Graphics.Rendering.Chart.Axis.Types:hBufferRect from Chart-1.5.3

Percentage Accurate: 99.9% → 100.0%

Time: 8.0s

Alternatives: 8

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ x + \frac{x - y}{2} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (/ (- x y) 2.0)))

C

double code(double x, double y) {
	return x + ((x - y) / 2.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + ((x - y) / 2.0d0)
end function

Java

public static double code(double x, double y) {
	return x + ((x - y) / 2.0);
}

Python

def code(x, y):
	return x + ((x - y) / 2.0)

Julia

function code(x, y)
	return Float64(x + Float64(Float64(x - y) / 2.0))
end

MATLAB

function tmp = code(x, y)
	tmp = x + ((x - y) / 2.0);
end

Wolfram

code[x_, y_] := N[(x + N[(N[(x - y), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{x - y}{2} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (/ (- x y) 2.0)))

C

double code(double x, double y) {
	return x + ((x - y) / 2.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + ((x - y) / 2.0d0)
end function

Java

public static double code(double x, double y) {
	return x + ((x - y) / 2.0);
}

Python

def code(x, y):
	return x + ((x - y) / 2.0)

Julia

function code(x, y)
	return Float64(x + Float64(Float64(x - y) / 2.0))
end

MATLAB

function tmp = code(x, y)
	tmp = x + ((x - y) / 2.0);
end

Wolfram

code[x_, y_] := N[(x + N[(N[(x - y), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 1.5, \frac{y}{-2}\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma x 1.5 (/ y -2.0)))

C

double code(double x, double y) {
	return fma(x, 1.5, (y / -2.0));
}

Julia

function code(x, y)
	return fma(x, 1.5, Float64(y / -2.0))
end

Wolfram

code[x_, y_] := N[(x * 1.5 + N[(y / -2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x + \frac{x - y}{2} \]
2. Step-by-step derivation

   1. div-sub N/A

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

   2. sub-neg N/A

      \[\leadsto x + \left(\frac{x}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{y}{2}\right)\right)}\right) \]

   3. associate-+r+ N/A

      \[\leadsto \left(x + \frac{x}{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{y}{2}\right)\right)} \]

   4. +-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{y}{2}\right)\right) + \color{blue}{\left(x + \frac{x}{2}\right)} \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{y}{2}\right)\right), \color{blue}{\left(x + \frac{x}{2}\right)}\right) \]

   6. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{y}{\mathsf{neg}\left(2\right)}\right), \left(\color{blue}{x} + \frac{x}{2}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(2\right)\right)\right), \left(\color{blue}{x} + \frac{x}{2}\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \frac{x}{2}\right)\right) \]

   9. remove-double-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)}{2}\right)\right) \]

   10. neg-mul-1 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \frac{-1 \cdot \left(\mathsf{neg}\left(x\right)\right)}{2}\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \frac{\left(\mathsf{neg}\left(x\right)\right) \cdot -1}{2}\right)\right) \]

   12. associate-/l* N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\frac{-1}{2}}\right)\right) \]

   13. cancel-sign-sub-inv N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x - \color{blue}{x \cdot \frac{-1}{2}}\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x - \frac{-1}{2} \cdot \color{blue}{x}\right)\right) \]

   15. cancel-sign-sub-inv N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot x}\right)\right) \]

   16. *-lft-identity N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(1 \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)} \cdot x\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x + \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{2}}\right)\right) \cdot x\right)\right) \]

   18. distribute-rgt-out N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right)}\right)\right) \]

   19. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right)}\right)\right) \]

   20. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \mathsf{*.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{2}}\right)\right)\right)\right)\right) \]

   21. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \mathsf{*.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right)\right)\right) \]

   22. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \mathsf{*.f64}\left(x, \left(1 + \frac{1}{2}\right)\right)\right) \]

   23. metadata-eval 99.9%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \mathsf{*.f64}\left(x, \frac{3}{2}\right)\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{y}{-2} + x \cdot 1.5} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \frac{3}{2} + \color{blue}{\frac{y}{-2}} \]

   2. fma-define N/A

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

   3. fma-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\frac{3}{2}}, \left(\frac{y}{-2}\right)\right) \]

   4. /-lowering-/.f64 100.0%

      \[\leadsto \mathsf{fma.f64}\left(x, \frac{3}{2}, \mathsf{/.f64}\left(y, -2\right)\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1.5, \frac{y}{-2}\right)} \]
7. Add Preprocessing

Alternative 2: 73.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+67}:\\ \;\;\;\;\frac{1.5}{\frac{1}{x}}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-133}:\\ \;\;\;\;x + y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.6e+67)
   (/ 1.5 (/ 1.0 x))
   (if (<= x 6e-133) (+ x (* y -0.5)) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.6e+67) {
		tmp = 1.5 / (1.0 / x);
	} else if (x <= 6e-133) {
		tmp = x + (y * -0.5);
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.6d+67)) then
        tmp = 1.5d0 / (1.0d0 / x)
    else if (x <= 6d-133) then
        tmp = x + (y * (-0.5d0))
    else
        tmp = x * 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.6e+67) {
		tmp = 1.5 / (1.0 / x);
	} else if (x <= 6e-133) {
		tmp = x + (y * -0.5);
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.6e+67:
		tmp = 1.5 / (1.0 / x)
	elif x <= 6e-133:
		tmp = x + (y * -0.5)
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.6e+67)
		tmp = Float64(1.5 / Float64(1.0 / x));
	elseif (x <= 6e-133)
		tmp = Float64(x + Float64(y * -0.5));
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.6e+67)
		tmp = 1.5 / (1.0 / x);
	elseif (x <= 6e-133)
		tmp = x + (y * -0.5);
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.6e+67], N[(1.5 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6e-133], N[(x + N[(y * -0.5), $MachinePrecision]), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.59999999999999991e67

   1. Initial program 99.7%

      \[x + \frac{x - y}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{3}{2} \cdot x} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 80.1%

         \[\leadsto \mathsf{*.f64}\left(\frac{3}{2}, \color{blue}{x}\right) \]

   5. Simplified 80.1%

      \[\leadsto \color{blue}{1.5 \cdot x} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{2}{3}} \cdot x \]

      2. associate-/r/ N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{2}{3}}{x}}} \]

      3. div-inv N/A

         \[\leadsto \frac{1}{\frac{2}{3} \cdot \color{blue}{\frac{1}{x}}} \]

      4. associate-/r* N/A

         \[\leadsto \frac{\frac{1}{\frac{2}{3}}}{\color{blue}{\frac{1}{x}}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\frac{3}{2}}{\frac{\color{blue}{1}}{x}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{3}{2}, \color{blue}{\left(\frac{1}{x}\right)}\right) \]

      7. /-lowering-/.f64 80.2%

         \[\leadsto \mathsf{/.f64}\left(\frac{3}{2}, \mathsf{/.f64}\left(1, \color{blue}{x}\right)\right) \]

   7. Applied egg-rr 80.2%

      \[\leadsto \color{blue}{\frac{1.5}{\frac{1}{x}}} \]

   if -1.59999999999999991e67 < x < 6.00000000000000038e-133

   1. Initial program 99.9%

      \[x + \frac{x - y}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot y\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 85.4%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{y}\right)\right) \]

   5. Simplified 85.4%

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

   if 6.00000000000000038e-133 < x

   1. Initial program 99.8%

      \[x + \frac{x - y}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{3}{2} \cdot x} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 74.8%

         \[\leadsto \mathsf{*.f64}\left(\frac{3}{2}, \color{blue}{x}\right) \]

   5. Simplified 74.8%

      \[\leadsto \color{blue}{1.5 \cdot x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+67}:\\ \;\;\;\;\frac{1.5}{\frac{1}{x}}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-133}:\\ \;\;\;\;x + y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{+67}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-133}:\\ \;\;\;\;x + y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.55e+67)
   (* x 1.5)
   (if (<= x 5.8e-133) (+ x (* y -0.5)) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.55e+67) {
		tmp = x * 1.5;
	} else if (x <= 5.8e-133) {
		tmp = x + (y * -0.5);
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.55d+67)) then
        tmp = x * 1.5d0
    else if (x <= 5.8d-133) then
        tmp = x + (y * (-0.5d0))
    else
        tmp = x * 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.55e+67) {
		tmp = x * 1.5;
	} else if (x <= 5.8e-133) {
		tmp = x + (y * -0.5);
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.55e+67:
		tmp = x * 1.5
	elif x <= 5.8e-133:
		tmp = x + (y * -0.5)
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.55e+67)
		tmp = Float64(x * 1.5);
	elseif (x <= 5.8e-133)
		tmp = Float64(x + Float64(y * -0.5));
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.55e+67)
		tmp = x * 1.5;
	elseif (x <= 5.8e-133)
		tmp = x + (y * -0.5);
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.55e+67], N[(x * 1.5), $MachinePrecision], If[LessEqual[x, 5.8e-133], N[(x + N[(y * -0.5), $MachinePrecision]), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5500000000000001e67 or 5.7999999999999997e-133 < x

   1. Initial program 99.8%

      \[x + \frac{x - y}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{3}{2} \cdot x} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 76.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{3}{2}, \color{blue}{x}\right) \]

   5. Simplified 76.6%

      \[\leadsto \color{blue}{1.5 \cdot x} \]

   if -2.5500000000000001e67 < x < 5.7999999999999997e-133

   1. Initial program 99.9%

      \[x + \frac{x - y}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot y\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 85.4%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{y}\right)\right) \]

   5. Simplified 85.4%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{+67}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-133}:\\ \;\;\;\;x + y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+67}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-133}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.8e+67) (* x 1.5) (if (<= x 5.7e-133) (* y -0.5) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.8e+67) {
		tmp = x * 1.5;
	} else if (x <= 5.7e-133) {
		tmp = y * -0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.8d+67)) then
        tmp = x * 1.5d0
    else if (x <= 5.7d-133) then
        tmp = y * (-0.5d0)
    else
        tmp = x * 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.8e+67) {
		tmp = x * 1.5;
	} else if (x <= 5.7e-133) {
		tmp = y * -0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.8e+67:
		tmp = x * 1.5
	elif x <= 5.7e-133:
		tmp = y * -0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.8e+67)
		tmp = Float64(x * 1.5);
	elseif (x <= 5.7e-133)
		tmp = Float64(y * -0.5);
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.8e+67)
		tmp = x * 1.5;
	elseif (x <= 5.7e-133)
		tmp = y * -0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.8e+67], N[(x * 1.5), $MachinePrecision], If[LessEqual[x, 5.7e-133], N[(y * -0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.8000000000000002e67 or 5.6999999999999997e-133 < x

   1. Initial program 99.8%

      \[x + \frac{x - y}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{3}{2} \cdot x} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 76.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{3}{2}, \color{blue}{x}\right) \]

   5. Simplified 76.6%

      \[\leadsto \color{blue}{1.5 \cdot x} \]

   if -3.8000000000000002e67 < x < 5.6999999999999997e-133

   1. Initial program 99.9%

      \[x + \frac{x - y}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 82.9%

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{y}\right) \]

   5. Simplified 82.9%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+67}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-133}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{+176}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 1.95e+176) (* y -0.5) x))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.95e+176) {
		tmp = y * -0.5;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.95d+176) then
        tmp = y * (-0.5d0)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.95e+176) {
		tmp = y * -0.5;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.95e+176:
		tmp = y * -0.5
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.95e+176)
		tmp = Float64(y * -0.5);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.95e+176)
		tmp = y * -0.5;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.95e+176], N[(y * -0.5), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < 1.9500000000000001e176

   1. Initial program 99.9%

      \[x + \frac{x - y}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 60.1%

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{y}\right) \]

   5. Simplified 60.1%

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

   if 1.9500000000000001e176 < x

   1. Initial program 99.8%

      \[x + \frac{x - y}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot y\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 24.4%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{y}\right)\right) \]

   5. Simplified 24.4%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 18.8%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{+176}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{y}{-2} + x \cdot 1.5 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (/ y -2.0) (* x 1.5)))

C

double code(double x, double y) {
	return (y / -2.0) + (x * 1.5);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (y / (-2.0d0)) + (x * 1.5d0)
end function

Java

public static double code(double x, double y) {
	return (y / -2.0) + (x * 1.5);
}

Python

def code(x, y):
	return (y / -2.0) + (x * 1.5)

Julia

function code(x, y)
	return Float64(Float64(y / -2.0) + Float64(x * 1.5))
end

MATLAB

function tmp = code(x, y)
	tmp = (y / -2.0) + (x * 1.5);
end

Wolfram

code[x_, y_] := N[(N[(y / -2.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x + \frac{x - y}{2} \]
2. Step-by-step derivation

   1. div-sub N/A

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

   2. sub-neg N/A

      \[\leadsto x + \left(\frac{x}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{y}{2}\right)\right)}\right) \]

   3. associate-+r+ N/A

      \[\leadsto \left(x + \frac{x}{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{y}{2}\right)\right)} \]

   4. +-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{y}{2}\right)\right) + \color{blue}{\left(x + \frac{x}{2}\right)} \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{y}{2}\right)\right), \color{blue}{\left(x + \frac{x}{2}\right)}\right) \]

   6. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{y}{\mathsf{neg}\left(2\right)}\right), \left(\color{blue}{x} + \frac{x}{2}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(2\right)\right)\right), \left(\color{blue}{x} + \frac{x}{2}\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \frac{x}{2}\right)\right) \]

   9. remove-double-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)}{2}\right)\right) \]

   10. neg-mul-1 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \frac{-1 \cdot \left(\mathsf{neg}\left(x\right)\right)}{2}\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \frac{\left(\mathsf{neg}\left(x\right)\right) \cdot -1}{2}\right)\right) \]

   12. associate-/l* N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\frac{-1}{2}}\right)\right) \]

   13. cancel-sign-sub-inv N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x - \color{blue}{x \cdot \frac{-1}{2}}\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x - \frac{-1}{2} \cdot \color{blue}{x}\right)\right) \]

   15. cancel-sign-sub-inv N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot x}\right)\right) \]

   16. *-lft-identity N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(1 \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)} \cdot x\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x + \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{2}}\right)\right) \cdot x\right)\right) \]

   18. distribute-rgt-out N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \left(x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right)}\right)\right) \]

   19. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right)}\right)\right) \]

   20. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \mathsf{*.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{2}}\right)\right)\right)\right)\right) \]

   21. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \mathsf{*.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right)\right)\right) \]

   22. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \mathsf{*.f64}\left(x, \left(1 + \frac{1}{2}\right)\right)\right) \]

   23. metadata-eval 99.9%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, -2\right), \mathsf{*.f64}\left(x, \frac{3}{2}\right)\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{y}{-2} + x \cdot 1.5} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 7: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{x - y}{2} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (/ (- x y) 2.0)))

C

double code(double x, double y) {
	return x + ((x - y) / 2.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + ((x - y) / 2.0d0)
end function

Java

public static double code(double x, double y) {
	return x + ((x - y) / 2.0);
}

Python

def code(x, y):
	return x + ((x - y) / 2.0)

Julia

function code(x, y)
	return Float64(x + Float64(Float64(x - y) / 2.0))
end

MATLAB

function tmp = code(x, y)
	tmp = x + ((x - y) / 2.0);
end

Wolfram

code[x_, y_] := N[(x + N[(N[(x - y), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x + \frac{x - y}{2} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 8: 11.6% accurate, 7.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 99.9%

   \[x + \frac{x - y}{2} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot y\right)}\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 60.3%

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{y}\right)\right) \]

5. Simplified 60.3%

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

6. Taylor expanded in x around inf

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

8. Simplified 11.4%

   \[\leadsto \color{blue}{x} \]
9. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1.5 \cdot x - 0.5 \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (* 1.5 x) (* 0.5 y)))

C

double code(double x, double y) {
	return (1.5 * x) - (0.5 * y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.5d0 * x) - (0.5d0 * y)
end function

Java

public static double code(double x, double y) {
	return (1.5 * x) - (0.5 * y);
}

Python

def code(x, y):
	return (1.5 * x) - (0.5 * y)

Julia

function code(x, y)
	return Float64(Float64(1.5 * x) - Float64(0.5 * y))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.5 * x) - (0.5 * y);
end

Wolfram

code[x_, y_] := N[(N[(1.5 * x), $MachinePrecision] - N[(0.5 * y), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024155 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Axis.Types:hBufferRect from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* 3/2 x) (* 1/2 y)))

  (+ x (/ (- x y) 2.0)))