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

Percentage Accurate: 91.0% → 94.1%

Time: 10.3s

Alternatives: 8

Speedup: 0.8×

• 49.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (* x x) (* (* y 4.0) (- (* z z) t))))

C

double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * x) - ((y * 4.0d0) * ((z * z) - t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}

Python

def code(x, y, z, t):
	return (x * x) - ((y * 4.0) * ((z * z) - t))

Julia

function code(x, y, z, t)
	return Float64(Float64(x * x) - Float64(Float64(y * 4.0) * Float64(Float64(z * z) - t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * x) - ((y * 4.0) * ((z * z) - t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 91.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (* x x) (* (* y 4.0) (- (* z z) t))))

C

double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * x) - ((y * 4.0d0) * ((z * z) - t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}

Python

def code(x, y, z, t):
	return (x * x) - ((y * 4.0) * ((z * z) - t))

Julia

function code(x, y, z, t)
	return Float64(Float64(x * x) - Float64(Float64(y * 4.0) * Float64(Float64(z * z) - t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * x) - ((y * 4.0) * ((z * z) - t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 94.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{+201}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-t, y \cdot -4, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= x 1.2e+201)
   (fma x x (* (- (* z z) t) (* y -4.0)))
   (fma (- t) (* y -4.0) (* x x))))

C

x = abs(x);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.2e+201) {
		tmp = fma(x, x, (((z * z) - t) * (y * -4.0)));
	} else {
		tmp = fma(-t, (y * -4.0), (x * x));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x, y, z, t)
	tmp = 0.0
	if (x <= 1.2e+201)
		tmp = fma(x, x, Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0)));
	else
		tmp = fma(Float64(-t), Float64(y * -4.0), Float64(x * x));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[x, 1.2e+201], N[(x * x + N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-t) * N[(y * -4.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.19999999999999996e201

   1. Initial program 88.3%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Step-by-step derivation

      1. fma-neg 90.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right)} \]

      2. *-commutative 90.9%

         \[\leadsto \mathsf{fma}\left(x, x, -\color{blue}{\left(z \cdot z - t\right) \cdot \left(y \cdot 4\right)}\right) \]

      3. distribute-rgt-neg-in 90.9%

         \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(z \cdot z - t\right) \cdot \left(-y \cdot 4\right)}\right) \]

      4. distribute-rgt-neg-in 90.9%

         \[\leadsto \mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \color{blue}{\left(y \cdot \left(-4\right)\right)}\right) \]

      5. metadata-eval 90.9%

         \[\leadsto \mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot \color{blue}{-4}\right)\right) \]

   3. Simplified 90.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)} \]

   if 1.19999999999999996e201 < x

   1. Initial program 88.0%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in x around 0 88.0%

      \[\leadsto \color{blue}{-4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right) + {x}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 88.0%

         \[\leadsto -4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right) + \color{blue}{x \cdot x} \]

      2. *-commutative 88.0%

         \[\leadsto \color{blue}{\left(y \cdot \left({z}^{2} - t\right)\right) \cdot -4} + x \cdot x \]

      3. *-commutative 88.0%

         \[\leadsto \color{blue}{\left(\left({z}^{2} - t\right) \cdot y\right)} \cdot -4 + x \cdot x \]

      4. unpow2 88.0%

         \[\leadsto \left(\left(\color{blue}{z \cdot z} - t\right) \cdot y\right) \cdot -4 + x \cdot x \]

      5. associate-*l* 88.0%

         \[\leadsto \color{blue}{\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)} + x \cdot x \]

      6. rem-log-exp 88.0%

         \[\leadsto \left(z \cdot z - t\right) \cdot \left(y \cdot \color{blue}{\log \left(e^{-4}\right)}\right) + x \cdot x \]

      7. log-pow 68.0%

         \[\leadsto \left(z \cdot z - t\right) \cdot \color{blue}{\log \left({\left(e^{-4}\right)}^{y}\right)} + x \cdot x \]

      8. fma-udef 68.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot z - t, \log \left({\left(e^{-4}\right)}^{y}\right), x \cdot x\right)} \]

      9. log-pow 88.0%

         \[\leadsto \mathsf{fma}\left(z \cdot z - t, \color{blue}{y \cdot \log \left(e^{-4}\right)}, x \cdot x\right) \]

      10. rem-log-exp 88.0%

          \[\leadsto \mathsf{fma}\left(z \cdot z - t, y \cdot \color{blue}{-4}, x \cdot x\right) \]

   4. Simplified 88.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot z - t, y \cdot -4, x \cdot x\right)} \]

   5. Taylor expanded in z around 0 100.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot t}, y \cdot -4, x \cdot x\right) \]
   6. Step-by-step derivation

      1. neg-mul-1 100.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{-t}, y \cdot -4, x \cdot x\right) \]

   7. Simplified 100.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{-t}, y \cdot -4, x \cdot x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{+201}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-t, y \cdot -4, x \cdot x\right)\\ \end{array} \]

Alternative 2: 93.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{+148}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-t, y \cdot -4, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= x 4.5e+148)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (fma (- t) (* y -4.0) (* x x))))

C

x = abs(x);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 4.5e+148) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = fma(-t, (y * -4.0), (x * x));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x, y, z, t)
	tmp = 0.0
	if (x <= 4.5e+148)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = fma(Float64(-t), Float64(y * -4.0), Float64(x * x));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[x, 4.5e+148], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-t) * N[(y * -4.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.49999999999999994e148

   1. Initial program 89.5%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   if 4.49999999999999994e148 < x

   1. Initial program 81.1%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in x around 0 81.1%

      \[\leadsto \color{blue}{-4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right) + {x}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 81.1%

         \[\leadsto -4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right) + \color{blue}{x \cdot x} \]

      2. *-commutative 81.1%

         \[\leadsto \color{blue}{\left(y \cdot \left({z}^{2} - t\right)\right) \cdot -4} + x \cdot x \]

      3. *-commutative 81.1%

         \[\leadsto \color{blue}{\left(\left({z}^{2} - t\right) \cdot y\right)} \cdot -4 + x \cdot x \]

      4. unpow2 81.1%

         \[\leadsto \left(\left(\color{blue}{z \cdot z} - t\right) \cdot y\right) \cdot -4 + x \cdot x \]

      5. associate-*l* 81.1%

         \[\leadsto \color{blue}{\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)} + x \cdot x \]

      6. rem-log-exp 81.1%

         \[\leadsto \left(z \cdot z - t\right) \cdot \left(y \cdot \color{blue}{\log \left(e^{-4}\right)}\right) + x \cdot x \]

      7. log-pow 59.5%

         \[\leadsto \left(z \cdot z - t\right) \cdot \color{blue}{\log \left({\left(e^{-4}\right)}^{y}\right)} + x \cdot x \]

      8. fma-udef 59.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot z - t, \log \left({\left(e^{-4}\right)}^{y}\right), x \cdot x\right)} \]

      9. log-pow 83.8%

         \[\leadsto \mathsf{fma}\left(z \cdot z - t, \color{blue}{y \cdot \log \left(e^{-4}\right)}, x \cdot x\right) \]

      10. rem-log-exp 83.8%

          \[\leadsto \mathsf{fma}\left(z \cdot z - t, y \cdot \color{blue}{-4}, x \cdot x\right) \]

   4. Simplified 83.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot z - t, y \cdot -4, x \cdot x\right)} \]

   5. Taylor expanded in z around 0 91.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot t}, y \cdot -4, x \cdot x\right) \]
   6. Step-by-step derivation

      1. neg-mul-1 91.9%

         \[\leadsto \mathsf{fma}\left(\color{blue}{-t}, y \cdot -4, x \cdot x\right) \]

   7. Simplified 91.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{-t}, y \cdot -4, x \cdot x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{+148}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-t, y \cdot -4, x \cdot x\right)\\ \end{array} \]

Alternative 3: 58.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 3.8 \cdot 10^{-44}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{elif}\;x \cdot x \leq 9000000000000:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \cdot x \leq 1.76 \cdot 10^{+152}:\\ \;\;\;\;y \cdot \left(\left(z \cdot z\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 3.8e-44)
   (* y (* t 4.0))
   (if (<= (* x x) 9000000000000.0)
     (* x x)
     (if (<= (* x x) 1.76e+152) (* y (* (* z z) -4.0)) (* x x)))))

C

x = abs(x);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 3.8e-44) {
		tmp = y * (t * 4.0);
	} else if ((x * x) <= 9000000000000.0) {
		tmp = x * x;
	} else if ((x * x) <= 1.76e+152) {
		tmp = y * ((z * z) * -4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x * x) <= 3.8d-44) then
        tmp = y * (t * 4.0d0)
    else if ((x * x) <= 9000000000000.0d0) then
        tmp = x * x
    else if ((x * x) <= 1.76d+152) then
        tmp = y * ((z * z) * (-4.0d0))
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 3.8e-44) {
		tmp = y * (t * 4.0);
	} else if ((x * x) <= 9000000000000.0) {
		tmp = x * x;
	} else if ((x * x) <= 1.76e+152) {
		tmp = y * ((z * z) * -4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 3.8e-44:
		tmp = y * (t * 4.0)
	elif (x * x) <= 9000000000000.0:
		tmp = x * x
	elif (x * x) <= 1.76e+152:
		tmp = y * ((z * z) * -4.0)
	else:
		tmp = x * x
	return tmp

Julia

x = abs(x)
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 3.8e-44)
		tmp = Float64(y * Float64(t * 4.0));
	elseif (Float64(x * x) <= 9000000000000.0)
		tmp = Float64(x * x);
	elseif (Float64(x * x) <= 1.76e+152)
		tmp = Float64(y * Float64(Float64(z * z) * -4.0));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 3.8e-44)
		tmp = y * (t * 4.0);
	elseif ((x * x) <= 9000000000000.0)
		tmp = x * x;
	elseif ((x * x) <= 1.76e+152)
		tmp = y * ((z * z) * -4.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 3.8e-44], N[(y * N[(t * 4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 9000000000000.0], N[(x * x), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 1.76e+152], N[(y * N[(N[(z * z), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 3.8000000000000001e-44

   1. Initial program 90.7%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in t around inf 54.6%

      \[\leadsto \color{blue}{4 \cdot \left(t \cdot y\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 54.6%

         \[\leadsto \color{blue}{\left(4 \cdot t\right) \cdot y} \]

      2. *-commutative 54.6%

         \[\leadsto \color{blue}{y \cdot \left(4 \cdot t\right)} \]

   4. Simplified 54.6%

      \[\leadsto \color{blue}{y \cdot \left(4 \cdot t\right)} \]

   if 3.8000000000000001e-44 < (*.f64 x x) < 9e12 or 1.76000000000000005e152 < (*.f64 x x)

   1. Initial program 84.7%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in x around inf 79.0%

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

      1. unpow2 79.0%

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

   4. Simplified 79.0%

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

   if 9e12 < (*.f64 x x) < 1.76000000000000005e152

   1. Initial program 92.1%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in z around inf 55.7%

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

      1. *-commutative 55.7%

         \[\leadsto \color{blue}{\left(y \cdot {z}^{2}\right) \cdot -4} \]

      2. unpow2 55.7%

         \[\leadsto \left(y \cdot \color{blue}{\left(z \cdot z\right)}\right) \cdot -4 \]

      3. associate-*l* 55.7%

         \[\leadsto \color{blue}{y \cdot \left(\left(z \cdot z\right) \cdot -4\right)} \]

   4. Simplified 55.7%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 3.8 \cdot 10^{-44}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{elif}\;x \cdot x \leq 9000000000000:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \cdot x \leq 1.76 \cdot 10^{+152}:\\ \;\;\;\;y \cdot \left(\left(z \cdot z\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 4: 86.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 8 \cdot 10^{-73}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{elif}\;z \cdot z \leq 2.65 \cdot 10^{+306}:\\ \;\;\;\;x \cdot x - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(z \cdot z\right) \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 8e-73)
   (- (* x x) (* y (* t -4.0)))
   (if (<= (* z z) 2.65e+306)
     (- (* x x) (* (* z z) (* y 4.0)))
     (* y (* (* z z) -4.0)))))

C

x = abs(x);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 8e-73) {
		tmp = (x * x) - (y * (t * -4.0));
	} else if ((z * z) <= 2.65e+306) {
		tmp = (x * x) - ((z * z) * (y * 4.0));
	} else {
		tmp = y * ((z * z) * -4.0);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z * z) <= 8d-73) then
        tmp = (x * x) - (y * (t * (-4.0d0)))
    else if ((z * z) <= 2.65d+306) then
        tmp = (x * x) - ((z * z) * (y * 4.0d0))
    else
        tmp = y * ((z * z) * (-4.0d0))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 8e-73) {
		tmp = (x * x) - (y * (t * -4.0));
	} else if ((z * z) <= 2.65e+306) {
		tmp = (x * x) - ((z * z) * (y * 4.0));
	} else {
		tmp = y * ((z * z) * -4.0);
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 8e-73:
		tmp = (x * x) - (y * (t * -4.0))
	elif (z * z) <= 2.65e+306:
		tmp = (x * x) - ((z * z) * (y * 4.0))
	else:
		tmp = y * ((z * z) * -4.0)
	return tmp

Julia

x = abs(x)
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 8e-73)
		tmp = Float64(Float64(x * x) - Float64(y * Float64(t * -4.0)));
	elseif (Float64(z * z) <= 2.65e+306)
		tmp = Float64(Float64(x * x) - Float64(Float64(z * z) * Float64(y * 4.0)));
	else
		tmp = Float64(y * Float64(Float64(z * z) * -4.0));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 8e-73)
		tmp = (x * x) - (y * (t * -4.0));
	elseif ((z * z) <= 2.65e+306)
		tmp = (x * x) - ((z * z) * (y * 4.0));
	else
		tmp = y * ((z * z) * -4.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 8e-73], N[(N[(x * x), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 2.65e+306], N[(N[(x * x), $MachinePrecision] - N[(N[(z * z), $MachinePrecision] * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(z * z), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 7.99999999999999998e-73

   1. Initial program 98.2%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in z around 0 97.4%

      \[\leadsto x \cdot x - \color{blue}{-4 \cdot \left(t \cdot y\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 97.4%

         \[\leadsto x \cdot x - \color{blue}{\left(-4 \cdot t\right) \cdot y} \]

   4. Simplified 97.4%

      \[\leadsto x \cdot x - \color{blue}{\left(-4 \cdot t\right) \cdot y} \]

   if 7.99999999999999998e-73 < (*.f64 z z) < 2.65000000000000013e306

   1. Initial program 95.9%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in z around inf 81.6%

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

      1. unpow2 81.6%

         \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \color{blue}{\left(z \cdot z\right)} \]

   4. Simplified 81.6%

      \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \color{blue}{\left(z \cdot z\right)} \]

   if 2.65000000000000013e306 < (*.f64 z z)

   1. Initial program 63.5%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in z around inf 69.3%

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

      1. *-commutative 69.3%

         \[\leadsto \color{blue}{\left(y \cdot {z}^{2}\right) \cdot -4} \]

      2. unpow2 69.3%

         \[\leadsto \left(y \cdot \color{blue}{\left(z \cdot z\right)}\right) \cdot -4 \]

      3. associate-*l* 69.3%

         \[\leadsto \color{blue}{y \cdot \left(\left(z \cdot z\right) \cdot -4\right)} \]

   4. Simplified 69.3%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 8 \cdot 10^{-73}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{elif}\;z \cdot z \leq 2.65 \cdot 10^{+306}:\\ \;\;\;\;x \cdot x - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(z \cdot z\right) \cdot -4\right)\\ \end{array} \]

Alternative 5: 93.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+301}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 5e+301) (+ (* x x) (* (* y 4.0) (- t (* z z)))) (* x x)))

C

x = abs(x);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 5e+301) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x * x) <= 5d+301) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 5e+301) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 5e+301:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = x * x
	return tmp

Julia

x = abs(x)
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 5e+301)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 5e+301)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e+301], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5.0000000000000004e301

   1. Initial program 92.7%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   if 5.0000000000000004e301 < (*.f64 x x)

   1. Initial program 75.0%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in x around inf 90.6%

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

      1. unpow2 90.6%

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

   4. Simplified 90.6%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+301}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 6: 74.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 2.2 \cdot 10^{+84}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(z \cdot z\right) \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= z 2.2e+84) (- (* x x) (* y (* t -4.0))) (* y (* (* z z) -4.0))))

C

x = abs(x);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 2.2e+84) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = y * ((z * z) * -4.0);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 2.2d+84) then
        tmp = (x * x) - (y * (t * (-4.0d0)))
    else
        tmp = y * ((z * z) * (-4.0d0))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 2.2e+84) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = y * ((z * z) * -4.0);
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y, z, t):
	tmp = 0
	if z <= 2.2e+84:
		tmp = (x * x) - (y * (t * -4.0))
	else:
		tmp = y * ((z * z) * -4.0)
	return tmp

Julia

x = abs(x)
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 2.2e+84)
		tmp = Float64(Float64(x * x) - Float64(y * Float64(t * -4.0)));
	else
		tmp = Float64(y * Float64(Float64(z * z) * -4.0));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 2.2e+84)
		tmp = (x * x) - (y * (t * -4.0));
	else
		tmp = y * ((z * z) * -4.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[z, 2.2e+84], N[(N[(x * x), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(z * z), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.1999999999999998e84

   1. Initial program 93.4%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in z around 0 77.2%

      \[\leadsto x \cdot x - \color{blue}{-4 \cdot \left(t \cdot y\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 77.2%

         \[\leadsto x \cdot x - \color{blue}{\left(-4 \cdot t\right) \cdot y} \]

   4. Simplified 77.2%

      \[\leadsto x \cdot x - \color{blue}{\left(-4 \cdot t\right) \cdot y} \]

   if 2.1999999999999998e84 < z

   1. Initial program 66.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in z around inf 70.8%

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

      1. *-commutative 70.8%

         \[\leadsto \color{blue}{\left(y \cdot {z}^{2}\right) \cdot -4} \]

      2. unpow2 70.8%

         \[\leadsto \left(y \cdot \color{blue}{\left(z \cdot z\right)}\right) \cdot -4 \]

      3. associate-*l* 70.8%

         \[\leadsto \color{blue}{y \cdot \left(\left(z \cdot z\right) \cdot -4\right)} \]

   4. Simplified 70.8%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.2 \cdot 10^{+84}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(z \cdot z\right) \cdot -4\right)\\ \end{array} \]

Alternative 7: 58.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.48 \cdot 10^{-52}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 1.48e-52) (* y (* t 4.0)) (* x x)))

C

x = abs(x);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 1.48e-52) {
		tmp = y * (t * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x * x) <= 1.48d-52) then
        tmp = y * (t * 4.0d0)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 1.48e-52) {
		tmp = y * (t * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 1.48e-52:
		tmp = y * (t * 4.0)
	else:
		tmp = x * x
	return tmp

Julia

x = abs(x)
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 1.48e-52)
		tmp = Float64(y * Float64(t * 4.0));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 1.48e-52)
		tmp = y * (t * 4.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 1.48e-52], N[(y * N[(t * 4.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.47999999999999993e-52

   1. Initial program 90.7%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in t around inf 54.6%

      \[\leadsto \color{blue}{4 \cdot \left(t \cdot y\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 54.6%

         \[\leadsto \color{blue}{\left(4 \cdot t\right) \cdot y} \]

      2. *-commutative 54.6%

         \[\leadsto \color{blue}{y \cdot \left(4 \cdot t\right)} \]

   4. Simplified 54.6%

      \[\leadsto \color{blue}{y \cdot \left(4 \cdot t\right)} \]

   if 1.47999999999999993e-52 < (*.f64 x x)

   1. Initial program 86.5%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in x around inf 64.9%

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

      1. unpow2 64.9%

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

   4. Simplified 64.9%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.48 \cdot 10^{-52}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 8: 40.4% accurate, 4.3× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ x \cdot x \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z t) :precision binary64 (* x x))

C

x = abs(x);
double code(double x, double y, double z, double t) {
	return x * x;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x * x
end function

Java

x = Math.abs(x);
public static double code(double x, double y, double z, double t) {
	return x * x;
}

Python

x = abs(x)
def code(x, y, z, t):
	return x * x

Julia

x = abs(x)
function code(x, y, z, t)
	return Float64(x * x)
end

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_, t_] := N[(x * x), $MachinePrecision]

Derivation

1. Initial program 88.3%

   \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

2. Taylor expanded in x around inf 43.0%

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

   1. unpow2 43.0%

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

4. Simplified 43.0%

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

5. Final simplification 43.0%

   \[\leadsto x \cdot x \]

Developer target: 91.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (* x x) (* 4.0 (* y (- (* z z) t)))))

C

double code(double x, double y, double z, double t) {
	return (x * x) - (4.0 * (y * ((z * z) - t)));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * x) - (4.0d0 * (y * ((z * z) - t)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * x) - (4.0 * (y * ((z * z) - t)));
}

Python

def code(x, y, z, t):
	return (x * x) - (4.0 * (y * ((z * z) - t)))

Julia

function code(x, y, z, t)
	return Float64(Float64(x * x) - Float64(4.0 * Float64(y * Float64(Float64(z * z) - t))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * x) - (4.0 * (y * ((z * z) - t)));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * x), $MachinePrecision] - N[(4.0 * N[(y * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023287 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

  :herbie-target
  (- (* x x) (* 4.0 (* y (- (* z z) t))))

  (- (* x x) (* (* y 4.0) (- (* z z) t))))