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

Percentage Accurate: 100.0% → 100.0%

Time: 5.5s

Alternatives: 9

Speedup: 1.5×

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, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(\frac{918938533204673}{1000000000000000} + x \cdot \left(y - 1\right)\right) - \frac{1}{2} \cdot y} \]

5. Applied rewrites 100.0%

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

Alternative 2: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y - 1\right) - y \cdot 0.5\\ \mathbf{if}\;t\_0 \leq -1000000 \lor \neg \left(t\_0 \leq 100\right):\\ \;\;\;\;\mathsf{fma}\left(-0.5 + x, y, -x\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (* x (- y 1.0)) (* y 0.5))))
   (if (or (<= t_0 -1000000.0) (not (<= t_0 100.0)))
     (fma (+ -0.5 x) y (- x))
     (- 0.918938533204673 x))))

C

double code(double x, double y) {
	double t_0 = (x * (y - 1.0)) - (y * 0.5);
	double tmp;
	if ((t_0 <= -1000000.0) || !(t_0 <= 100.0)) {
		tmp = fma((-0.5 + x), y, -x);
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(y - 1.0)) - Float64(y * 0.5))
	tmp = 0.0
	if ((t_0 <= -1000000.0) || !(t_0 <= 100.0))
		tmp = fma(Float64(-0.5 + x), y, Float64(-x));
	else
		tmp = Float64(0.918938533204673 - x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(y - 1.0), $MachinePrecision]), $MachinePrecision] - N[(y * 0.5), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1000000.0], N[Not[LessEqual[t$95$0, 100.0]], $MachinePrecision]], N[(N[(-0.5 + x), $MachinePrecision] * y + (-x)), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x (-.f64 y #s(literal 1 binary64))) (*.f64 y #s(literal 1/2 binary64))) < -1e6 or 100 < (-.f64 (*.f64 x (-.f64 y #s(literal 1 binary64))) (*.f64 y #s(literal 1/2 binary64)))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{918938533204673}{1000000000000000} + x \cdot \left(y - 1\right)\right) - \frac{1}{2} \cdot y} \]

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2} + x, y, -1 \cdot x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.3%

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

   if -1e6 < (-.f64 (*.f64 x (-.f64 y #s(literal 1 binary64))) (*.f64 y #s(literal 1/2 binary64))) < 100

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
   4. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - \left(\mathsf{neg}\left(-1\right)\right) \cdot x} \]

      2. metadata-eval N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{1} \cdot x \]

      3. *-lft-identity N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{x} \]

      4. lower--.f64 98.2

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

   5. Applied rewrites 98.2%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(y - 1\right) - y \cdot 0.5 \leq -1000000 \lor \neg \left(x \cdot \left(y - 1\right) - y \cdot 0.5 \leq 100\right):\\ \;\;\;\;\mathsf{fma}\left(-0.5 + x, y, -x\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.06 \cdot 10^{+97}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -2.85 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-10}:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+133}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.06e+97)
   (* y x)
   (if (<= y -2.85e-8)
     (fma -0.5 y 0.918938533204673)
     (if (<= y 3.8e-10)
       (- 0.918938533204673 x)
       (if (<= y 2.8e+133) (fma -0.5 y 0.918938533204673) (* y x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.06e+97) {
		tmp = y * x;
	} else if (y <= -2.85e-8) {
		tmp = fma(-0.5, y, 0.918938533204673);
	} else if (y <= 3.8e-10) {
		tmp = 0.918938533204673 - x;
	} else if (y <= 2.8e+133) {
		tmp = fma(-0.5, y, 0.918938533204673);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.06e+97)
		tmp = Float64(y * x);
	elseif (y <= -2.85e-8)
		tmp = fma(-0.5, y, 0.918938533204673);
	elseif (y <= 3.8e-10)
		tmp = Float64(0.918938533204673 - x);
	elseif (y <= 2.8e+133)
		tmp = fma(-0.5, y, 0.918938533204673);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.06e+97], N[(y * x), $MachinePrecision], If[LessEqual[y, -2.85e-8], N[(-0.5 * y + 0.918938533204673), $MachinePrecision], If[LessEqual[y, 3.8e-10], N[(0.918938533204673 - x), $MachinePrecision], If[LessEqual[y, 2.8e+133], N[(-0.5 * y + 0.918938533204673), $MachinePrecision], N[(y * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.05999999999999994e97 or 2.80000000000000016e133 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(x - \frac{1}{2}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot y} \]

      3. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 62.1%

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

   if -1.05999999999999994e97 < y < -2.85000000000000004e-8 or 3.7999999999999998e-10 < y < 2.80000000000000016e133

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - \frac{1}{2} \cdot y} \]
   4. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot y} \]

      2. metadata-eval N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\frac{-1}{2}} \cdot y \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot y + \frac{918938533204673}{1000000000000000}} \]

      4. lower-fma.f64 65.9

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

   5. Applied rewrites 65.9%

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

   if -2.85000000000000004e-8 < y < 3.7999999999999998e-10

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
   4. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - \left(\mathsf{neg}\left(-1\right)\right) \cdot x} \]

      2. metadata-eval N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{1} \cdot x \]

      3. *-lft-identity N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{x} \]

      4. lower--.f64 99.4

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

   5. Applied rewrites 99.4%

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

Alternative 4: 72.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.06 \cdot 10^{+97}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -102:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq 1.65:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+133}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.06e+97)
   (* y x)
   (if (<= y -102.0)
     (* -0.5 y)
     (if (<= y 1.65)
       (- 0.918938533204673 x)
       (if (<= y 2.8e+133) (* -0.5 y) (* y x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.06e+97) {
		tmp = y * x;
	} else if (y <= -102.0) {
		tmp = -0.5 * y;
	} else if (y <= 1.65) {
		tmp = 0.918938533204673 - x;
	} else if (y <= 2.8e+133) {
		tmp = -0.5 * y;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.06d+97)) then
        tmp = y * x
    else if (y <= (-102.0d0)) then
        tmp = (-0.5d0) * y
    else if (y <= 1.65d0) then
        tmp = 0.918938533204673d0 - x
    else if (y <= 2.8d+133) then
        tmp = (-0.5d0) * y
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.06e+97) {
		tmp = y * x;
	} else if (y <= -102.0) {
		tmp = -0.5 * y;
	} else if (y <= 1.65) {
		tmp = 0.918938533204673 - x;
	} else if (y <= 2.8e+133) {
		tmp = -0.5 * y;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.06e+97:
		tmp = y * x
	elif y <= -102.0:
		tmp = -0.5 * y
	elif y <= 1.65:
		tmp = 0.918938533204673 - x
	elif y <= 2.8e+133:
		tmp = -0.5 * y
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.06e+97)
		tmp = Float64(y * x);
	elseif (y <= -102.0)
		tmp = Float64(-0.5 * y);
	elseif (y <= 1.65)
		tmp = Float64(0.918938533204673 - x);
	elseif (y <= 2.8e+133)
		tmp = Float64(-0.5 * y);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.06e+97)
		tmp = y * x;
	elseif (y <= -102.0)
		tmp = -0.5 * y;
	elseif (y <= 1.65)
		tmp = 0.918938533204673 - x;
	elseif (y <= 2.8e+133)
		tmp = -0.5 * y;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.06e+97], N[(y * x), $MachinePrecision], If[LessEqual[y, -102.0], N[(-0.5 * y), $MachinePrecision], If[LessEqual[y, 1.65], N[(0.918938533204673 - x), $MachinePrecision], If[LessEqual[y, 2.8e+133], N[(-0.5 * y), $MachinePrecision], N[(y * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.05999999999999994e97 or 2.80000000000000016e133 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(x - \frac{1}{2}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot y} \]

      3. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 62.1%

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

   if -1.05999999999999994e97 < y < -102 or 1.6499999999999999 < y < 2.80000000000000016e133

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(x - \frac{1}{2}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot y} \]

      3. lower--.f64 93.4

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

   5. Applied rewrites 93.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{-1}{2} \cdot \color{blue}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 63.6%

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

   if -102 < y < 1.6499999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
   4. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - \left(\mathsf{neg}\left(-1\right)\right) \cdot x} \]

      2. metadata-eval N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{1} \cdot x \]

      3. *-lft-identity N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{x} \]

      4. lower--.f64 98.1

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

   5. Applied rewrites 98.1%

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

Alternative 5: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.65 \lor \neg \left(x \leq 0.6\right):\\ \;\;\;\;\left(-1 + y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.65) (not (<= x 0.6)))
   (* (+ -1.0 y) x)
   (fma -0.5 y 0.918938533204673)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.65) || !(x <= 0.6)) {
		tmp = (-1.0 + y) * x;
	} else {
		tmp = fma(-0.5, y, 0.918938533204673);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.65) || !(x <= 0.6))
		tmp = Float64(Float64(-1.0 + y) * x);
	else
		tmp = fma(-0.5, y, 0.918938533204673);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.65], N[Not[LessEqual[x, 0.6]], $MachinePrecision]], N[(N[(-1.0 + y), $MachinePrecision] * x), $MachinePrecision], N[(-0.5 * y + 0.918938533204673), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.650000000000000022 or 0.599999999999999978 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{918938533204673}{1000000000000000} + x \cdot \left(y - 1\right)\right) - \frac{1}{2} \cdot y} \]

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

      1. distribute-lft-out-- N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. distribute-lft-neg-in N/A

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

      6. fp-cancel-sub-sign N/A

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

      7. mul-1-neg N/A

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

      8. mul-1-neg N/A

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

      9. distribute-lft-in N/A

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

      10. +-commutative N/A

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

      11. associate-*r* N/A

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

      12. mul-1-neg N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. distribute-neg-in N/A

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

      17. metadata-eval N/A

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

      18. mul-1-neg N/A

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

      19. remove-double-neg N/A

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

      20. lower-+.f64 98.8

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

   8. Applied rewrites 98.8%

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

   if -0.650000000000000022 < x < 0.599999999999999978

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - \frac{1}{2} \cdot y} \]
   4. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot y} \]

      2. metadata-eval N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\frac{-1}{2}} \cdot y \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot y + \frac{918938533204673}{1000000000000000}} \]

      4. lower-fma.f64 97.2

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

   5. Applied rewrites 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.65 \lor \neg \left(x \leq 0.6\right):\\ \;\;\;\;\left(-1 + y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -102 \lor \neg \left(y \leq 1.65\right):\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -102.0) (not (<= y 1.65))) (* -0.5 y) (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -102.0) || !(y <= 1.65)) {
		tmp = -0.5 * y;
	} 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 <= (-102.0d0)) .or. (.not. (y <= 1.65d0))) then
        tmp = (-0.5d0) * y
    else
        tmp = 0.918938533204673d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -102.0) || !(y <= 1.65)) {
		tmp = -0.5 * y;
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -102.0) or not (y <= 1.65):
		tmp = -0.5 * y
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -102.0) || !(y <= 1.65))
		tmp = Float64(-0.5 * y);
	else
		tmp = Float64(0.918938533204673 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -102.0) || ~((y <= 1.65)))
		tmp = -0.5 * y;
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -102.0], N[Not[LessEqual[y, 1.65]], $MachinePrecision]], N[(-0.5 * y), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -102 or 1.6499999999999999 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(x - \frac{1}{2}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot y} \]

      3. lower--.f64 97.6

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{-1}{2} \cdot \color{blue}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.2%

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

   if -102 < y < 1.6499999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
   4. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - \left(\mathsf{neg}\left(-1\right)\right) \cdot x} \]

      2. metadata-eval N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{1} \cdot x \]

      3. *-lft-identity N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{x} \]

      4. lower--.f64 98.1

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

   5. Applied rewrites 98.1%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -102 \lor \neg \left(y \leq 1.65\right):\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.92 \lor \neg \left(x \leq 770000000000\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.92) (not (<= x 770000000000.0))) (- x) 0.918938533204673))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.92) || !(x <= 770000000000.0)) {
		tmp = -x;
	} else {
		tmp = 0.918938533204673;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-0.92d0)) .or. (.not. (x <= 770000000000.0d0))) then
        tmp = -x
    else
        tmp = 0.918938533204673d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -0.92) || !(x <= 770000000000.0)) {
		tmp = -x;
	} else {
		tmp = 0.918938533204673;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -0.92) or not (x <= 770000000000.0):
		tmp = -x
	else:
		tmp = 0.918938533204673
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.92) || !(x <= 770000000000.0))
		tmp = Float64(-x);
	else
		tmp = 0.918938533204673;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -0.92) || ~((x <= 770000000000.0)))
		tmp = -x;
	else
		tmp = 0.918938533204673;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.92], N[Not[LessEqual[x, 770000000000.0]], $MachinePrecision]], (-x), 0.918938533204673]

Derivation

1. Split input into 2 regimes

2. if x < -0.92000000000000004 or 7.7e11 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
   4. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - \left(\mathsf{neg}\left(-1\right)\right) \cdot x} \]

      2. metadata-eval N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{1} \cdot x \]

      3. *-lft-identity N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{x} \]

      4. lower--.f64 50.1

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

   5. Applied rewrites 50.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 49.5%

      \[\leadsto -x \]

   if -0.92000000000000004 < x < 7.7e11

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
   4. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - \left(\mathsf{neg}\left(-1\right)\right) \cdot x} \]

      2. metadata-eval N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{1} \cdot x \]

      3. *-lft-identity N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{x} \]

      4. lower--.f64 46.3

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

   5. Applied rewrites 46.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{918938533204673}{1000000000000000} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.3%

      \[\leadsto 0.918938533204673 \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.92 \lor \neg \left(x \leq 770000000000\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.8% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 0.918938533204673 - x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
4. Step-by-step derivation

   1. fp-cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - \left(\mathsf{neg}\left(-1\right)\right) \cdot x} \]

   2. metadata-eval N/A

      \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{1} \cdot x \]

   3. *-lft-identity N/A

      \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{x} \]

   4. lower--.f64 48.3

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

5. Applied rewrites 48.3%

   \[\leadsto \color{blue}{0.918938533204673 - x} \]
6. Add Preprocessing

Alternative 9: 26.7% accurate, 20.0× 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. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
4. Step-by-step derivation

   1. fp-cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - \left(\mathsf{neg}\left(-1\right)\right) \cdot x} \]

   2. metadata-eval N/A

      \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{1} \cdot x \]

   3. *-lft-identity N/A

      \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{x} \]

   4. lower--.f64 48.3

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

5. Applied rewrites 48.3%

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

6. Taylor expanded in x around 0

   \[\leadsto \frac{918938533204673}{1000000000000000} \]
7. Step-by-step derivation

8. Applied rewrites 23.0%

   \[\leadsto 0.918938533204673 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024320 
(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))