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

Percentage Accurate: 100.0% → 100.0%

Time: 4.9s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

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

   2. cancel-sign-sub-inv 100.0%

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

   3. +-commutative 100.0%

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

   4. associate-+r+ 100.0%

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

   5. cancel-sign-sub-inv 100.0%

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

   6. associate-+l- 100.0%

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

   7. sub-neg 100.0%

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

   8. distribute-rgt-in 100.0%

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

   9. metadata-eval 100.0%

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

   10. neg-mul-1 100.0%

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

   11. associate--r+ 100.0%

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

   12. distribute-lft-out-- 100.0%

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

   13. unsub-neg 100.0%

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

   14. fma-neg 100.0%

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

   15. unsub-neg 100.0%

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

   16. remove-double-neg 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -620000000000:\\ \;\;\;\;y \cdot x - x\\ \mathbf{elif}\;x \leq 1800000000:\\ \;\;\;\;0.918938533204673 + \left(y \cdot -0.5 - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -620000000000.0)
   (- (* y x) x)
   (if (<= x 1800000000.0)
     (+ 0.918938533204673 (- (* y -0.5) x))
     (* x (+ y -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -620000000000.0) {
		tmp = (y * x) - x;
	} else if (x <= 1800000000.0) {
		tmp = 0.918938533204673 + ((y * -0.5) - x);
	} else {
		tmp = x * (y + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-620000000000.0d0)) then
        tmp = (y * x) - x
    else if (x <= 1800000000.0d0) then
        tmp = 0.918938533204673d0 + ((y * (-0.5d0)) - x)
    else
        tmp = x * (y + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -620000000000.0) {
		tmp = (y * x) - x;
	} else if (x <= 1800000000.0) {
		tmp = 0.918938533204673 + ((y * -0.5) - x);
	} else {
		tmp = x * (y + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -620000000000.0:
		tmp = (y * x) - x
	elif x <= 1800000000.0:
		tmp = 0.918938533204673 + ((y * -0.5) - x)
	else:
		tmp = x * (y + -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -620000000000.0)
		tmp = Float64(Float64(y * x) - x);
	elseif (x <= 1800000000.0)
		tmp = Float64(0.918938533204673 + Float64(Float64(y * -0.5) - x));
	else
		tmp = Float64(x * Float64(y + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -620000000000.0)
		tmp = (y * x) - x;
	elseif (x <= 1800000000.0)
		tmp = 0.918938533204673 + ((y * -0.5) - x);
	else
		tmp = x * (y + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -620000000000.0], N[(N[(y * x), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[x, 1800000000.0], N[(0.918938533204673 + N[(N[(y * -0.5), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.2e11

   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. Taylor expanded in x around inf 99.5%

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

      1. sub-neg 99.5%

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

      2. distribute-rgt-in 99.5%

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

      3. metadata-eval 99.5%

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

      4. neg-mul-1 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. unsub-neg 99.5%

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

   8. Applied egg-rr 99.5%

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

   if -6.2e11 < x < 1.8e9

   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 0 100.0%

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

      1. cancel-sign-sub-inv 100.0%

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

      2. metadata-eval 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-+r+ 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. *-commutative 100.0%

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

      8. neg-mul-1 100.0%

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

      9. +-commutative 100.0%

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

      10. associate-+l+ 100.0%

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

      11. distribute-rgt-in 100.0%

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

      12. metadata-eval 100.0%

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

      13. sub-neg 100.0%

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

      14. +-commutative 100.0%

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

      15. unsub-neg 100.0%

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

      16. sub-neg 100.0%

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

      17. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 99.1%

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

      1. *-commutative 99.1%

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

   9. Simplified 99.1%

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

   if 1.8e9 < x

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -620000000000:\\ \;\;\;\;y \cdot x - x\\ \mathbf{elif}\;x \leq 1800000000:\\ \;\;\;\;0.918938533204673 + \left(y \cdot -0.5 - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \end{array} \]

Alternative 3: 72.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-11} \lor \neg \left(y \leq 2 \cdot 10^{-38}\right):\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.1e-11) (not (<= y 2e-38)))
   (* x (+ y -1.0))
   (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.1e-11) || !(y <= 2e-38)) {
		tmp = x * (y + -1.0);
	} 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 <= (-1.1d-11)) .or. (.not. (y <= 2d-38))) then
        tmp = x * (y + (-1.0d0))
    else
        tmp = 0.918938533204673d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.1e-11) || !(y <= 2e-38)) {
		tmp = x * (y + -1.0);
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.1e-11) or not (y <= 2e-38):
		tmp = x * (y + -1.0)
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.1e-11) || !(y <= 2e-38))
		tmp = Float64(x * Float64(y + -1.0));
	else
		tmp = Float64(0.918938533204673 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.1e-11) || ~((y <= 2e-38)))
		tmp = x * (y + -1.0);
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.1e-11], N[Not[LessEqual[y, 2e-38]], $MachinePrecision]], N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.1000000000000001e-11 or 1.9999999999999999e-38 < y

   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. Taylor expanded in x around inf 56.4%

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

   if -1.1000000000000001e-11 < y < 1.9999999999999999e-38

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-11} \lor \neg \left(y \leq 2 \cdot 10^{-38}\right):\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]

Alternative 4: 98.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.5 \lor \neg \left(y \leq 1.15\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.5) (not (<= y 1.15)))
   (* y (- x 0.5))
   (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.5) || !(y <= 1.15)) {
		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 <= (-1.5d0)) .or. (.not. (y <= 1.15d0))) 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 <= -1.5) || !(y <= 1.15)) {
		tmp = y * (x - 0.5);
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.5) or not (y <= 1.15):
		tmp = y * (x - 0.5)
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.5) || !(y <= 1.15))
		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 <= -1.5) || ~((y <= 1.15)))
		tmp = y * (x - 0.5);
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.5], N[Not[LessEqual[y, 1.15]], $MachinePrecision]], N[(y * N[(x - 0.5), $MachinePrecision]), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.5 or 1.1499999999999999 < 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 98.1%

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

   if -1.5 < y < 1.1499999999999999

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 97.5%

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

      1. neg-mul-1 97.5%

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

      2. unsub-neg 97.5%

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

   6. Simplified 97.5%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \lor \neg \left(y \leq 1.15\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]

Alternative 5: 97.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.72 \lor \neg \left(x \leq 0.6\right):\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.72) (not (<= x 0.6)))
   (* x (+ y -1.0))
   (- 0.918938533204673 (* y 0.5))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.72) || !(x <= 0.6)) {
		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 <= 0.6d0))) 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 <= 0.6)) {
		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 <= 0.6):
		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 <= 0.6))
		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 <= 0.6)))
		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, 0.6]], $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 0.599999999999999978 < x

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 99.6%

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

   if -0.71999999999999997 < x < 0.599999999999999978

   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 98.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.72 \lor \neg \left(x \leq 0.6\right):\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - y \cdot 0.5\\ \end{array} \]

Alternative 6: 97.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.8:\\ \;\;\;\;y \cdot x - x\\ \mathbf{elif}\;x \leq 0.65:\\ \;\;\;\;0.918938533204673 - y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.8)
   (- (* y x) x)
   (if (<= x 0.65) (- 0.918938533204673 (* y 0.5)) (* x (+ y -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.8) {
		tmp = (y * x) - x;
	} else if (x <= 0.65) {
		tmp = 0.918938533204673 - (y * 0.5);
	} else {
		tmp = x * (y + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.8d0)) then
        tmp = (y * x) - x
    else if (x <= 0.65d0) then
        tmp = 0.918938533204673d0 - (y * 0.5d0)
    else
        tmp = x * (y + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.8) {
		tmp = (y * x) - x;
	} else if (x <= 0.65) {
		tmp = 0.918938533204673 - (y * 0.5);
	} else {
		tmp = x * (y + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.8:
		tmp = (y * x) - x
	elif x <= 0.65:
		tmp = 0.918938533204673 - (y * 0.5)
	else:
		tmp = x * (y + -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.8)
		tmp = Float64(Float64(y * x) - x);
	elseif (x <= 0.65)
		tmp = Float64(0.918938533204673 - Float64(y * 0.5));
	else
		tmp = Float64(x * Float64(y + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.8)
		tmp = (y * x) - x;
	elseif (x <= 0.65)
		tmp = 0.918938533204673 - (y * 0.5);
	else
		tmp = x * (y + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.8], N[(N[(y * x), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[x, 0.65], N[(0.918938533204673 - N[(y * 0.5), $MachinePrecision]), $MachinePrecision], N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.80000000000000004

   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. Taylor expanded in x around inf 99.2%

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

      1. sub-neg 99.2%

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

      2. distribute-rgt-in 99.2%

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

      3. metadata-eval 99.2%

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

      4. neg-mul-1 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. unsub-neg 99.2%

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

   8. Applied egg-rr 99.2%

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

   if -0.80000000000000004 < x < 0.650000000000000022

   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 98.8%

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

   if 0.650000000000000022 < x

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.8:\\ \;\;\;\;y \cdot x - x\\ \mathbf{elif}\;x \leq 0.65:\\ \;\;\;\;0.918938533204673 - y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \end{array} \]

Alternative 7: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. 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 0 100.0%

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

   1. cancel-sign-sub-inv 100.0%

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

   2. metadata-eval 100.0%

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

   3. sub-neg 100.0%

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

   4. metadata-eval 100.0%

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

   5. associate-+r+ 100.0%

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

   6. distribute-lft-in 100.0%

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

   7. *-commutative 100.0%

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

   8. neg-mul-1 100.0%

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

   9. +-commutative 100.0%

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

   10. associate-+l+ 100.0%

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

   11. distribute-rgt-in 100.0%

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

   12. metadata-eval 100.0%

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

   13. sub-neg 100.0%

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

   14. +-commutative 100.0%

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

   15. unsub-neg 100.0%

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

   16. sub-neg 100.0%

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

   17. metadata-eval 100.0%

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

6. Simplified 100.0%

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

7. Final simplification 100.0%

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

Alternative 8: 48.6% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.0))) (* y x) (- x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = y * x;
	} 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 ((y <= (-1.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = y * x
    else
        tmp = -x
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 53.5%

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

      1. sub-neg 53.5%

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

      2. distribute-rgt-in 53.4%

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

      3. metadata-eval 53.4%

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

      4. neg-mul-1 53.4%

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

   6. Applied egg-rr 53.4%

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

   7. Taylor expanded in y around inf 52.7%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 54.1%

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

      1. sub-neg 54.1%

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

      2. distribute-rgt-in 54.2%

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

      3. metadata-eval 54.2%

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

      4. neg-mul-1 54.2%

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

   6. Applied egg-rr 54.2%

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

   7. Taylor expanded in y around 0 52.1%

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

      1. mul-1-neg 52.1%

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

   9. Simplified 52.1%

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

4. Final simplification 52.4%

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

Alternative 9: 72.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -70000000 \lor \neg \left(y \leq 1.65\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -70000000.0) (not (<= y 1.65)))
   (* y x)
   (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -70000000.0) || !(y <= 1.65)) {
		tmp = y * x;
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-70000000.0d0)) .or. (.not. (y <= 1.65d0))) then
        tmp = y * x
    else
        tmp = 0.918938533204673d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -70000000.0) || !(y <= 1.65)) {
		tmp = y * x;
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -70000000.0) or not (y <= 1.65):
		tmp = y * x
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -70000000.0) || !(y <= 1.65))
		tmp = Float64(y * x);
	else
		tmp = Float64(0.918938533204673 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -70000000.0) || ~((y <= 1.65)))
		tmp = y * x;
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -70000000.0], N[Not[LessEqual[y, 1.65]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7e7 or 1.6499999999999999 < y

   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. Taylor expanded in x around inf 53.9%

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

      1. sub-neg 53.9%

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

      2. distribute-rgt-in 53.8%

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

      3. metadata-eval 53.8%

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

      4. neg-mul-1 53.8%

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

   6. Applied egg-rr 53.8%

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

   7. Taylor expanded in y around inf 53.1%

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

   if -7e7 < y < 1.6499999999999999

   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 96.9%

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

      1. neg-mul-1 96.9%

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

      2. unsub-neg 96.9%

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

   6. Simplified 96.9%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -70000000 \lor \neg \left(y \leq 1.65\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]

Alternative 10: 25.9% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ -x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- x))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return -x

Julia

function code(x, y)
	return Float64(-x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
2. 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 inf 53.8%

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

   1. sub-neg 53.8%

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

   2. distribute-rgt-in 53.8%

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

   3. metadata-eval 53.8%

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

   4. neg-mul-1 53.8%

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

6. Applied egg-rr 53.8%

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

7. Taylor expanded in y around 0 29.2%

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

   1. mul-1-neg 29.2%

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

9. Simplified 29.2%

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

10. Final simplification 29.2%

    \[\leadsto -x \]

Reproduce

herbie shell --seed 2023318 
(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))