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

Percentage Accurate: 100.0% → 100.0%

Time: 8.3s

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, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
2. 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%

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

   1. neg-mul-1 100.0%

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

   2. associate-+r+ 100.0%

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

   3. sub-neg 100.0%

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

   4. mul-1-neg 100.0%

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

   5. unsub-neg 100.0%

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

   6. neg-mul-1 100.0%

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

   7. unsub-neg 100.0%

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

6. Simplified 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 49.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{-8}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -1.58 \cdot 10^{-286}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-277}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-159}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.75e-8)
   (* y -0.5)
   (if (<= y -1.58e-286)
     0.918938533204673
     (if (<= y 8e-277)
       (- x)
       (if (<= y 1.2e-159)
         0.918938533204673
         (if (<= y 1.85) (- x) (* y -0.5)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.75e-8) {
		tmp = y * -0.5;
	} else if (y <= -1.58e-286) {
		tmp = 0.918938533204673;
	} else if (y <= 8e-277) {
		tmp = -x;
	} else if (y <= 1.2e-159) {
		tmp = 0.918938533204673;
	} else if (y <= 1.85) {
		tmp = -x;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.75d-8)) then
        tmp = y * (-0.5d0)
    else if (y <= (-1.58d-286)) then
        tmp = 0.918938533204673d0
    else if (y <= 8d-277) then
        tmp = -x
    else if (y <= 1.2d-159) then
        tmp = 0.918938533204673d0
    else if (y <= 1.85d0) then
        tmp = -x
    else
        tmp = y * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.75e-8) {
		tmp = y * -0.5;
	} else if (y <= -1.58e-286) {
		tmp = 0.918938533204673;
	} else if (y <= 8e-277) {
		tmp = -x;
	} else if (y <= 1.2e-159) {
		tmp = 0.918938533204673;
	} else if (y <= 1.85) {
		tmp = -x;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.75e-8:
		tmp = y * -0.5
	elif y <= -1.58e-286:
		tmp = 0.918938533204673
	elif y <= 8e-277:
		tmp = -x
	elif y <= 1.2e-159:
		tmp = 0.918938533204673
	elif y <= 1.85:
		tmp = -x
	else:
		tmp = y * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.75e-8)
		tmp = Float64(y * -0.5);
	elseif (y <= -1.58e-286)
		tmp = 0.918938533204673;
	elseif (y <= 8e-277)
		tmp = Float64(-x);
	elseif (y <= 1.2e-159)
		tmp = 0.918938533204673;
	elseif (y <= 1.85)
		tmp = Float64(-x);
	else
		tmp = Float64(y * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.75e-8)
		tmp = y * -0.5;
	elseif (y <= -1.58e-286)
		tmp = 0.918938533204673;
	elseif (y <= 8e-277)
		tmp = -x;
	elseif (y <= 1.2e-159)
		tmp = 0.918938533204673;
	elseif (y <= 1.85)
		tmp = -x;
	else
		tmp = y * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.75e-8], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, -1.58e-286], 0.918938533204673, If[LessEqual[y, 8e-277], (-x), If[LessEqual[y, 1.2e-159], 0.918938533204673, If[LessEqual[y, 1.85], (-x), N[(y * -0.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.75000000000000012e-8 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 96.9%

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

   5. Taylor expanded in x around 0 54.0%

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

      1. *-commutative 54.0%

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

   7. Simplified 54.0%

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

   if -1.75000000000000012e-8 < y < -1.57999999999999989e-286 or 7.99999999999999975e-277 < y < 1.19999999999999999e-159

   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 x around 0 68.2%

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

   5. Taylor expanded in y around 0 67.4%

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

   if -1.57999999999999989e-286 < y < 7.99999999999999975e-277 or 1.19999999999999999e-159 < 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. 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. distribute-lft-out-- 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-+r+ 100.0%

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

      7. associate-+l+ 100.0%

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

      8. +-commutative 100.0%

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

      9. unsub-neg 100.0%

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

      10. associate-+r- 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-lft-neg-out 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. distribute-lft-out 100.0%

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

      15. fma-def 100.0%

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

      16. +-commutative 100.0%

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

      17. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 63.9%

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

      1. mul-1-neg 63.9%

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

      2. *-commutative 63.9%

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

      3. distribute-rgt-neg-in 63.9%

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

      4. mul-1-neg 63.9%

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

      5. unsub-neg 63.9%

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

   6. Simplified 63.9%

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

   7. Taylor expanded in y around 0 60.8%

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

      1. mul-1-neg 60.8%

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

   9. Simplified 60.8%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{-8}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -1.58 \cdot 10^{-286}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-277}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-159}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]

Alternative 3: 74.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -17:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{-19}:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 1250000000000:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -17.0)
   (* y -0.5)
   (if (<= y 1.06e-19)
     (- 0.918938533204673 x)
     (if (<= y 1250000000000.0) (* x (+ y -1.0)) (* y -0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -17.0) {
		tmp = y * -0.5;
	} else if (y <= 1.06e-19) {
		tmp = 0.918938533204673 - x;
	} else if (y <= 1250000000000.0) {
		tmp = x * (y + -1.0);
	} 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 <= (-17.0d0)) then
        tmp = y * (-0.5d0)
    else if (y <= 1.06d-19) then
        tmp = 0.918938533204673d0 - x
    else if (y <= 1250000000000.0d0) then
        tmp = x * (y + (-1.0d0))
    else
        tmp = y * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -17.0) {
		tmp = y * -0.5;
	} else if (y <= 1.06e-19) {
		tmp = 0.918938533204673 - x;
	} else if (y <= 1250000000000.0) {
		tmp = x * (y + -1.0);
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -17.0:
		tmp = y * -0.5
	elif y <= 1.06e-19:
		tmp = 0.918938533204673 - x
	elif y <= 1250000000000.0:
		tmp = x * (y + -1.0)
	else:
		tmp = y * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -17.0)
		tmp = Float64(y * -0.5);
	elseif (y <= 1.06e-19)
		tmp = Float64(0.918938533204673 - x);
	elseif (y <= 1250000000000.0)
		tmp = Float64(x * Float64(y + -1.0));
	else
		tmp = Float64(y * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -17.0)
		tmp = y * -0.5;
	elseif (y <= 1.06e-19)
		tmp = 0.918938533204673 - x;
	elseif (y <= 1250000000000.0)
		tmp = x * (y + -1.0);
	else
		tmp = y * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -17.0], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 1.06e-19], N[(0.918938533204673 - x), $MachinePrecision], If[LessEqual[y, 1250000000000.0], N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision], N[(y * -0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -17 or 1.25e12 < y

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 99.3%

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

   5. Taylor expanded in x around 0 55.8%

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

      1. *-commutative 55.8%

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

   7. Simplified 55.8%

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

   if -17 < y < 1.06e-19

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 99.1%

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

      1. neg-mul-1 99.1%

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

      2. sub-neg 99.1%

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

   6. Simplified 99.1%

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

   if 1.06e-19 < y < 1.25e12

   1. Initial program 99.8%

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

      1. cancel-sign-sub-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-lft-out-- 99.6%

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

      4. cancel-sign-sub-inv 99.6%

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

      5. *-rgt-identity 99.6%

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

      6. associate-+r+ 99.6%

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

      7. associate-+l+ 99.6%

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

      8. +-commutative 99.6%

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

      9. unsub-neg 99.6%

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

      10. associate-+r- 99.6%

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

      11. *-commutative 99.6%

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

      12. distribute-lft-neg-out 99.6%

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

      13. distribute-rgt-neg-in 99.6%

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

      14. distribute-lft-out 99.6%

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

      15. fma-def 99.6%

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

      16. +-commutative 99.6%

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

      17. sub-neg 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around -inf 86.5%

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

      1. mul-1-neg 86.5%

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

      2. *-commutative 86.5%

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

      3. distribute-rgt-neg-in 86.5%

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

      4. mul-1-neg 86.5%

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

      5. unsub-neg 86.5%

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

   6. Simplified 86.5%

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

   7. Taylor expanded in y around 0 86.3%

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

   8. Taylor expanded in x around 0 86.5%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -17:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{-19}:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 1250000000000:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]

Alternative 4: 97.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \lor \neg \left(y \leq 1.2\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.3) (not (<= y 1.2)))
   (* y (- x 0.5))
   (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.3) || !(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 <= (-1.3d0)) .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 <= -1.3) || !(y <= 1.2)) {
		tmp = y * (x - 0.5);
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.3) 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 <= -1.3) || !(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 <= -1.3) || ~((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, -1.3], 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 < -1.30000000000000004 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. Taylor expanded in y around inf 97.6%

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

   if -1.30000000000000004 < 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. Taylor expanded in y around 0 98.0%

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

      1. neg-mul-1 98.0%

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

      2. sub-neg 98.0%

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

   6. Simplified 98.0%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \lor \neg \left(y \leq 1.2\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]

Alternative 5: 97.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.72 \lor \neg \left(x \leq 0.7\right):\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.72) (not (<= x 0.7)))
   (* x (+ y -1.0))
   (- 0.918938533204673 (* y 0.5))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.72) || !(x <= 0.7)) {
		tmp = 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 <= (-0.72d0)) .or. (.not. (x <= 0.7d0))) then
        tmp = 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 <= -0.72) || !(x <= 0.7)) {
		tmp = x * (y + -1.0);
	} else {
		tmp = 0.918938533204673 - (y * 0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -0.72) or not (x <= 0.7):
		tmp = x * (y + -1.0)
	else:
		tmp = 0.918938533204673 - (y * 0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.72) || !(x <= 0.7))
		tmp = 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 <= -0.72) || ~((x <= 0.7)))
		tmp = x * (y + -1.0);
	else
		tmp = 0.918938533204673 - (y * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.72], N[Not[LessEqual[x, 0.7]], $MachinePrecision]], N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision], N[(0.918938533204673 - N[(y * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.71999999999999997 or 0.69999999999999996 < 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. 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. distribute-lft-out-- 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-+r+ 100.0%

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

      7. associate-+l+ 100.0%

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

      8. +-commutative 100.0%

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

      9. unsub-neg 100.0%

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

      10. associate-+r- 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-lft-neg-out 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. distribute-lft-out 100.0%

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

      15. fma-def 100.0%

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

      16. +-commutative 100.0%

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

      17. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 98.4%

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

      1. mul-1-neg 98.4%

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

      2. *-commutative 98.4%

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

      3. distribute-rgt-neg-in 98.4%

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

      4. mul-1-neg 98.4%

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

      5. unsub-neg 98.4%

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

   6. Simplified 98.4%

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

   7. Taylor expanded in y around 0 98.4%

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

   8. Taylor expanded in x around 0 98.4%

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

   if -0.71999999999999997 < x < 0.69999999999999996

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.5%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.72 \lor \neg \left(x \leq 0.7\right):\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - y \cdot 0.5\\ \end{array} \]

Alternative 6: 97.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.68:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;0.918938533204673 - y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.68)
   (* x (+ y -1.0))
   (if (<= x 0.9) (- 0.918938533204673 (* y 0.5)) (- (* x y) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.68) {
		tmp = x * (y + -1.0);
	} else if (x <= 0.9) {
		tmp = 0.918938533204673 - (y * 0.5);
	} else {
		tmp = (x * 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 (x <= (-0.68d0)) then
        tmp = x * (y + (-1.0d0))
    else if (x <= 0.9d0) then
        tmp = 0.918938533204673d0 - (y * 0.5d0)
    else
        tmp = (x * y) - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.68) {
		tmp = x * (y + -1.0);
	} else if (x <= 0.9) {
		tmp = 0.918938533204673 - (y * 0.5);
	} else {
		tmp = (x * y) - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.68:
		tmp = x * (y + -1.0)
	elif x <= 0.9:
		tmp = 0.918938533204673 - (y * 0.5)
	else:
		tmp = (x * y) - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.68)
		tmp = Float64(x * Float64(y + -1.0));
	elseif (x <= 0.9)
		tmp = Float64(0.918938533204673 - Float64(y * 0.5));
	else
		tmp = Float64(Float64(x * y) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.68)
		tmp = x * (y + -1.0);
	elseif (x <= 0.9)
		tmp = 0.918938533204673 - (y * 0.5);
	else
		tmp = (x * y) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.68], N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.9], N[(0.918938533204673 - N[(y * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] - x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.680000000000000049

   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. distribute-lft-out-- 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-+r+ 100.0%

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

      7. associate-+l+ 100.0%

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

      8. +-commutative 100.0%

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

      9. unsub-neg 100.0%

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

      10. associate-+r- 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-lft-neg-out 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. distribute-lft-out 100.0%

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

      15. fma-def 100.0%

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

      16. +-commutative 100.0%

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

      17. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. *-commutative 98.7%

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

      3. distribute-rgt-neg-in 98.7%

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

      4. mul-1-neg 98.7%

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

      5. unsub-neg 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in y around 0 98.7%

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

   8. Taylor expanded in x around 0 98.7%

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

   if -0.680000000000000049 < x < 0.900000000000000022

   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 x around 0 99.5%

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

   if 0.900000000000000022 < 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. cancel-sign-sub-inv 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-lft-out-- 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-+r+ 100.0%

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

      7. associate-+l+ 100.0%

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

      8. +-commutative 100.0%

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

      9. unsub-neg 100.0%

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

      10. associate-+r- 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-lft-neg-out 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. distribute-lft-out 100.0%

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

      15. fma-def 100.0%

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

      16. +-commutative 100.0%

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

      17. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 98.2%

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

      1. mul-1-neg 98.2%

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

      2. *-commutative 98.2%

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

      3. distribute-rgt-neg-in 98.2%

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

      4. mul-1-neg 98.2%

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

      5. unsub-neg 98.2%

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

   6. Simplified 98.2%

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

   7. Taylor expanded in y around 0 98.2%

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

   8. Taylor expanded in x around 0 98.2%

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

      1. sub-neg 98.2%

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

      2. metadata-eval 98.2%

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

      3. distribute-lft-in 98.2%

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

      4. *-commutative 98.2%

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

      5. mul-1-neg 98.2%

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

      6. sub-neg 98.2%

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

   10. Simplified 98.2%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.68:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;0.918938533204673 - y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y - x\\ \end{array} \]

Alternative 7: 74.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -56:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -56.0)
   (* y -0.5)
   (if (<= y 1.85) (- 0.918938533204673 x) (* y -0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -56.0) {
		tmp = y * -0.5;
	} else if (y <= 1.85) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-56.0d0)) then
        tmp = y * (-0.5d0)
    else if (y <= 1.85d0) then
        tmp = 0.918938533204673d0 - x
    else
        tmp = y * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -56.0) {
		tmp = y * -0.5;
	} else if (y <= 1.85) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -56.0:
		tmp = y * -0.5
	elif y <= 1.85:
		tmp = 0.918938533204673 - x
	else:
		tmp = y * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -56.0)
		tmp = Float64(y * -0.5);
	elseif (y <= 1.85)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = Float64(y * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -56.0)
		tmp = y * -0.5;
	elseif (y <= 1.85)
		tmp = 0.918938533204673 - x;
	else
		tmp = y * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -56.0], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 1.85], N[(0.918938533204673 - x), $MachinePrecision], N[(y * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -56 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 97.6%

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

   5. Taylor expanded in x around 0 54.4%

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

      1. *-commutative 54.4%

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

   7. Simplified 54.4%

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

   if -56 < 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. Taylor expanded in y around 0 98.0%

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

      1. neg-mul-1 98.0%

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

      2. sub-neg 98.0%

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

   6. Simplified 98.0%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -56:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]

Alternative 8: 50.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.92:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 0.92:\\ \;\;\;\;0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.92) (- x) (if (<= x 0.92) 0.918938533204673 (- x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.92) {
		tmp = -x;
	} else if (x <= 0.92) {
		tmp = 0.918938533204673;
	} else {
		tmp = -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 <= (-0.92d0)) then
        tmp = -x
    else if (x <= 0.92d0) then
        tmp = 0.918938533204673d0
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.92) {
		tmp = -x;
	} else if (x <= 0.92) {
		tmp = 0.918938533204673;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.92:
		tmp = -x
	elif x <= 0.92:
		tmp = 0.918938533204673
	else:
		tmp = -x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.92)
		tmp = Float64(-x);
	elseif (x <= 0.92)
		tmp = 0.918938533204673;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.92)
		tmp = -x;
	elseif (x <= 0.92)
		tmp = 0.918938533204673;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.92], (-x), If[LessEqual[x, 0.92], 0.918938533204673, (-x)]]

Derivation

1. Split input into 2 regimes

2. if x < -0.92000000000000004 or 0.92000000000000004 < 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. 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. distribute-lft-out-- 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-+r+ 100.0%

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

      7. associate-+l+ 100.0%

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

      8. +-commutative 100.0%

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

      9. unsub-neg 100.0%

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

      10. associate-+r- 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-lft-neg-out 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. distribute-lft-out 100.0%

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

      15. fma-def 100.0%

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

      16. +-commutative 100.0%

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

      17. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 98.4%

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

      1. mul-1-neg 98.4%

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

      2. *-commutative 98.4%

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

      3. distribute-rgt-neg-in 98.4%

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

      4. mul-1-neg 98.4%

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

      5. unsub-neg 98.4%

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

   6. Simplified 98.4%

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

   7. Taylor expanded in y around 0 47.2%

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

      1. mul-1-neg 47.2%

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

   9. Simplified 47.2%

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

   if -0.92000000000000004 < x < 0.92000000000000004

   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 x around 0 99.5%

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

   5. Taylor expanded in y around 0 49.8%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.92:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 0.92:\\ \;\;\;\;0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 9: 26.1% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 0.918938533204673)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 0.918938533204673

Julia

function code(x, y)
	return 0.918938533204673
end

MATLAB

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

Wolfram

code[x_, y_] := 0.918938533204673

Derivation

1. Initial program 100.0%

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

   1. associate-+l- 100.0%

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

   2. sub-neg 100.0%

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

   3. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 55.9%

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

5. Taylor expanded in y around 0 28.4%

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

6. Final simplification 28.4%

   \[\leadsto 0.918938533204673 \]

Reproduce

herbie shell --seed 2023285 
(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))