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

Percentage Accurate: 100.0% → 100.0%

Time: 2.5s

Alternatives: 3

Speedup: 1.0×

• 25.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot x + 1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return (x * x) + 1.0

Julia

function code(x)
	return Float64(Float64(x * x) + 1.0)
end

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot x + 1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return (x * x) + 1.0

Julia

function code(x)
	return Float64(Float64(x * x) + 1.0)
end

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot x + 1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return (x * x) + 1.0

Julia

function code(x)
	return Float64(Float64(x * x) + 1.0)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x \cdot x + 1 \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= (* x x) 0.5) 1.0 (* x x)))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 0.5) {
		tmp = 1.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x * x) <= 0.5d0) then
        tmp = 1.0d0
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 0.5) {
		tmp = 1.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 0.5:
		tmp = 1.0
	else:
		tmp = x * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 0.5)
		tmp = 1.0;
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 0.5)
		tmp = 1.0;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 0.5], 1.0, N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 0.5

   1. Initial program 100.0%

      \[x \cdot x + 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 98.9%

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

   if 0.5 < (*.f64 x x)

   1. Initial program 100.0%

      \[x \cdot x + 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 99.5%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

   5. Simplified 99.5%

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

Alternative 3: 51.6% accurate, 5.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 100.0%

   \[x \cdot x + 1 \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Simplified 54.7%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024163 
(FPCore (x)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, A"
  :precision binary64
  (+ (* x x) 1.0))