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

Percentage Accurate: 100.0% → 100.0%

Time: 4.6s

Alternatives: 5

Speedup: 1.0×

• 25.8% 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

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

   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: 85.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{-118}:\\ \;\;\;\;\mathsf{fma}\left(y, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;-y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 1e-118) (fma y y x) (- (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1e-118) {
		tmp = fma(y, y, x);
	} else {
		tmp = -(y * y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 1e-118)
		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-118], N[(y * y + x), $MachinePrecision], (-N[(y * y), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 9.99999999999999985e-119

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      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

   6. Applied rewrites 91.5%

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

   if 9.99999999999999985e-119 < (*.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. lower-neg.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 81.7

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

   5. Applied rewrites 81.7%

      \[\leadsto \color{blue}{-y \cdot y} \]
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: 49.4% 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

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

   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

6. Applied rewrites 50.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, y, x\right)} \]
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. lower-neg.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 50.6

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

5. Applied rewrites 50.6%

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

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

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

   2. neg-sub0 N/A

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

   3. flip-- N/A

      \[\leadsto y \cdot \color{blue}{\frac{0 \cdot 0 - y \cdot y}{0 + y}} \]

   4. metadata-eval N/A

      \[\leadsto y \cdot \frac{\color{blue}{0} - y \cdot y}{0 + y} \]

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

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

   6. lift-neg.f64 N/A

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

   7. +-lft-identity N/A

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

   8. +-lft-identity N/A

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

   9. associate-*r/ N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. sqr-neg N/A

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

   13. lift-neg.f64 N/A

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

   14. lift-neg.f64 N/A

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

   15. cube-mult N/A

       \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(y\right)\right)}^{3}}}{y} \]

   16. sqr-pow N/A

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

   17. unpow-prod-down N/A

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

   18. lift-neg.f64 N/A

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

   19. lift-neg.f64 N/A

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

   20. sqr-neg N/A

       \[\leadsto \frac{{\color{blue}{\left(y \cdot y\right)}}^{\left(\frac{3}{2}\right)}}{y} \]

   21. unpow-prod-down N/A

       \[\leadsto \frac{\color{blue}{{y}^{\left(\frac{3}{2}\right)} \cdot {y}^{\left(\frac{3}{2}\right)}}}{y} \]

   22. sqr-pow N/A

       \[\leadsto \frac{\color{blue}{{y}^{3}}}{y} \]

   23. unpow1 N/A

       \[\leadsto \frac{{y}^{3}}{\color{blue}{{y}^{1}}} \]

   24. pow-div N/A

       \[\leadsto \color{blue}{{y}^{\left(3 - 1\right)}} \]

   25. metadata-eval N/A

       \[\leadsto {y}^{\color{blue}{2}} \]

7. Applied rewrites 2.1%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (x y)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1"
  :precision binary64
  (- x (* y y)))