tmp = y * -0.5
	elif y <= 1.85:
		tmp = 0.918938533204673 - x
	elif (y <= 2.3e+75) or (not (y <= 1.65e+146) and (y <= 7.2e+229)):
		tmp = y * -0.5
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -140.0)
		tmp = Float64(y * -0.5);
	elseif (y <= 1.85)
		tmp = Float64(0.918938533204673 - x);
	elseif ((y <= 2.3e+75) || (!(y <= 1.65e+146) && (y <= 7.2e+229)))
		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 <= -140.0)
		tmp = y * -0.5;
	elseif (y <= 1.85)
		tmp = 0.918938533204673 - x;
	elseif ((y <= 2.3e+75) || (~((y <= 1.65e+146)) && (y <= 7.2e+229)))
		tmp = y * -0.5;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -140.0], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 1.85], N[(0.918938533204673 - x), $MachinePrecision], If[Or[LessEqual[y, 2.3e+75], And[N[Not[LessEqual[y, 1.65e+146]], $MachinePrecision], LessEqual[y, 7.2e+229]]], N[(y * -0.5), $MachinePrecision], N[(y * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -140 or 1.8500000000000001 < y < 2.2999999999999999e75 or 1.65000000000000008e146 < y < 7.19999999999999973e229

   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. Add Preprocessing

   5. Taylor expanded in y around inf 96.6%

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

   6. Taylor expanded in x around 0 60.4%

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

   if -140 < 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. 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. Add Preprocessing

   5. Taylor expanded in y around 0 97.9%

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

      1. neg-mul-1 97.9%

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

      2. unsub-neg 97.9%

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

   7. Simplified 97.9%

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

   if 2.2999999999999999e75 < y < 1.65000000000000008e146 or 7.19999999999999973e229 < 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. Add Preprocessing

   5. Taylor expanded in y around inf 100.0%

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

   6. Taylor expanded in x around inf 69.5%

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

      1. *-commutative 69.5%

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

   8. Simplified 69.5%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -140:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+75} \lor \neg \left(y \leq 1.65 \cdot 10^{+146}\right) \land y \leq 7.2 \cdot 10^{+229}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{-8}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-7}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;y \leq 1.38 \cdot 10^{+75} \lor \neg \left(y \leq 2.2 \cdot 10^{+144}\right) \land y \leq 4 \cdot 10^{+229}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.8e-8)
   (* y -0.5)
   (if (<= y 3.7e-7)
     0.918938533204673
     (if (or (<= y 1.38e+75) (and (not (<= y 2.2e+144)) (<= y 4e+229)))
       (* y -0.5)
       (* y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.8e-8) {
		tmp = y * -0.5;
	} else if (y <= 3.7e-7) {
		tmp = 0.918938533204673;
	} else if ((y <= 1.38e+75) || (!(y <= 2.2e+144) && (y <= 4e+229))) {
		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 <= (-7.8d-8)) then
        tmp = y * (-0.5d0)
    else if (y <= 3.7d-7) then
        tmp = 0.918938533204673d0
    else if ((y <= 1.38d+75) .or. (.not. (y <= 2.2d+144)) .and. (y <= 4d+229)) 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 <= -7.8e-8) {
		tmp = y * -0.5;
	} else if (y <= 3.7e-7) {
		tmp = 0.918938533204673;
	} else if ((y <= 1.38e+75) || (!(y <= 2.2e+144) && (y <= 4e+229))) {
		tmp = y * -0.5;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.8e-8:
		tmp = y * -0.5
	elif y <= 3.7e-7:
		tmp = 0.918938533204673
	elif (y <= 1.38e+75) or (not (y <= 2.2e+144) and (y <= 4e+229)):
		tmp = y * -0.5
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.8e-8)
		tmp = Float64(y * -0.5);
	elseif (y <= 3.7e-7)
		tmp = 0.918938533204673;
	elseif ((y <= 1.38e+75) || (!(y <= 2.2e+144) && (y <= 4e+229)))
		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 <= -7.8e-8)
		tmp = y * -0.5;
	elseif (y <= 3.7e-7)
		tmp = 0.918938533204673;
	elseif ((y <= 1.38e+75) || (~((y <= 2.2e+144)) && (y <= 4e+229)))
		tmp = y * -0.5;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.8e-8], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 3.7e-7], 0.918938533204673, If[Or[LessEqual[y, 1.38e+75], And[N[Not[LessEqual[y, 2.2e+144]], $MachinePrecision], LessEqual[y, 4e+229]]], N[(y * -0.5), $MachinePrecision], N[(y * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.7999999999999997e-8 or 3.70000000000000004e-7 < y < 1.3799999999999999e75 or 2.19999999999999988e144 < y < 4e229

   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. Add Preprocessing

   5. Taylor expanded in y around inf 93.7%

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

   6. Taylor expanded in x around 0 58.5%

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

   if -7.7999999999999997e-8 < y < 3.70000000000000004e-7

   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. Add Preprocessing

   5. Taylor expanded in y around inf 50.1%

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

   6. Taylor expanded in y around 0 49.4%

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

   if 1.3799999999999999e75 < y < 2.19999999999999988e144 or 4e229 < 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. Add Preprocessing

   5. Taylor expanded in y around inf 100.0%

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

   6. Taylor expanded in x around inf 69.5%

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

      1. *-commutative 69.5%

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

   8. Simplified 69.5%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{-8}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-7}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;y \leq 1.38 \cdot 10^{+75} \lor \neg \left(y \leq 2.2 \cdot 10^{+144}\right) \land y \leq 4 \cdot 10^{+229}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-9} \lor \neg \left(x \leq 1.55 \cdot 10^{-7}\right):\\ \;\;\;\;0.918938533204673 + x \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.55e-9) (not (<= x 1.55e-7)))
   (+ 0.918938533204673 (* x (+ y -1.0)))
   (- 0.918938533204673 (* y 0.5))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.55e-9) || !(x <= 1.55e-7)) {
		tmp = 0.918938533204673 + (x * (y + -1.0));
	} 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 <= (-1.55d-9)) .or. (.not. (x <= 1.55d-7))) then
        tmp = 0.918938533204673d0 + (x * (y + (-1.0d0)))
    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 <= -1.55e-9) || !(x <= 1.55e-7)) {
		tmp = 0.918938533204673 + (x * (y + -1.0));
	} else {
		tmp = 0.918938533204673 - (y * 0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.55e-9) or not (x <= 1.55e-7):
		tmp = 0.918938533204673 + (x * (y + -1.0))
	else:
		tmp = 0.918938533204673 - (y * 0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.55e-9) || !(x <= 1.55e-7))
		tmp = Float64(0.918938533204673 + Float64(x * Float64(y + -1.0)));
	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 <= -1.55e-9) || ~((x <= 1.55e-7)))
		tmp = 0.918938533204673 + (x * (y + -1.0));
	else
		tmp = 0.918938533204673 - (y * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.55e-9], N[Not[LessEqual[x, 1.55e-7]], $MachinePrecision]], N[(0.918938533204673 + N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.918938533204673 - N[(y * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000002e-9 or 1.55e-7 < 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. Add Preprocessing

   5. Taylor expanded in y around 0 100.0%

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

   6. Taylor expanded in x around inf 98.8%

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

   if -1.55000000000000002e-9 < x < 1.55e-7

   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. Add Preprocessing

   5. Taylor expanded in x around 0 99.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-9} \lor \neg \left(x \leq 1.55 \cdot 10^{-7}\right):\\ \;\;\;\;0.918938533204673 + x \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;0.918938533204673 + x \cdot \left(y + -1\right)\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-8}:\\ \;\;\;\;0.918938533204673 + y \cdot \left(x + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 + \left(y \cdot x - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.0)
   (+ 0.918938533204673 (* x (+ y -1.0)))
   (if (<= x 5.5e-8)
     (+ 0.918938533204673 (* y (+ x -0.5)))
     (+ 0.918938533204673 (- (* y x) x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.0) {
		tmp = 0.918938533204673 + (x * (y + -1.0));
	} else if (x <= 5.5e-8) {
		tmp = 0.918938533204673 + (y * (x + -0.5));
	} else {
		tmp = 0.918938533204673 + ((y * x) - x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5.0d0)) then
        tmp = 0.918938533204673d0 + (x * (y + (-1.0d0)))
    else if (x <= 5.5d-8) then
        tmp = 0.918938533204673d0 + (y * (x + (-0.5d0)))
    else
        tmp = 0.918938533204673d0 + ((y * x) - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5.0) {
		tmp = 0.918938533204673 + (x * (y + -1.0));
	} else if (x <= 5.5e-8) {
		tmp = 0.918938533204673 + (y * (x + -0.5));
	} else {
		tmp = 0.918938533204673 + ((y * x) - x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.0:
		tmp = 0.918938533204673 + (x * (y + -1.0))
	elif x <= 5.5e-8:
		tmp = 0.918938533204673 + (y * (x + -0.5))
	else:
		tmp = 0.918938533204673 + ((y * x) - x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.0)
		tmp = Float64(0.918938533204673 + Float64(x * Float64(y + -1.0)));
	elseif (x <= 5.5e-8)
		tmp = Float64(0.918938533204673 + Float64(y * Float64(x + -0.5)));
	else
		tmp = Float64(0.918938533204673 + Float64(Float64(y * x) - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.0)
		tmp = 0.918938533204673 + (x * (y + -1.0));
	elseif (x <= 5.5e-8)
		tmp = 0.918938533204673 + (y * (x + -0.5));
	else
		tmp = 0.918938533204673 + ((y * x) - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.0], N[(0.918938533204673 + N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.5e-8], N[(0.918938533204673 + N[(y * N[(x + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.918938533204673 + N[(N[(y * x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 100.0%

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

   6. Taylor expanded in x around inf 99.6%

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

   if -5 < x < 5.5000000000000003e-8

   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. Add Preprocessing

   5. Taylor expanded in y around inf 99.6%

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

      1. associate-+l- 99.6%

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

      2. *-commutative 99.6%

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

      3. distribute-lft-out-- 99.6%

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

      4. *-un-lft-identity 99.6%

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

      5. sub-neg 99.6%

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

      6. *-un-lft-identity 99.6%

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

      7. metadata-eval 99.6%

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

   7. Applied egg-rr 99.6%

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

   if 5.5000000000000003e-8 < x

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. sub-neg 100.0%

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

      6. *-un-lft-identity 100.0%

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

      7. metadata-eval 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around inf 99.1%

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

      1. *-commutative 99.1%

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

   10. Simplified 99.1%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;0.918938533204673 + x \cdot \left(y + -1\right)\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-8}:\\ \;\;\;\;0.918938533204673 + y \cdot \left(x + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 + \left(y \cdot x - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-10}:\\ \;\;\;\;0.918938533204673 + x \cdot \left(y + -1\right)\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-9}:\\ \;\;\;\;0.918938533204673 - y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 + \left(y \cdot x - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.95e-10)
   (+ 0.918938533204673 (* x (+ y -1.0)))
   (if (<= x 6.6e-9)
     (- 0.918938533204673 (* y 0.5))
     (+ 0.918938533204673 (- (* y x) x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.95e-10) {
		tmp = 0.918938533204673 + (x * (y + -1.0));
	} else if (x <= 6.6e-9) {
		tmp = 0.918938533204673 - (y * 0.5);
	} else {
		tmp = 0.918938533204673 + ((y * x) - x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.95d-10)) then
        tmp = 0.918938533204673d0 + (x * (y + (-1.0d0)))
    else if (x <= 6.6d-9) then
        tmp = 0.918938533204673d0 - (y * 0.5d0)
    else
        tmp = 0.918938533204673d0 + ((y * x) - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.95e-10) {
		tmp = 0.918938533204673 + (x * (y + -1.0));
	} else if (x <= 6.6e-9) {
		tmp = 0.918938533204673 - (y * 0.5);
	} else {
		tmp = 0.918938533204673 + ((y * x) - x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.95e-10:
		tmp = 0.918938533204673 + (x * (y + -1.0))
	elif x <= 6.6e-9:
		tmp = 0.918938533204673 - (y * 0.5)
	else:
		tmp = 0.918938533204673 + ((y * x) - x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.95e-10)
		tmp = Float64(0.918938533204673 + Float64(x * Float64(y + -1.0)));
	elseif (x <= 6.6e-9)
		tmp = Float64(0.918938533204673 - Float64(y * 0.5));
	else
		tmp = Float64(0.918938533204673 + Float64(Float64(y * x) - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.95e-10)
		tmp = 0.918938533204673 + (x * (y + -1.0));
	elseif (x <= 6.6e-9)
		tmp = 0.918938533204673 - (y * 0.5);
	else
		tmp = 0.918938533204673 + ((y * x) - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.95e-10], N[(0.918938533204673 + N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.6e-9], N[(0.918938533204673 - N[(y * 0.5), $MachinePrecision]), $MachinePrecision], N[(0.918938533204673 + N[(N[(y * x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.95e-10

   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. Add Preprocessing

   5. Taylor expanded in y around 0 100.0%

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

   6. Taylor expanded in x around inf 98.6%

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

   if -1.95e-10 < x < 6.60000000000000037e-9

   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. Add Preprocessing

   5. Taylor expanded in x around 0 99.5%

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

   if 6.60000000000000037e-9 < x

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. sub-neg 100.0%

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

      6. *-un-lft-identity 100.0%

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

      7. metadata-eval 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around inf 99.1%

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

      1. *-commutative 99.1%

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

   10. Simplified 99.1%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-10}:\\ \;\;\;\;0.918938533204673 + x \cdot \left(y + -1\right)\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-9}:\\ \;\;\;\;0.918938533204673 - y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 + \left(y \cdot x - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.35 \lor \neg \left(y \leq 1.2\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.35) (not (<= y 1.2)))
   (* y (- x 0.5))
   (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.35) || !(y <= 1.2)) {
		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 <= (-1.35d0)) .or. (.not. (y <= 1.2d0))) 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 <= -1.35) || !(y <= 1.2)) {
		tmp = y * (x - 0.5);
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.35) or not (y <= 1.2):
		tmp = y * (x - 0.5)
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.35) || !(y <= 1.2))
		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 <= -1.35) || ~((y <= 1.2)))
		tmp = y * (x - 0.5);
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.35], N[Not[LessEqual[y, 1.2]], $MachinePrecision]], N[(y * N[(x - 0.5), $MachinePrecision]), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3500000000000001 or 1.19999999999999996 < 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. Add Preprocessing

   5. Taylor expanded in y around inf 97.5%

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

   if -1.3500000000000001 < y < 1.19999999999999996

   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. Add Preprocessing

   5. Taylor expanded in y around 0 97.9%

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

      1. neg-mul-1 97.9%

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

      2. unsub-neg 97.9%

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

   7. Simplified 97.9%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35 \lor \neg \left(y \leq 1.2\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{-8} \lor \neg \left(y \leq 3.7 \cdot 10^{-7}\right):\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -7.8e-8) (not (<= y 3.7e-7))) (* y -0.5) 0.918938533204673))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -7.8e-8) || !(y <= 3.7e-7)) {
		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 <= (-7.8d-8)) .or. (.not. (y <= 3.7d-7))) 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 <= -7.8e-8) || !(y <= 3.7e-7)) {
		tmp = y * -0.5;
	} else {
		tmp = 0.918938533204673;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -7.8e-8) or not (y <= 3.7e-7):
		tmp = y * -0.5
	else:
		tmp = 0.918938533204673
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -7.8e-8) || !(y <= 3.7e-7))
		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 <= -7.8e-8) || ~((y <= 3.7e-7)))
		tmp = y * -0.5;
	else
		tmp = 0.918938533204673;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -7.8e-8], N[Not[LessEqual[y, 3.7e-7]], $MachinePrecision]], N[(y * -0.5), $MachinePrecision], 0.918938533204673]

Derivation

1. Split input into 2 regimes

2. if y < -7.7999999999999997e-8 or 3.70000000000000004e-7 < 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. Add Preprocessing

   5. Taylor expanded in y around inf 95.4%

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

   6. Taylor expanded in x around 0 51.6%

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

   if -7.7999999999999997e-8 < y < 3.70000000000000004e-7

   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. Add Preprocessing

   5. Taylor expanded in y around inf 50.1%

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

   6. Taylor expanded in y around 0 49.4%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{-8} \lor \neg \left(y \leq 3.7 \cdot 10^{-7}\right):\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 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. 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. Add Preprocessing

5. Taylor expanded in y around 0 100.0%

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

   1. +-commutative 100.0%

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

   2. mul-1-neg 100.0%

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

   3. unsub-neg 100.0%

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

   4. *-un-lft-identity 100.0%

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

   5. sub-neg 100.0%

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

   6. *-un-lft-identity 100.0%

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

   7. metadata-eval 100.0%

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

7. Applied egg-rr 100.0%

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

Alternative 10: 26.9% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 0.918938533204673)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 0.918938533204673

Julia

function code(x, y)
	return 0.918938533204673
end

MATLAB

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

Wolfram

code[x_, y_] := 0.918938533204673

Derivation

1. Initial program 100.0%

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

   1. associate-+l- 100.0%

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

   2. sub-neg 100.0%

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

   3. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in y around inf 72.2%

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

6. Taylor expanded in y around 0 27.1%

   \[\leadsto \color{blue}{0.918938533204673} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024108 
(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))