Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1

Percentage Accurate: 100.0% → 100.0%

Time: 2.2s

Alternatives: 4

Speedup: 1.0×

• 24.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x - y \cdot y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - y \cdot y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x - y \cdot y \]
2. Step-by-step derivation

   1. add-cube-cbrt 98.9%

      \[\leadsto \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}} - y \cdot y \]

   2. fma-neg 98.9%

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

   3. pow2 98.9%

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

3. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{2}, \sqrt[3]{x}, -y \cdot y\right)} \]
4. Step-by-step derivation

   1. add-sqr-sqrt 98.9%

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

   2. sqrt-unprod 86.0%

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

   3. unpow2 86.0%

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

   4. associate-*r* 86.0%

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

   5. unpow2 86.0%

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

   6. add-cube-cbrt 86.3%

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

5. Applied egg-rr 86.3%

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

6. Taylor expanded in y around 0 100.0%

   \[\leadsto \color{blue}{-1 \cdot {y}^{2} + {1}^{0.3333333333333333} \cdot x} \]
7. Step-by-step derivation

   1. unpow2 100.0%

      \[\leadsto -1 \cdot \color{blue}{\left(y \cdot y\right)} + {1}^{0.3333333333333333} \cdot x \]

   2. mul-1-neg 100.0%

      \[\leadsto \color{blue}{\left(-y \cdot y\right)} + {1}^{0.3333333333333333} \cdot x \]

   3. distribute-rgt-neg-in 100.0%

      \[\leadsto \color{blue}{y \cdot \left(-y\right)} + {1}^{0.3333333333333333} \cdot x \]

   4. pow-base-1 100.0%

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

   5. *-lft-identity 100.0%

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

   6. fma-def 100.0%

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

8. Simplified 100.0%

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

9. Final simplification 100.0%

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

Alternative 2: 69.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{+47}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.15e+47) x (* y (- y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.15e+47) {
		tmp = x;
	} else {
		tmp = y * -y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.15d+47) then
        tmp = x
    else
        tmp = y * -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.15e+47) {
		tmp = x;
	} else {
		tmp = y * -y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.15e+47:
		tmp = x
	else:
		tmp = y * -y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.15e+47)
		tmp = x;
	else
		tmp = Float64(y * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.15e+47)
		tmp = x;
	else
		tmp = y * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.15e+47], x, N[(y * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.1499999999999999e47

   1. Initial program 100.0%

      \[x - y \cdot y \]

   2. Taylor expanded in x around inf 65.1%

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

   if 1.1499999999999999e47 < y

   1. Initial program 100.0%

      \[x - y \cdot y \]

   2. Taylor expanded in x around 0 91.8%

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

      1. unpow2 91.8%

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

      2. neg-mul-1 91.8%

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

      3. distribute-rgt-neg-in 91.8%

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

   4. Simplified 91.8%

      \[\leadsto \color{blue}{y \cdot \left(-y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{+47}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \end{array} \]

Alternative 3: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - y \cdot y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x - y \cdot y \]

2. Final simplification 100.0%

   \[\leadsto x - y \cdot y \]

Alternative 4: 51.0% accurate, 5.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 100.0%

   \[x - y \cdot y \]

2. Taylor expanded in x around inf 53.3%

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

3. Final simplification 53.3%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023271 
(FPCore (x y)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1"
  :precision binary64
  (- x (* y y)))