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

Percentage Accurate: 100.0% → 100.0%

Time: 1.2s

Alternatives: 4

Speedup: 1.0×

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

Specification

Math

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

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, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, x, 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma x x 1.0))

C

double code(double x) {
	return fma(x, x, 1.0);
}

Julia

function code(x)
	return fma(x, x, 1.0)
end

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x \cdot x + 1 \]
2. Step-by-step derivation

   1. fma-def 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(x, x, 1\right) \]

Alternative 2: 98.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x * x) <= 1.0) {
		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) <= 1.0d0) 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) <= 1.0) {
		tmp = 1.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1

   1. Initial program 100.0%

      \[x \cdot x + 1 \]

   2. Taylor expanded in x around 0 98.7%

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

   if 1 < (*.f64 x x)

   1. Initial program 100.0%

      \[x \cdot x + 1 \]

   2. Taylor expanded in x around inf 97.6%

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

      1. unpow2 97.6%

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

   4. Simplified 97.6%

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

4. Final simplification 98.2%

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

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x \cdot x + 1 \]

2. Final simplification 100.0%

   \[\leadsto 1 + x \cdot x \]

Alternative 4: 52.3% 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. Taylor expanded in x around 0 50.9%

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

3. Final simplification 50.9%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023196 
(FPCore (x)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, A"
  :precision binary64
  (+ (* x x) 1.0))