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

Percentage Accurate: 99.9% → 100.0%

Time: 3.4s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. distribute-rgt1-in N/A

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

   4. metadata-eval N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 100.0

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

5. Applied rewrites 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 76.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -400:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-96}:\\ \;\;\;\;-0.5 \cdot y + x\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -400.0) (* 1.5 x) (if (<= x 1.05e-96) (+ (* -0.5 y) x) (* 1.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -400.0) {
		tmp = 1.5 * x;
	} else if (x <= 1.05e-96) {
		tmp = (-0.5 * y) + x;
	} else {
		tmp = 1.5 * 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 <= (-400.0d0)) then
        tmp = 1.5d0 * x
    else if (x <= 1.05d-96) then
        tmp = ((-0.5d0) * y) + x
    else
        tmp = 1.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -400.0) {
		tmp = 1.5 * x;
	} else if (x <= 1.05e-96) {
		tmp = (-0.5 * y) + x;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -400.0:
		tmp = 1.5 * x
	elif x <= 1.05e-96:
		tmp = (-0.5 * y) + x
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -400.0)
		tmp = Float64(1.5 * x);
	elseif (x <= 1.05e-96)
		tmp = Float64(Float64(-0.5 * y) + x);
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -400.0)
		tmp = 1.5 * x;
	elseif (x <= 1.05e-96)
		tmp = (-0.5 * y) + x;
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -400.0], N[(1.5 * x), $MachinePrecision], If[LessEqual[x, 1.05e-96], N[(N[(-0.5 * y), $MachinePrecision] + x), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -400 or 1.05000000000000001e-96 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 80.8

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

   5. Applied rewrites 80.8%

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

   if -400 < x < 1.05000000000000001e-96

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 85.6

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

   5. Applied rewrites 85.6%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -400:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-96}:\\ \;\;\;\;-0.5 \cdot y + x\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -85:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{elif}\;x \leq 10^{-96}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -85.0) (* 1.5 x) (if (<= x 1e-96) (* -0.5 y) (* 1.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -85.0) {
		tmp = 1.5 * x;
	} else if (x <= 1e-96) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * 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 <= (-85.0d0)) then
        tmp = 1.5d0 * x
    else if (x <= 1d-96) then
        tmp = (-0.5d0) * y
    else
        tmp = 1.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -85.0) {
		tmp = 1.5 * x;
	} else if (x <= 1e-96) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -85.0:
		tmp = 1.5 * x
	elif x <= 1e-96:
		tmp = -0.5 * y
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -85.0)
		tmp = Float64(1.5 * x);
	elseif (x <= 1e-96)
		tmp = Float64(-0.5 * y);
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -85.0)
		tmp = 1.5 * x;
	elseif (x <= 1e-96)
		tmp = -0.5 * y;
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -85.0], N[(1.5 * x), $MachinePrecision], If[LessEqual[x, 1e-96], N[(-0.5 * y), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -85 or 9.9999999999999991e-97 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 80.8

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

   5. Applied rewrites 80.8%

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

   if -85 < x < 9.9999999999999991e-97

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 83.2

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

   5. Applied rewrites 83.2%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -85:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{elif}\;x \leq 10^{-96}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \]
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: 51.1% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

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

   1. distribute-rgt1-in N/A

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

   2. metadata-eval N/A

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

   3. lower-*.f64 56.0

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

5. Applied rewrites 56.0%

   \[\leadsto \color{blue}{1.5 \cdot x} \]
6. 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 2024255 
(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)))