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

Percentage Accurate: 99.9% → 100.0%

Time: 5.3s

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 99.8%

   \[x + \frac{x - y}{2} \]
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. flip-+ N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. div-inv N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. clear-num N/A

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

   12. associate-/r/ N/A

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

   13. associate-*r* N/A

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

   14. lower-*.f64 N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

       \[\leadsto \frac{\left(x - y\right) \cdot \left(\color{blue}{\frac{1}{4}} \cdot \left(x - y\right)\right) - x \cdot x}{\frac{x - y}{2} - x} \]

   18. lower-*.f64 N/A

       \[\leadsto \frac{\left(x - y\right) \cdot \left(\frac{1}{4} \cdot \left(x - y\right)\right) - \color{blue}{x \cdot x}}{\frac{x - y}{2} - x} \]

   19. sub-neg N/A

       \[\leadsto \frac{\left(x - y\right) \cdot \left(\frac{1}{4} \cdot \left(x - y\right)\right) - x \cdot x}{\color{blue}{\frac{x - y}{2} + \left(\mathsf{neg}\left(x\right)\right)}} \]

4. Applied rewrites 52.7%

   \[\leadsto \color{blue}{\frac{\left(x - y\right) \cdot \left(0.25 \cdot \left(x - y\right)\right) - x \cdot x}{\mathsf{fma}\left(x - y, 0.5, -x\right)}} \]

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. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 100.0

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

7. Applied rewrites 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 75.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1900:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-89}:\\ \;\;\;\;x + y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1900.0) (* x 1.5) (if (<= x 2e-89) (+ x (* y -0.5)) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1900.0) {
		tmp = x * 1.5;
	} else if (x <= 2e-89) {
		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 <= (-1900.0d0)) then
        tmp = x * 1.5d0
    else if (x <= 2d-89) 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 <= -1900.0) {
		tmp = x * 1.5;
	} else if (x <= 2e-89) {
		tmp = x + (y * -0.5);
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1900.0:
		tmp = x * 1.5
	elif x <= 2e-89:
		tmp = x + (y * -0.5)
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1900.0)
		tmp = Float64(x * 1.5);
	elseif (x <= 2e-89)
		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 <= -1900.0)
		tmp = x * 1.5;
	elseif (x <= 2e-89)
		tmp = x + (y * -0.5);
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1900.0], N[(x * 1.5), $MachinePrecision], If[LessEqual[x, 2e-89], N[(x + N[(y * -0.5), $MachinePrecision]), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1900 or 2.00000000000000008e-89 < 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 70.6

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

   5. Applied rewrites 70.6%

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

   if -1900 < x < 2.00000000000000008e-89

   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 81.5

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

   5. Applied rewrites 81.5%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1900:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-89}:\\ \;\;\;\;x + y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. 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 2024228 
(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)))