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

Percentage Accurate: 99.9% → 100.0%

Time: 6.8s

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. div-sub N/A

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

   2. associate-+r- N/A

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

   3. lower--.f64 N/A

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

   4. +-commutative N/A

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

   5. div-inv N/A

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

   6. lower-fma.f64 N/A

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

   7. metadata-eval N/A

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

   8. div-inv N/A

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

   9. lower-*.f64 N/A

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

   10. metadata-eval 99.9

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

4. Applied egg-rr 99.9%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. sub-neg N/A

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

   4. lift-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. distribute-lft1-in N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. metadata-eval N/A

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

   11. lift-*.f64 N/A

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

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

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

   13. metadata-eval N/A

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

   14. lower-*.f64 100.0

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

6. Applied egg-rr 100.0%

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

Alternative 2: 77.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-41}:\\ \;\;\;\;\mathsf{fma}\left(y, -0.5, x\right)\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{-36}:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.4e-41)
   (fma y -0.5 x)
   (if (<= y 7.4e-36) (* 1.5 x) (fma y -0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.4e-41) {
		tmp = fma(y, -0.5, x);
	} else if (y <= 7.4e-36) {
		tmp = 1.5 * x;
	} else {
		tmp = fma(y, -0.5, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.4e-41)
		tmp = fma(y, -0.5, x);
	elseif (y <= 7.4e-36)
		tmp = Float64(1.5 * x);
	else
		tmp = fma(y, -0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.4e-41], N[(y * -0.5 + x), $MachinePrecision], If[LessEqual[y, 7.4e-36], N[(1.5 * x), $MachinePrecision], N[(y * -0.5 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.40000000000000022e-41 or 7.40000000000000003e-36 < y

   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 83.6

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

   5. Simplified 83.6%

      \[\leadsto x + \color{blue}{-0.5 \cdot y} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 83.6

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

   7. Applied egg-rr 83.6%

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

   if -2.40000000000000022e-41 < y < 7.40000000000000003e-36

   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 81.2

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

   5. Simplified 81.2%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-41}:\\ \;\;\;\;\mathsf{fma}\left(y, -0.5, x\right)\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{-36}:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -0.5, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{-21}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{-36}:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6.6e-21) (* y -0.5) (if (<= y 7.4e-36) (* 1.5 x) (* y -0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6.6e-21) {
		tmp = y * -0.5;
	} else if (y <= 7.4e-36) {
		tmp = 1.5 * x;
	} 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 <= (-6.6d-21)) then
        tmp = y * (-0.5d0)
    else if (y <= 7.4d-36) then
        tmp = 1.5d0 * x
    else
        tmp = y * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -6.6e-21) {
		tmp = y * -0.5;
	} else if (y <= 7.4e-36) {
		tmp = 1.5 * x;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -6.6e-21:
		tmp = y * -0.5
	elif y <= 7.4e-36:
		tmp = 1.5 * x
	else:
		tmp = y * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6.6e-21)
		tmp = Float64(y * -0.5);
	elseif (y <= 7.4e-36)
		tmp = Float64(1.5 * x);
	else
		tmp = Float64(y * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -6.6e-21)
		tmp = y * -0.5;
	elseif (y <= 7.4e-36)
		tmp = 1.5 * x;
	else
		tmp = y * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6.6e-21], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 7.4e-36], N[(1.5 * x), $MachinePrecision], N[(y * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.60000000000000018e-21 or 7.40000000000000003e-36 < y

   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 80.7

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

   5. Simplified 80.7%

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

   if -6.60000000000000018e-21 < y < 7.40000000000000003e-36

   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 80.8

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

   5. Simplified 80.8%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{-21}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{-36}:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]
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

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

   2. lift-/.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. div-inv N/A

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

   7. lower-fma.f64 N/A

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

   8. neg-sub0 N/A

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

   9. lift--.f64 N/A

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

   10. sub-neg N/A

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

   11. +-commutative N/A

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

   12. associate--r+ N/A

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

   13. neg-sub0 N/A

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

   14. remove-double-neg N/A

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

   15. lower--.f64 N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 5: 50.0% 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 51.6

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

5. Simplified 51.6%

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

6. Final simplification 51.6%

   \[\leadsto y \cdot -0.5 \]
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 2024207 
(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)))