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

Percentage Accurate: 99.9% → 100.0%

Time: 6.9s

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(x * 1.5 + 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. flip3-+ N/A

      \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{x - y}{2}\right)}^{3}}{x \cdot x + \left(\frac{x - y}{2} \cdot \frac{x - y}{2} - x \cdot \frac{x - y}{2}\right)}} \]

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip3-+ N/A

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

   7. lift-+.f64 N/A

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

   8. lower-/.f64 99.7

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-/.f64 N/A

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

   12. div-inv N/A

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

   13. lower-fma.f64 N/A

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

   14. metadata-eval 99.7

       \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(x - y, \color{blue}{0.5}, x\right)}} \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(x - y, 0.5, x\right)}}} \]

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. metadata-eval N/A

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

   2. distribute-rgt1-in N/A

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

   3. lower-fma.f64 N/A

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

   4. distribute-rgt1-in N/A

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

   5. metadata-eval N/A

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

   6. lower-*.f64 99.9

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

7. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, y, 1.5 \cdot x\right)} \]
8. Step-by-step derivation

9. Applied rewrites 100.0%

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

10. Final simplification 100.0%

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

Alternative 2: 75.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{+105}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-23}:\\ \;\;\;\;x + y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.15e+105)
   (* x 1.5)
   (if (<= x 1.5e-23) (+ x (* y -0.5)) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.15e+105) {
		tmp = x * 1.5;
	} else if (x <= 1.5e-23) {
		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 <= (-2.15d+105)) then
        tmp = x * 1.5d0
    else if (x <= 1.5d-23) 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 <= -2.15e+105) {
		tmp = x * 1.5;
	} else if (x <= 1.5e-23) {
		tmp = x + (y * -0.5);
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.15e+105:
		tmp = x * 1.5
	elif x <= 1.5e-23:
		tmp = x + (y * -0.5)
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.15e+105)
		tmp = Float64(x * 1.5);
	elseif (x <= 1.5e-23)
		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 <= -2.15e+105)
		tmp = x * 1.5;
	elseif (x <= 1.5e-23)
		tmp = x + (y * -0.5);
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.15e+105], N[(x * 1.5), $MachinePrecision], If[LessEqual[x, 1.5e-23], N[(x + N[(y * -0.5), $MachinePrecision]), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1500000000000001e105 or 1.50000000000000001e-23 < 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. *-commutative N/A

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

      2. lower-*.f64 78.7

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

   5. Applied rewrites 78.7%

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

   if -2.1500000000000001e105 < x < 1.50000000000000001e-23

   1. Initial program 99.9%

      \[x + \frac{x - y}{2} \]
   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 80.6

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

   5. Applied rewrites 80.6%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{+105}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-23}:\\ \;\;\;\;x + y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+65}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-35}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.8e+65) (* x 1.5) (if (<= x 4.4e-35) (* y -0.5) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.8e+65) {
		tmp = x * 1.5;
	} else if (x <= 4.4e-35) {
		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+65)) then
        tmp = x * 1.5d0
    else if (x <= 4.4d-35) 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+65) {
		tmp = x * 1.5;
	} else if (x <= 4.4e-35) {
		tmp = y * -0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.8e+65:
		tmp = x * 1.5
	elif x <= 4.4e-35:
		tmp = y * -0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.8e+65)
		tmp = Float64(x * 1.5);
	elseif (x <= 4.4e-35)
		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+65)
		tmp = x * 1.5;
	elseif (x <= 4.4e-35)
		tmp = y * -0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.8e+65], N[(x * 1.5), $MachinePrecision], If[LessEqual[x, 4.4e-35], N[(y * -0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.80000000000000011e65 or 4.39999999999999987e-35 < 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. *-commutative N/A

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

      2. lower-*.f64 76.6

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

   5. Applied rewrites 76.6%

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

   if -3.80000000000000011e65 < x < 4.39999999999999987e-35

   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 78.8

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

   5. Applied rewrites 78.8%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+65}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-35}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
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: 50.6% 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 50.9

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

5. Applied rewrites 50.9%

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

6. Final simplification 50.9%

   \[\leadsto y \cdot -0.5 \]
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 2024220 
(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)))