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

Percentage Accurate: 77.9% → 100.0%

Time: 2.6s

Alternatives: 4

Speedup: 1.3×

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: 77.9% 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, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

   \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
2. Step-by-step derivation

   1. +-commutative 77.7%

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

   2. sub-neg 77.7%

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

   3. distribute-rgt-in 77.8%

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

   4. *-lft-identity 77.8%

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

   5. associate-+l+ 77.8%

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

   6. +-commutative 77.8%

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

   7. distribute-lft-neg-out 77.8%

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

   8. sub-neg 77.8%

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

   9. associate--l+ 88.9%

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

   10. associate-+l- 100.0%

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

   11. +-inverses 100.0%

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

   12. metadata-eval 100.0%

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

   13. *-commutative 100.0%

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

3. Simplified 100.0%

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

   1. *-commutative 100.0%

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

   2. sub-neg 100.0%

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

   3. distribute-rgt-in 100.0%

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

   4. *-un-lft-identity 100.0%

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

5. Applied egg-rr 100.0%

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

   1. distribute-lft-neg-out 100.0%

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

   2. unsub-neg 100.0%

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

   3. *-commutative 100.0%

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

7. Applied egg-rr 100.0%

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

8. Final simplification 100.0%

   \[\leadsto 1 + \left(y \cdot x - y\right) \]

Alternative 2: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -38000 \lor \neg \left(x \leq 0.000155\right):\\ \;\;\;\;1 + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -38000.0) (not (<= x 0.000155))) (+ 1.0 (* y x)) (- 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -38000.0) || !(x <= 0.000155)) {
		tmp = 1.0 + (y * x);
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-38000.0d0)) .or. (.not. (x <= 0.000155d0))) then
        tmp = 1.0d0 + (y * x)
    else
        tmp = 1.0d0 - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -38000.0) || !(x <= 0.000155)) {
		tmp = 1.0 + (y * x);
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -38000.0) or not (x <= 0.000155):
		tmp = 1.0 + (y * x)
	else:
		tmp = 1.0 - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -38000.0) || !(x <= 0.000155))
		tmp = Float64(1.0 + Float64(y * x));
	else
		tmp = Float64(1.0 - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -38000.0) || ~((x <= 0.000155)))
		tmp = 1.0 + (y * x);
	else
		tmp = 1.0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -38000.0], N[Not[LessEqual[x, 0.000155]], $MachinePrecision]], N[(1.0 + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(1.0 - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -38000 or 1.55e-4 < x

   1. Initial program 57.2%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   2. Step-by-step derivation

      1. +-commutative 57.2%

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

      2. sub-neg 57.2%

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

      3. distribute-rgt-in 57.2%

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

      4. *-lft-identity 57.2%

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

      5. associate-+l+ 57.2%

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

      6. +-commutative 57.2%

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

      7. distribute-lft-neg-out 57.2%

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

      8. sub-neg 57.2%

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

      9. associate--l+ 78.6%

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

      10. associate-+l- 100.0%

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

      11. +-inverses 100.0%

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

      12. metadata-eval 100.0%

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

      13. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 99.4%

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

      1. mul-1-neg 99.4%

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

      2. distribute-lft-neg-out 99.4%

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

      3. *-commutative 99.4%

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

   6. Simplified 99.4%

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

   if -38000 < x < 1.55e-4

   1. Initial program 100.0%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. *-lft-identity 100.0%

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

      5. associate-+l+ 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-lft-neg-out 100.0%

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

      8. sub-neg 100.0%

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

      9. associate--l+ 100.0%

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

      10. associate-+l- 100.0%

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

      11. +-inverses 100.0%

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

      12. metadata-eval 100.0%

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

      13. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 98.6%

      \[\leadsto 1 - \color{blue}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -38000 \lor \neg \left(x \leq 0.000155\right):\\ \;\;\;\;1 + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]

Alternative 3: 100.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

   \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
2. Step-by-step derivation

   1. +-commutative 77.7%

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

   2. sub-neg 77.7%

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

   3. distribute-rgt-in 77.8%

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

   4. *-lft-identity 77.8%

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

   5. associate-+l+ 77.8%

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

   6. +-commutative 77.8%

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

   7. distribute-lft-neg-out 77.8%

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

   8. sub-neg 77.8%

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

   9. associate--l+ 88.9%

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

   10. associate-+l- 100.0%

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

   11. +-inverses 100.0%

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

   12. metadata-eval 100.0%

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

   13. *-commutative 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto 1 + y \cdot \left(x + -1\right) \]

Alternative 4: 62.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ 1 - y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

   \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
2. Step-by-step derivation

   1. +-commutative 77.7%

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

   2. sub-neg 77.7%

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

   3. distribute-rgt-in 77.8%

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

   4. *-lft-identity 77.8%

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

   5. associate-+l+ 77.8%

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

   6. +-commutative 77.8%

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

   7. distribute-lft-neg-out 77.8%

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

   8. sub-neg 77.8%

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

   9. associate--l+ 88.9%

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

   10. associate-+l- 100.0%

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

   11. +-inverses 100.0%

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

   12. metadata-eval 100.0%

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

   13. *-commutative 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 62.5%

   \[\leadsto 1 - \color{blue}{y} \]

5. Final simplification 62.5%

   \[\leadsto 1 - y \]

Developer target: 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 2023336 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.Vectors:renderPlotVectors from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (- (* y x) (- y 1.0))

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