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

Percentage Accurate: 99.9% → 100.0%

Time: 5.1s

Alternatives: 5

Speedup: 1.8×

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, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(1.5, x, y \cdot -0.5\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. div-sub N/A

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

   6. sub-neg N/A

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

   7. associate-+l+ N/A

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

   8. div-inv N/A

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

   9. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, \left(\mathsf{neg}\left(\frac{y}{2}\right)\right) + x\right)} \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{2}}, \left(\mathsf{neg}\left(\frac{y}{2}\right)\right) + x\right) \]

   11. div-inv N/A

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

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

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{\mathsf{fma}\left(y, \frac{1}{\mathsf{neg}\left(2\right)}, x\right)}\right) \]

   18. metadata-eval N/A

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

   19. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot y + \frac{3}{2} \cdot x} \]
6. Step-by-step derivation

   1. +-commutative N/A

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

   2. metadata-eval N/A

      \[\leadsto \frac{3}{2} \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot y \]

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

      \[\leadsto \frac{3}{2} \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot y\right)\right)} \]

   4. lower-fma.f64 N/A

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

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

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

   6. metadata-eval N/A

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

   7. lower-*.f64 100.0

      \[\leadsto \mathsf{fma}\left(1.5, x, \color{blue}{-0.5 \cdot y}\right) \]

7. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(1.5, x, -0.5 \cdot y\right)} \]

8. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(1.5, x, y \cdot -0.5\right) \]
9. Add Preprocessing

Alternative 2: 77.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + y \cdot -0.5\\ \mathbf{if}\;y \leq -1.85 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-25}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (* y -0.5))))
   (if (<= y -1.85e-6) t_0 (if (<= y 1.15e-25) (* x 1.5) t_0))))

C

double code(double x, double y) {
	double t_0 = x + (y * -0.5);
	double tmp;
	if (y <= -1.85e-6) {
		tmp = t_0;
	} else if (y <= 1.15e-25) {
		tmp = x * 1.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (y * (-0.5d0))
    if (y <= (-1.85d-6)) then
        tmp = t_0
    else if (y <= 1.15d-25) then
        tmp = x * 1.5d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x + (y * -0.5);
	double tmp;
	if (y <= -1.85e-6) {
		tmp = t_0;
	} else if (y <= 1.15e-25) {
		tmp = x * 1.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + (y * -0.5)
	tmp = 0
	if y <= -1.85e-6:
		tmp = t_0
	elif y <= 1.15e-25:
		tmp = x * 1.5
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(y * -0.5))
	tmp = 0.0
	if (y <= -1.85e-6)
		tmp = t_0;
	elseif (y <= 1.15e-25)
		tmp = Float64(x * 1.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + (y * -0.5);
	tmp = 0.0;
	if (y <= -1.85e-6)
		tmp = t_0;
	elseif (y <= 1.15e-25)
		tmp = x * 1.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(y * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.85e-6], t$95$0, If[LessEqual[y, 1.15e-25], N[(x * 1.5), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8500000000000001e-6 or 1.15e-25 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 76.8

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

   5. Applied rewrites 76.8%

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

   if -1.8500000000000001e-6 < y < 1.15e-25

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 80.8

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

   5. Applied rewrites 80.8%

      \[\leadsto \color{blue}{x \cdot 1.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-6}:\\ \;\;\;\;x + y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-25}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{-5}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 880000000000:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.35e-5)
   (* y -0.5)
   (if (<= y 880000000000.0) (* x 1.5) (* y -0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.35e-5) {
		tmp = y * -0.5;
	} else if (y <= 880000000000.0) {
		tmp = x * 1.5;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.35d-5)) then
        tmp = y * (-0.5d0)
    else if (y <= 880000000000.0d0) then
        tmp = x * 1.5d0
    else
        tmp = y * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.35e-5) {
		tmp = y * -0.5;
	} else if (y <= 880000000000.0) {
		tmp = x * 1.5;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.35e-5:
		tmp = y * -0.5
	elif y <= 880000000000.0:
		tmp = x * 1.5
	else:
		tmp = y * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.35e-5)
		tmp = Float64(y * -0.5);
	elseif (y <= 880000000000.0)
		tmp = Float64(x * 1.5);
	else
		tmp = Float64(y * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.35e-5)
		tmp = y * -0.5;
	elseif (y <= 880000000000.0)
		tmp = x * 1.5;
	else
		tmp = y * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.35e-5], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 880000000000.0], N[(x * 1.5), $MachinePrecision], N[(y * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3499999999999999e-5 or 8.8e11 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

   if -1.3499999999999999e-5 < y < 8.8e11

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 78.5

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

   5. Applied rewrites 78.5%

      \[\leadsto \color{blue}{x \cdot 1.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{-5}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 880000000000:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. frac-2neg N/A

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

   5. div-inv N/A

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

   6. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\left(x - y\right)\right), \frac{1}{\mathsf{neg}\left(2\right)}, x\right)} \]

   7. neg-sub0 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \left(x - y\right)}, \frac{1}{\mathsf{neg}\left(2\right)}, x\right) \]

   8. lift--.f64 N/A

      \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(x - y\right)}, \frac{1}{\mathsf{neg}\left(2\right)}, x\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(x + \left(\mathsf{neg}\left(y\right)\right)\right)}, \frac{1}{\mathsf{neg}\left(2\right)}, x\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + x\right)}, \frac{1}{\mathsf{neg}\left(2\right)}, x\right) \]

   11. associate--r+ N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - x}, \frac{1}{\mathsf{neg}\left(2\right)}, x\right) \]

   12. neg-sub0 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)} - x, \frac{1}{\mathsf{neg}\left(2\right)}, x\right) \]

   13. remove-double-neg N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{y} - x, \frac{1}{\mathsf{neg}\left(2\right)}, x\right) \]

   14. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{1}{\mathsf{neg}\left(2\right)}, x\right) \]

   15. metadata-eval N/A

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

   16. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

Alternative 5: 50.7% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. lower-*.f64 47.5

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

5. Applied rewrites 47.5%

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

6. Final simplification 47.5%

   \[\leadsto y \cdot -0.5 \]
7. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.3× 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 2024234 
(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)))