Data.Colour.CIE.Chromaticity:chromaCoords from colour-2.3.3

Percentage Accurate: 100.0% → 100.0%

Time: 3.2s

Alternatives: 6

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(1 - x\right) - y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (- 1.0 x) y))

C

double code(double x, double y) {
	return (1.0 - x) - y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - x) - y
end function

Java

public static double code(double x, double y) {
	return (1.0 - x) - y;
}

Python

def code(x, y):
	return (1.0 - x) - y

Julia

function code(x, y)
	return Float64(Float64(1.0 - x) - y)
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - x) - y;
end

Wolfram

code[x_, y_] := N[(N[(1.0 - x), $MachinePrecision] - y), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - x\right) - y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (- 1.0 x) y))

C

double code(double x, double y) {
	return (1.0 - x) - y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - x) - y
end function

Java

public static double code(double x, double y) {
	return (1.0 - x) - y;
}

Python

def code(x, y):
	return (1.0 - x) - y

Julia

function code(x, y)
	return Float64(Float64(1.0 - x) - y)
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - x) - y;
end

Wolfram

code[x_, y_] := N[(N[(1.0 - x), $MachinePrecision] - y), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - x\right) - y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (- 1.0 x) y))

C

double code(double x, double y) {
	return (1.0 - x) - y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - x) - y
end function

Java

public static double code(double x, double y) {
	return (1.0 - x) - y;
}

Python

def code(x, y):
	return (1.0 - x) - y

Julia

function code(x, y)
	return Float64(Float64(1.0 - x) - y)
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - x) - y;
end

Wolfram

code[x_, y_] := N[(N[(1.0 - x), $MachinePrecision] - y), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(1 - x\right) - y \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 50.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.019:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-79} \lor \neg \left(x \leq 5.1 \cdot 10^{-222}\right):\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.019)
   (- x)
   (if (or (<= x -3.4e-79) (not (<= x 5.1e-222))) (- y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.019) {
		tmp = -x;
	} else if ((x <= -3.4e-79) || !(x <= 5.1e-222)) {
		tmp = -y;
	} else {
		tmp = 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 <= (-0.019d0)) then
        tmp = -x
    else if ((x <= (-3.4d-79)) .or. (.not. (x <= 5.1d-222))) then
        tmp = -y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.019) {
		tmp = -x;
	} else if ((x <= -3.4e-79) || !(x <= 5.1e-222)) {
		tmp = -y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.019:
		tmp = -x
	elif (x <= -3.4e-79) or not (x <= 5.1e-222):
		tmp = -y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.019)
		tmp = Float64(-x);
	elseif ((x <= -3.4e-79) || !(x <= 5.1e-222))
		tmp = Float64(-y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.019)
		tmp = -x;
	elseif ((x <= -3.4e-79) || ~((x <= 5.1e-222)))
		tmp = -y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.019], (-x), If[Or[LessEqual[x, -3.4e-79], N[Not[LessEqual[x, 5.1e-222]], $MachinePrecision]], (-y), 1.0]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0189999999999999995

   1. Initial program 100.0%

      \[\left(1 - x\right) - y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 97.4%

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

      1. neg-mul-1 97.4%

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

   5. Simplified 97.4%

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

      1. neg-mul-1 97.4%

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

      2. *-commutative 97.4%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 12.9%

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

      5. sqr-neg 12.9%

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

      6. sqrt-unprod 23.7%

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

      7. add-sqr-sqrt 23.7%

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

      8. add-sqr-sqrt 12.9%

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

      9. sqrt-unprod 2.8%

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

      10. sqr-neg 2.8%

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

      11. sqrt-unprod 0.2%

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

      12. add-sqr-sqrt 0.6%

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

      13. add-cube-cbrt 0.6%

          \[\leadsto \left(-x\right) \cdot -1 - \left(-\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}\right) \]

      14. distribute-rgt-neg-in 0.6%

          \[\leadsto \left(-x\right) \cdot -1 - \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(-\sqrt[3]{y}\right)} \]

      15. prod-diff 0.6%

          \[\leadsto \color{blue}{\mathsf{fma}\left(-x, -1, -\left(-\sqrt[3]{y}\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) + \mathsf{fma}\left(-\left(-\sqrt[3]{y}\right), \sqrt[3]{y} \cdot \sqrt[3]{y}, \left(-\sqrt[3]{y}\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right)} \]

   7. Applied egg-rr 71.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -1, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \mathsf{fma}\left(-\left(-\sqrt[3]{y}\right), {\left(\sqrt[3]{y}\right)}^{2}, \left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right)} \]
   8. Step-by-step derivation

      1. fma-undefine 71.3%

         \[\leadsto \color{blue}{\left(x \cdot -1 + \left(-\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right)\right)} + \mathsf{fma}\left(-\left(-\sqrt[3]{y}\right), {\left(\sqrt[3]{y}\right)}^{2}, \left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) \]

      2. *-commutative 71.3%

         \[\leadsto \left(\color{blue}{-1 \cdot x} + \left(-\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right)\right) + \mathsf{fma}\left(-\left(-\sqrt[3]{y}\right), {\left(\sqrt[3]{y}\right)}^{2}, \left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) \]

      3. fma-undefine 71.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right)} + \mathsf{fma}\left(-\left(-\sqrt[3]{y}\right), {\left(\sqrt[3]{y}\right)}^{2}, \left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) \]

      4. remove-double-neg 71.3%

         \[\leadsto \mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \mathsf{fma}\left(\color{blue}{\sqrt[3]{y}}, {\left(\sqrt[3]{y}\right)}^{2}, \left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) \]

      5. distribute-lft-neg-out 71.3%

         \[\leadsto \mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \mathsf{fma}\left(\sqrt[3]{y}, {\left(\sqrt[3]{y}\right)}^{2}, \color{blue}{-\sqrt[3]{y} \cdot {\left(\sqrt[3]{y}\right)}^{2}}\right) \]

      6. unpow2 71.3%

         \[\leadsto \mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \mathsf{fma}\left(\sqrt[3]{y}, {\left(\sqrt[3]{y}\right)}^{2}, -\sqrt[3]{y} \cdot \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}\right) \]

      7. rem-3cbrt-rft 71.3%

         \[\leadsto \mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \mathsf{fma}\left(\sqrt[3]{y}, {\left(\sqrt[3]{y}\right)}^{2}, -\color{blue}{y}\right) \]

      8. mul-1-neg 71.3%

         \[\leadsto \mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \mathsf{fma}\left(\sqrt[3]{y}, {\left(\sqrt[3]{y}\right)}^{2}, \color{blue}{-1 \cdot y}\right) \]

      9. fma-define 71.3%

         \[\leadsto \mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \color{blue}{\left(\sqrt[3]{y} \cdot {\left(\sqrt[3]{y}\right)}^{2} + -1 \cdot y\right)} \]

      10. unpow2 71.3%

          \[\leadsto \mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \left(\sqrt[3]{y} \cdot \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)} + -1 \cdot y\right) \]

      11. rem-3cbrt-rft 71.3%

          \[\leadsto \mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \left(\color{blue}{y} + -1 \cdot y\right) \]

      12. distribute-rgt1-in 71.3%

          \[\leadsto \mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \color{blue}{\left(-1 + 1\right) \cdot y} \]

      13. metadata-eval 71.3%

          \[\leadsto \mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \color{blue}{0} \cdot y \]

      14. mul0-lft 71.3%

          \[\leadsto \mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \color{blue}{0} \]

      15. +-rgt-identity 71.3%

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right)} \]

   9. Simplified 71.3%

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

   10. Taylor expanded in y around 0 72.2%

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

       1. neg-mul-1 72.2%

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

   12. Simplified 72.2%

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

   if -0.0189999999999999995 < x < -3.39999999999999976e-79 or 5.1000000000000002e-222 < x

   1. Initial program 100.0%

      \[\left(1 - x\right) - y \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 42.9%

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

      1. neg-mul-1 42.9%

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

   5. Simplified 42.9%

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

   if -3.39999999999999976e-79 < x < 5.1000000000000002e-222

   1. Initial program 100.0%

      \[\left(1 - x\right) - y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 100.0%

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

   4. Taylor expanded in y around 0 61.3%

      \[\leadsto \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.019:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-79} \lor \neg \left(x \leq 5.1 \cdot 10^{-222}\right):\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.5% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (if (<= x -1.0) (- (- x) y) (- 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = -x - y;
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = -x - y
    else
        tmp = 1.0d0 - y
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = -x - y
	else:
		tmp = 1.0 - y
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[((-x) - y), $MachinePrecision], N[(1.0 - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 100.0%

      \[\left(1 - x\right) - y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 98.8%

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

      1. neg-mul-1 98.8%

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

   5. Simplified 98.8%

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

   if -1 < x

   1. Initial program 100.0%

      \[\left(1 - x\right) - y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 76.3%

      \[\leadsto \color{blue}{1} - y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 74.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -65:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -65.0) (- x) (- 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -65.0) {
		tmp = -x;
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-65.0d0)) then
        tmp = -x
    else
        tmp = 1.0d0 - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -65.0) {
		tmp = -x;
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -65.0:
		tmp = -x
	else:
		tmp = 1.0 - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -65.0)
		tmp = Float64(-x);
	else
		tmp = Float64(1.0 - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -65.0)
		tmp = -x;
	else
		tmp = 1.0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -65.0], (-x), N[(1.0 - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -65

   1. Initial program 100.0%

      \[\left(1 - x\right) - y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 98.8%

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

      1. neg-mul-1 98.8%

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

   5. Simplified 98.8%

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

      1. neg-mul-1 98.8%

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

      2. *-commutative 98.8%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 13.1%

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

      5. sqr-neg 13.1%

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

      6. sqrt-unprod 24.1%

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

      7. add-sqr-sqrt 24.1%

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

      8. add-sqr-sqrt 13.1%

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

      9. sqrt-unprod 2.8%

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

      10. sqr-neg 2.8%

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

      11. sqrt-unprod 0.2%

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

      12. add-sqr-sqrt 0.6%

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

      13. add-cube-cbrt 0.6%

          \[\leadsto \left(-x\right) \cdot -1 - \left(-\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}\right) \]

      14. distribute-rgt-neg-in 0.6%

          \[\leadsto \left(-x\right) \cdot -1 - \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(-\sqrt[3]{y}\right)} \]

      15. prod-diff 0.6%

          \[\leadsto \color{blue}{\mathsf{fma}\left(-x, -1, -\left(-\sqrt[3]{y}\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) + \mathsf{fma}\left(-\left(-\sqrt[3]{y}\right), \sqrt[3]{y} \cdot \sqrt[3]{y}, \left(-\sqrt[3]{y}\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right)} \]

   7. Applied egg-rr 72.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -1, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \mathsf{fma}\left(-\left(-\sqrt[3]{y}\right), {\left(\sqrt[3]{y}\right)}^{2}, \left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right)} \]
   8. Step-by-step derivation

      1. fma-undefine 72.2%

         \[\leadsto \color{blue}{\left(x \cdot -1 + \left(-\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right)\right)} + \mathsf{fma}\left(-\left(-\sqrt[3]{y}\right), {\left(\sqrt[3]{y}\right)}^{2}, \left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) \]

      2. *-commutative 72.2%

         \[\leadsto \left(\color{blue}{-1 \cdot x} + \left(-\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right)\right) + \mathsf{fma}\left(-\left(-\sqrt[3]{y}\right), {\left(\sqrt[3]{y}\right)}^{2}, \left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) \]

      3. fma-undefine 72.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right)} + \mathsf{fma}\left(-\left(-\sqrt[3]{y}\right), {\left(\sqrt[3]{y}\right)}^{2}, \left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) \]

      4. remove-double-neg 72.2%

         \[\leadsto \mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \mathsf{fma}\left(\color{blue}{\sqrt[3]{y}}, {\left(\sqrt[3]{y}\right)}^{2}, \left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) \]

      5. distribute-lft-neg-out 72.2%

         \[\leadsto \mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \mathsf{fma}\left(\sqrt[3]{y}, {\left(\sqrt[3]{y}\right)}^{2}, \color{blue}{-\sqrt[3]{y} \cdot {\left(\sqrt[3]{y}\right)}^{2}}\right) \]

      6. unpow2 72.2%

         \[\leadsto \mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \mathsf{fma}\left(\sqrt[3]{y}, {\left(\sqrt[3]{y}\right)}^{2}, -\sqrt[3]{y} \cdot \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}\right) \]

      7. rem-3cbrt-rft 72.2%

         \[\leadsto \mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \mathsf{fma}\left(\sqrt[3]{y}, {\left(\sqrt[3]{y}\right)}^{2}, -\color{blue}{y}\right) \]

      8. mul-1-neg 72.2%

         \[\leadsto \mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \mathsf{fma}\left(\sqrt[3]{y}, {\left(\sqrt[3]{y}\right)}^{2}, \color{blue}{-1 \cdot y}\right) \]

      9. fma-define 72.2%

         \[\leadsto \mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \color{blue}{\left(\sqrt[3]{y} \cdot {\left(\sqrt[3]{y}\right)}^{2} + -1 \cdot y\right)} \]

      10. unpow2 72.2%

          \[\leadsto \mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \left(\sqrt[3]{y} \cdot \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)} + -1 \cdot y\right) \]

      11. rem-3cbrt-rft 72.2%

          \[\leadsto \mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \left(\color{blue}{y} + -1 \cdot y\right) \]

      12. distribute-rgt1-in 72.2%

          \[\leadsto \mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \color{blue}{\left(-1 + 1\right) \cdot y} \]

      13. metadata-eval 72.2%

          \[\leadsto \mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \color{blue}{0} \cdot y \]

      14. mul0-lft 72.2%

          \[\leadsto \mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \color{blue}{0} \]

      15. +-rgt-identity 72.2%

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right)} \]

   9. Simplified 72.2%

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

   10. Taylor expanded in y around 0 73.1%

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

       1. neg-mul-1 73.1%

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

   12. Simplified 73.1%

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

   if -65 < x

   1. Initial program 100.0%

      \[\left(1 - x\right) - y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 76.3%

      \[\leadsto \color{blue}{1} - y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 43.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -1.0) (- x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = -x;
	} else {
		tmp = 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 <= (-1.0d0)) then
        tmp = -x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = -x
	else:
		tmp = 1.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], (-x), 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 100.0%

      \[\left(1 - x\right) - y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 98.8%

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

      1. neg-mul-1 98.8%

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

   5. Simplified 98.8%

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

      1. neg-mul-1 98.8%

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

      2. *-commutative 98.8%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 13.1%

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

      5. sqr-neg 13.1%

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

      6. sqrt-unprod 24.1%

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

      7. add-sqr-sqrt 24.1%

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

      8. add-sqr-sqrt 13.1%

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

      9. sqrt-unprod 2.8%

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

      10. sqr-neg 2.8%

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

      11. sqrt-unprod 0.2%

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

      12. add-sqr-sqrt 0.6%

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

      13. add-cube-cbrt 0.6%

          \[\leadsto \left(-x\right) \cdot -1 - \left(-\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}\right) \]

      14. distribute-rgt-neg-in 0.6%

          \[\leadsto \left(-x\right) \cdot -1 - \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(-\sqrt[3]{y}\right)} \]

      15. prod-diff 0.6%

          \[\leadsto \color{blue}{\mathsf{fma}\left(-x, -1, -\left(-\sqrt[3]{y}\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) + \mathsf{fma}\left(-\left(-\sqrt[3]{y}\right), \sqrt[3]{y} \cdot \sqrt[3]{y}, \left(-\sqrt[3]{y}\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right)} \]

   7. Applied egg-rr 72.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -1, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \mathsf{fma}\left(-\left(-\sqrt[3]{y}\right), {\left(\sqrt[3]{y}\right)}^{2}, \left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right)} \]
   8. Step-by-step derivation

      1. fma-undefine 72.2%

         \[\leadsto \color{blue}{\left(x \cdot -1 + \left(-\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right)\right)} + \mathsf{fma}\left(-\left(-\sqrt[3]{y}\right), {\left(\sqrt[3]{y}\right)}^{2}, \left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) \]

      2. *-commutative 72.2%

         \[\leadsto \left(\color{blue}{-1 \cdot x} + \left(-\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right)\right) + \mathsf{fma}\left(-\left(-\sqrt[3]{y}\right), {\left(\sqrt[3]{y}\right)}^{2}, \left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) \]

      3. fma-undefine 72.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right)} + \mathsf{fma}\left(-\left(-\sqrt[3]{y}\right), {\left(\sqrt[3]{y}\right)}^{2}, \left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) \]

      4. remove-double-neg 72.2%

         \[\leadsto \mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \mathsf{fma}\left(\color{blue}{\sqrt[3]{y}}, {\left(\sqrt[3]{y}\right)}^{2}, \left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) \]

      5. distribute-lft-neg-out 72.2%

         \[\leadsto \mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \mathsf{fma}\left(\sqrt[3]{y}, {\left(\sqrt[3]{y}\right)}^{2}, \color{blue}{-\sqrt[3]{y} \cdot {\left(\sqrt[3]{y}\right)}^{2}}\right) \]

      6. unpow2 72.2%

         \[\leadsto \mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \mathsf{fma}\left(\sqrt[3]{y}, {\left(\sqrt[3]{y}\right)}^{2}, -\sqrt[3]{y} \cdot \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}\right) \]

      7. rem-3cbrt-rft 72.2%

         \[\leadsto \mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \mathsf{fma}\left(\sqrt[3]{y}, {\left(\sqrt[3]{y}\right)}^{2}, -\color{blue}{y}\right) \]

      8. mul-1-neg 72.2%

         \[\leadsto \mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \mathsf{fma}\left(\sqrt[3]{y}, {\left(\sqrt[3]{y}\right)}^{2}, \color{blue}{-1 \cdot y}\right) \]

      9. fma-define 72.2%

         \[\leadsto \mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \color{blue}{\left(\sqrt[3]{y} \cdot {\left(\sqrt[3]{y}\right)}^{2} + -1 \cdot y\right)} \]

      10. unpow2 72.2%

          \[\leadsto \mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \left(\sqrt[3]{y} \cdot \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)} + -1 \cdot y\right) \]

      11. rem-3cbrt-rft 72.2%

          \[\leadsto \mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \left(\color{blue}{y} + -1 \cdot y\right) \]

      12. distribute-rgt1-in 72.2%

          \[\leadsto \mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \color{blue}{\left(-1 + 1\right) \cdot y} \]

      13. metadata-eval 72.2%

          \[\leadsto \mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \color{blue}{0} \cdot y \]

      14. mul0-lft 72.2%

          \[\leadsto \mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right) + \color{blue}{0} \]

      15. +-rgt-identity 72.2%

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1, x, -\left(-\sqrt[3]{y}\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}\right)} \]

   9. Simplified 72.2%

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

   10. Taylor expanded in y around 0 73.1%

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

       1. neg-mul-1 73.1%

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

   12. Simplified 73.1%

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

   if -1 < x

   1. Initial program 100.0%

      \[\left(1 - x\right) - y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 76.3%

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

   4. Taylor expanded in y around 0 35.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 26.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

   \[\left(1 - x\right) - y \]
2. Add Preprocessing

3. Taylor expanded in x around 0 64.8%

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

4. Taylor expanded in y around 0 28.6%

   \[\leadsto \color{blue}{1} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024116 
(FPCore (x y)
  :name "Data.Colour.CIE.Chromaticity:chromaCoords from colour-2.3.3"
  :precision binary64
  (- (- 1.0 x) y))