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

Percentage Accurate: 100.0% → 100.0%

Time: 6.4s

Alternatives: 10

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: 97.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+28} \lor \neg \left(t\_0 \leq 1\right):\\ \;\;\;\;\mathsf{fma}\left(-0.5 + x, y, -x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (- (* x (- y 1.0)) (* y 0.5)) 0.918938533204673)))
   (if (or (<= t_0 -1e+28) (not (<= t_0 1.0)))
     (fma (+ -0.5 x) y (- x))
     (fma -0.5 y 0.918938533204673))))

C

double code(double x, double y) {
	double t_0 = ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
	double tmp;
	if ((t_0 <= -1e+28) || !(t_0 <= 1.0)) {
		tmp = fma((-0.5 + x), y, -x);
	} else {
		tmp = fma(-0.5, y, 0.918938533204673);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x * Float64(y - 1.0)) - Float64(y * 0.5)) + 0.918938533204673)
	tmp = 0.0
	if ((t_0 <= -1e+28) || !(t_0 <= 1.0))
		tmp = fma(Float64(-0.5 + x), y, Float64(-x));
	else
		tmp = fma(-0.5, y, 0.918938533204673);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x * N[(y - 1.0), $MachinePrecision]), $MachinePrecision] - N[(y * 0.5), $MachinePrecision]), $MachinePrecision] + 0.918938533204673), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e+28], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]], N[(N[(-0.5 + x), $MachinePrecision] * y + (-x)), $MachinePrecision], N[(-0.5 * y + 0.918938533204673), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (*.f64 x (-.f64 y #s(literal 1 binary64))) (*.f64 y #s(literal 1/2 binary64))) #s(literal 918938533204673/1000000000000000 binary64)) < -9.99999999999999958e27 or 1 < (+.f64 (-.f64 (*.f64 x (-.f64 y #s(literal 1 binary64))) (*.f64 y #s(literal 1/2 binary64))) #s(literal 918938533204673/1000000000000000 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 99.5%

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

   if -9.99999999999999958e27 < (+.f64 (-.f64 (*.f64 x (-.f64 y #s(literal 1 binary64))) (*.f64 y #s(literal 1/2 binary64))) #s(literal 918938533204673/1000000000000000 binary64)) < 1

   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 97.0

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

   5. Applied rewrites 97.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \leq -1 \cdot 10^{+28} \lor \neg \left(\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \leq 1\right):\\ \;\;\;\;\mathsf{fma}\left(-0.5 + x, y, -x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -46:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-12}:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -46.0)
   (* y x)
   (if (<= y 1.05e-12)
     (- 0.918938533204673 x)
     (if (<= y 1.5e+118) (fma -0.5 y 0.918938533204673) (* y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -46.0) {
		tmp = y * x;
	} else if (y <= 1.05e-12) {
		tmp = 0.918938533204673 - x;
	} else if (y <= 1.5e+118) {
		tmp = fma(-0.5, y, 0.918938533204673);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -46.0)
		tmp = Float64(y * x);
	elseif (y <= 1.05e-12)
		tmp = Float64(0.918938533204673 - x);
	elseif (y <= 1.5e+118)
		tmp = fma(-0.5, y, 0.918938533204673);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -46.0], N[(y * x), $MachinePrecision], If[LessEqual[y, 1.05e-12], N[(0.918938533204673 - x), $MachinePrecision], If[LessEqual[y, 1.5e+118], N[(-0.5 * y + 0.918938533204673), $MachinePrecision], N[(y * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -46 or 1.5e118 < 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 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 69.7%

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

   if -46 < y < 1.04999999999999997e-12

   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 98.0

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

   5. Applied rewrites 98.0%

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

   if 1.04999999999999997e-12 < y < 1.5e118

   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 64.6

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

   5. Applied rewrites 64.6%

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

Alternative 4: 73.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -46:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+118}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -46.0)
   (* y x)
   (if (<= y 1.85)
     (- 0.918938533204673 x)
     (if (<= y 1.5e+118) (* -0.5 y) (* y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -46.0) {
		tmp = y * x;
	} else if (y <= 1.85) {
		tmp = 0.918938533204673 - x;
	} else if (y <= 1.5e+118) {
		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 <= (-46.0d0)) then
        tmp = y * x
    else if (y <= 1.85d0) then
        tmp = 0.918938533204673d0 - x
    else if (y <= 1.5d+118) 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 <= -46.0) {
		tmp = y * x;
	} else if (y <= 1.85) {
		tmp = 0.918938533204673 - x;
	} else if (y <= 1.5e+118) {
		tmp = -0.5 * y;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -46.0:
		tmp = y * x
	elif y <= 1.85:
		tmp = 0.918938533204673 - x
	elif y <= 1.5e+118:
		tmp = -0.5 * y
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -46.0)
		tmp = Float64(y * x);
	elseif (y <= 1.85)
		tmp = Float64(0.918938533204673 - x);
	elseif (y <= 1.5e+118)
		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 <= -46.0)
		tmp = y * x;
	elseif (y <= 1.85)
		tmp = 0.918938533204673 - x;
	elseif (y <= 1.5e+118)
		tmp = -0.5 * y;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -46.0], N[(y * x), $MachinePrecision], If[LessEqual[y, 1.85], N[(0.918938533204673 - x), $MachinePrecision], If[LessEqual[y, 1.5e+118], N[(-0.5 * y), $MachinePrecision], N[(y * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -46 or 1.5e118 < 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 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 69.7%

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

   if -46 < y < 1.8500000000000001

   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 96.3

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

   5. Applied rewrites 96.3%

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

   if 1.8500000000000001 < y < 1.5e118

   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 95.0

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

   5. Applied rewrites 95.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 61.1%

      \[\leadsto -0.5 \cdot y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.66 \lor \neg \left(x \leq 0.72\right):\\ \;\;\;\;\left(-1 + y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.66) (not (<= x 0.72)))
   (* (+ -1.0 y) x)
   (fma -0.5 y 0.918938533204673)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.66) || !(x <= 0.72)) {
		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 <= -0.66) || !(x <= 0.72))
		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, -0.66], N[Not[LessEqual[x, 0.72]], $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 < -0.660000000000000031 or 0.71999999999999997 < 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 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 58.6

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

   5. Applied rewrites 58.6%

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

   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

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

      8. mul-1-neg N/A

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

      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. mul-1-neg N/A

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

      12. distribute-lft-neg-in 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 98.6

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

   8. Applied rewrites 98.6%

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

   if -0.660000000000000031 < x < 0.71999999999999997

   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 97.3

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

   5. Applied rewrites 97.3%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.66 \lor \neg \left(x \leq 0.72\right):\\ \;\;\;\;\left(-1 + y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.66:\\ \;\;\;\;\left(-1 + y\right) \cdot x\\ \mathbf{elif}\;x \leq 0.72:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, -x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.66)
   (* (+ -1.0 y) x)
   (if (<= x 0.72) (fma -0.5 y 0.918938533204673) (fma y x (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.66) {
		tmp = (-1.0 + y) * x;
	} else if (x <= 0.72) {
		tmp = fma(-0.5, y, 0.918938533204673);
	} else {
		tmp = fma(y, x, -x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.66)
		tmp = Float64(Float64(-1.0 + y) * x);
	elseif (x <= 0.72)
		tmp = fma(-0.5, y, 0.918938533204673);
	else
		tmp = fma(y, x, Float64(-x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.66], N[(N[(-1.0 + y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 0.72], N[(-0.5 * y + 0.918938533204673), $MachinePrecision], N[(y * x + (-x)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.660000000000000031

   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 55.2

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

   5. Applied rewrites 55.2%

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

   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

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

      8. mul-1-neg N/A

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

      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. mul-1-neg N/A

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

      12. distribute-lft-neg-in 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 97.6

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

   8. Applied rewrites 97.6%

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

   if -0.660000000000000031 < x < 0.71999999999999997

   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 97.3

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

   5. Applied rewrites 97.3%

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

   if 0.71999999999999997 < 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 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 62.4

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

   5. Applied rewrites 62.4%

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

   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

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

      8. mul-1-neg N/A

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

      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. mul-1-neg N/A

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

      12. distribute-lft-neg-in 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 99.7

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

   8. Applied rewrites 99.7%

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

   10. Applied rewrites 99.7%

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{x}, -x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 74.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -46 \lor \neg \left(y \leq 1.05\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -46.0) (not (<= y 1.05))) (* y x) (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -46.0) || !(y <= 1.05)) {
		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 <= (-46.0d0)) .or. (.not. (y <= 1.05d0))) 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 <= -46.0) || !(y <= 1.05)) {
		tmp = y * x;
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -46.0) or not (y <= 1.05):
		tmp = y * x
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -46.0) || !(y <= 1.05))
		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 <= -46.0) || ~((y <= 1.05)))
		tmp = y * x;
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -46.0], N[Not[LessEqual[y, 1.05]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -46 or 1.05000000000000004 < 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 inf

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

   8. Applied rewrites 60.2%

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

   if -46 < y < 1.05000000000000004

   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 96.3

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

   5. Applied rewrites 96.3%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -46 \lor \neg \left(y \leq 1.05\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 48.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.92 \lor \neg \left(x \leq 1.15 \cdot 10^{+24}\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.92) (not (<= x 1.15e+24))) (- x) 0.918938533204673))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.92) || !(x <= 1.15e+24)) {
		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 <= 1.15d+24))) 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 <= 1.15e+24)) {
		tmp = -x;
	} else {
		tmp = 0.918938533204673;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -0.92) or not (x <= 1.15e+24):
		tmp = -x
	else:
		tmp = 0.918938533204673
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.92) || !(x <= 1.15e+24))
		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 <= 1.15e+24)))
		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, 1.15e+24]], $MachinePrecision]], (-x), 0.918938533204673]

Derivation

1. Split input into 2 regimes

2. if x < -0.92000000000000004 or 1.15e24 < 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 44.1

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

   5. Applied rewrites 44.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 43.5%

      \[\leadsto -x \]

   if -0.92000000000000004 < x < 1.15e24

   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 46.0

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

   5. Applied rewrites 46.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 44.5%

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

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.92 \lor \neg \left(x \leq 1.15 \cdot 10^{+24}\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.4% 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 45.0

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

5. Applied rewrites 45.0%

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

Alternative 10: 25.7% 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 45.0

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

5. Applied rewrites 45.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 21.2%

   \[\leadsto 0.918938533204673 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024338 
(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))