Graphics.Rendering.Chart.Plot.Vectors:renderPlotVectors from Chart-1.5.3

Percentage Accurate: 78.2% → 100.0%

Time: 6.2s

Alternatives: 6

Speedup: 1.5×

Specification

Math

\[\begin{array}{l} \\ x + \left(1 - x\right) \cdot \left(1 - y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (* (- 1.0 x) (- 1.0 y))))

C

double code(double x, double y) {
	return x + ((1.0 - x) * (1.0 - y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + ((1.0d0 - x) * (1.0d0 - y))
end function

Java

public static double code(double x, double y) {
	return x + ((1.0 - x) * (1.0 - y));
}

Python

def code(x, y):
	return x + ((1.0 - x) * (1.0 - y))

Julia

function code(x, y)
	return Float64(x + Float64(Float64(1.0 - x) * Float64(1.0 - y)))
end

MATLAB

function tmp = code(x, y)
	tmp = x + ((1.0 - x) * (1.0 - y));
end

Wolfram

code[x_, y_] := N[(x + N[(N[(1.0 - x), $MachinePrecision] * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 78.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \left(1 - x\right) \cdot \left(1 - y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (* (- 1.0 x) (- 1.0 y))))

C

double code(double x, double y) {
	return x + ((1.0 - x) * (1.0 - y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + ((1.0d0 - x) * (1.0d0 - y))
end function

Java

public static double code(double x, double y) {
	return x + ((1.0 - x) * (1.0 - y));
}

Python

def code(x, y):
	return x + ((1.0 - x) * (1.0 - y))

Julia

function code(x, y)
	return Float64(x + Float64(Float64(1.0 - x) * Float64(1.0 - y)))
end

MATLAB

function tmp = code(x, y)
	tmp = x + ((1.0 - x) * (1.0 - y));
end

Wolfram

code[x_, y_] := N[(x + N[(N[(1.0 - x), $MachinePrecision] * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x - 1, y, 1\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma (- x 1.0) y 1.0))

C

double code(double x, double y) {
	return fma((x - 1.0), y, 1.0);
}

Julia

function code(x, y)
	return fma(Float64(x - 1.0), y, 1.0)
end

Wolfram

code[x_, y_] := N[(N[(x - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision]

Derivation

1. Initial program 74.2%

   \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. mul-1-neg N/A

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

   6. unsub-neg N/A

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

   7. sub-neg N/A

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

   8. associate--r+ N/A

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

   9. metadata-eval N/A

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

   10. neg-sub0 N/A

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

   11. cancel-sign-sub-inv N/A

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

   12. associate--r+ N/A

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

   13. *-rgt-identity N/A

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

   14. distribute-lft-neg-out N/A

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

   15. distribute-rgt-neg-out N/A

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

   16. distribute-lft-in N/A

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

   17. sub-neg N/A

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

   18. unsub-neg N/A

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

   19. mul-1-neg N/A

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

   20. +-commutative N/A

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

5. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x - 1, y, 1\right)} \]
6. Add Preprocessing

Alternative 2: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - 1\right) \cdot y\\ \mathbf{if}\;y \leq -7 \cdot 10^{-34}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-89}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- x 1.0) y)))
   (if (<= y -7e-34) t_0 (if (<= y 3.2e-89) 1.0 t_0))))

C

double code(double x, double y) {
	double t_0 = (x - 1.0) * y;
	double tmp;
	if (y <= -7e-34) {
		tmp = t_0;
	} else if (y <= 3.2e-89) {
		tmp = 1.0;
	} 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 - 1.0d0) * y
    if (y <= (-7d-34)) then
        tmp = t_0
    else if (y <= 3.2d-89) then
        tmp = 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - 1.0) * y;
	double tmp;
	if (y <= -7e-34) {
		tmp = t_0;
	} else if (y <= 3.2e-89) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - 1.0) * y
	tmp = 0
	if y <= -7e-34:
		tmp = t_0
	elif y <= 3.2e-89:
		tmp = 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - 1.0) * y)
	tmp = 0.0
	if (y <= -7e-34)
		tmp = t_0;
	elseif (y <= 3.2e-89)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - 1.0) * y;
	tmp = 0.0;
	if (y <= -7e-34)
		tmp = t_0;
	elseif (y <= 3.2e-89)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - 1.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -7e-34], t$95$0, If[LessEqual[y, 3.2e-89], 1.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -7e-34 or 3.19999999999999998e-89 < y

   1. Initial program 94.1%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 90.5

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

   5. Applied rewrites 90.5%

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

   if -7e-34 < y < 3.19999999999999998e-89

   1. Initial program 54.9%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 80.3%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ y \cdot x - \left(y - 1\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (* y x) (- y 1.0)))

C

double code(double x, double y) {
	return (y * x) - (y - 1.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (y * x) - (y - 1.0d0)
end function

Java

public static double code(double x, double y) {
	return (y * x) - (y - 1.0);
}

Python

def code(x, y):
	return (y * x) - (y - 1.0)

Julia

function code(x, y)
	return Float64(Float64(y * x) - Float64(y - 1.0))
end

MATLAB

function tmp = code(x, y)
	tmp = (y * x) - (y - 1.0);
end

Wolfram

code[x_, y_] := N[(N[(y * x), $MachinePrecision] - N[(y - 1.0), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024230 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.Vectors:renderPlotVectors from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* y x) (- y 1)))

  (+ x (* (- 1.0 x) (- 1.0 y))))