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

Percentage Accurate: 100.0% → 100.0%

Time: 3.4s

Alternatives: 5

Speedup: 1.0×

• 24.9% 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, 1.0× 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(Float64(-y), y, x)
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-lft-neg-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 86.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(y, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 1e+54) (fma y y x) (* (- y) y)))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1e+54) {
		tmp = fma(y, y, x);
	} else {
		tmp = -y * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 1e+54)
		tmp = fma(y, y, x);
	else
		tmp = Float64(Float64(-y) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 1e+54], N[(y * y + x), $MachinePrecision], N[((-y) * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1.0000000000000001e54

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-neg.f64 N/A

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

      2. neg-sub0 N/A

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

      3. flip3-- N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{0}^{3} - {y}^{3}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}}, y, x\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0} - {y}^{3}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}, y, x\right) \]

      5. sub0-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\mathsf{neg}\left({y}^{3}\right)}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}, y, x\right) \]

      6. cube-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{{\left(\mathsf{neg}\left(y\right)\right)}^{3}}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}, y, x\right) \]

      7. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{{\color{blue}{\left(-y\right)}}^{3}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}, y, x\right) \]

      8. sqr-pow N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{{\left(-y\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(-y\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}, y, x\right) \]

      9. unpow-prod-down N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{{\left(\left(-y\right) \cdot \left(-y\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}, y, x\right) \]

      10. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot \left(-y\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}, y, x\right) \]

      11. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}, y, x\right) \]

      12. sqr-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\color{blue}{\left(y \cdot y\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}, y, x\right) \]

      13. unpow-prod-down N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{{y}^{\left(\frac{3}{2}\right)} \cdot {y}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}, y, x\right) \]

      14. sqr-pow N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{{y}^{3}}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}, y, x\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{{y}^{3}}{\color{blue}{0} + \left(y \cdot y + 0 \cdot y\right)}, y, x\right) \]

      16. +-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{{y}^{3}}{\color{blue}{y \cdot y + 0 \cdot y}}, y, x\right) \]

      17. distribute-rgt-out N/A

          \[\leadsto \mathsf{fma}\left(\frac{{y}^{3}}{\color{blue}{y \cdot \left(y + 0\right)}}, y, x\right) \]

      18. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{{y}^{3}}{y \cdot \color{blue}{\left(0 + y\right)}}, y, x\right) \]

      19. +-lft-identity N/A

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

      20. pow2 N/A

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

      21. pow-div N/A

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

      22. metadata-eval N/A

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

      23. unpow1 89.7

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

   6. Applied rewrites 89.7%

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

   if 1.0000000000000001e54 < (*.f64 y y)

   1. Initial program 100.0%

      \[x - y \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({y}^{2}\right)} \]

      2. unpow2 N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 94.5

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

   5. Applied rewrites 94.5%

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

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. Add Preprocessing
3. Add Preprocessing

Alternative 4: 50.6% accurate, 1.3× 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, y, x)
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-lft-neg-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-neg.f64 N/A

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

   2. neg-sub0 N/A

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

   3. flip3-- N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{0}^{3} - {y}^{3}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}}, y, x\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0} - {y}^{3}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}, y, x\right) \]

   5. sub0-neg N/A

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\mathsf{neg}\left({y}^{3}\right)}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}, y, x\right) \]

   6. cube-neg N/A

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{{\left(\mathsf{neg}\left(y\right)\right)}^{3}}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}, y, x\right) \]

   7. lift-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{{\color{blue}{\left(-y\right)}}^{3}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}, y, x\right) \]

   8. sqr-pow N/A

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{{\left(-y\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(-y\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}, y, x\right) \]

   9. unpow-prod-down N/A

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{{\left(\left(-y\right) \cdot \left(-y\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}, y, x\right) \]

   10. lift-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{{\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot \left(-y\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}, y, x\right) \]

   11. lift-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{{\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}, y, x\right) \]

   12. sqr-neg N/A

       \[\leadsto \mathsf{fma}\left(\frac{{\color{blue}{\left(y \cdot y\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}, y, x\right) \]

   13. unpow-prod-down N/A

       \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{{y}^{\left(\frac{3}{2}\right)} \cdot {y}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}, y, x\right) \]

   14. sqr-pow N/A

       \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{{y}^{3}}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}, y, x\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{{y}^{3}}{\color{blue}{0} + \left(y \cdot y + 0 \cdot y\right)}, y, x\right) \]

   16. +-lft-identity N/A

       \[\leadsto \mathsf{fma}\left(\frac{{y}^{3}}{\color{blue}{y \cdot y + 0 \cdot y}}, y, x\right) \]

   17. distribute-rgt-out N/A

       \[\leadsto \mathsf{fma}\left(\frac{{y}^{3}}{\color{blue}{y \cdot \left(y + 0\right)}}, y, x\right) \]

   18. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{{y}^{3}}{y \cdot \color{blue}{\left(0 + y\right)}}, y, x\right) \]

   19. +-lft-identity N/A

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

   20. pow2 N/A

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

   21. pow-div N/A

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

   22. metadata-eval N/A

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

   23. unpow1 52.2

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

6. Applied rewrites 52.2%

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

Alternative 5: 2.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x - y \cdot y \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left({y}^{2}\right)} \]

   2. unpow2 N/A

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

   3. distribute-lft-neg-in N/A

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 49.4

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

5. Applied rewrites 49.4%

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

7. Applied rewrites 2.0%

   \[\leadsto \color{blue}{y \cdot y} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024298 
(FPCore (x y)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1"
  :precision binary64
  (- x (* y y)))