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

Percentage Accurate: 90.7% → 93.6%

Time: 6.8s

Alternatives: 7

Speedup: 0.9×

• 50.3% 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: 90.7% 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: 93.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+165}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= x 2e+165) (fma x x (* (- (* z z) t) (* y -4.0))) (* x x)))

C

x = abs(x);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 2e+165) {
		tmp = fma(x, x, (((z * z) - t) * (y * -4.0)));
	} else {
		tmp = x * x;
	}
	return tmp;
}

Julia

x = abs(x)
function code(x, y, z, t)
	tmp = 0.0
	if (x <= 2e+165)
		tmp = fma(x, x, Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0)));
	else
		tmp = 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, 2e+165], N[(x * x + N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.9999999999999998e165

   1. Initial program 93.4%

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

      1. fma-neg 94.8%

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

      2. *-commutative 94.8%

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

      3. distribute-rgt-neg-in 94.8%

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

      4. distribute-rgt-neg-in 94.8%

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

      5. metadata-eval 94.8%

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

   3. Simplified 94.8%

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

   if 1.9999999999999998e165 < x

   1. Initial program 81.3%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+165}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 2: 82.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_1 := x \cdot x - t \cdot \left(y \cdot -4\right)\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-73}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z - t\right) \cdot y\right)\\ \mathbf{elif}\;x \cdot x \leq 8 \cdot 10^{+204}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot x \leq 3.2 \cdot 10^{+214}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \mathbf{elif}\;x \cdot x \leq 1.9 \cdot 10^{+274}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x x) (* t (* y -4.0)))))
   (if (<= (* x x) 5e-73)
     (* -4.0 (* (- (* z z) t) y))
     (if (<= (* x x) 8e+204)
       t_1
       (if (<= (* x x) 3.2e+214)
         (* z (* z (* y -4.0)))
         (if (<= (* x x) 1.9e+274) t_1 (* x x)))))))

C

x = abs(x);
double code(double x, double y, double z, double t) {
	double t_1 = (x * x) - (t * (y * -4.0));
	double tmp;
	if ((x * x) <= 5e-73) {
		tmp = -4.0 * (((z * z) - t) * y);
	} else if ((x * x) <= 8e+204) {
		tmp = t_1;
	} else if ((x * x) <= 3.2e+214) {
		tmp = z * (z * (y * -4.0));
	} else if ((x * x) <= 1.9e+274) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = (x * x) - (t * (y * (-4.0d0)))
    if ((x * x) <= 5d-73) then
        tmp = (-4.0d0) * (((z * z) - t) * y)
    else if ((x * x) <= 8d+204) then
        tmp = t_1
    else if ((x * x) <= 3.2d+214) then
        tmp = z * (z * (y * (-4.0d0)))
    else if ((x * x) <= 1.9d+274) then
        tmp = t_1
    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 t_1 = (x * x) - (t * (y * -4.0));
	double tmp;
	if ((x * x) <= 5e-73) {
		tmp = -4.0 * (((z * z) - t) * y);
	} else if ((x * x) <= 8e+204) {
		tmp = t_1;
	} else if ((x * x) <= 3.2e+214) {
		tmp = z * (z * (y * -4.0));
	} else if ((x * x) <= 1.9e+274) {
		tmp = t_1;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y, z, t):
	t_1 = (x * x) - (t * (y * -4.0))
	tmp = 0
	if (x * x) <= 5e-73:
		tmp = -4.0 * (((z * z) - t) * y)
	elif (x * x) <= 8e+204:
		tmp = t_1
	elif (x * x) <= 3.2e+214:
		tmp = z * (z * (y * -4.0))
	elif (x * x) <= 1.9e+274:
		tmp = t_1
	else:
		tmp = x * x
	return tmp

Julia

x = abs(x)
function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) - Float64(t * Float64(y * -4.0)))
	tmp = 0.0
	if (Float64(x * x) <= 5e-73)
		tmp = Float64(-4.0 * Float64(Float64(Float64(z * z) - t) * y));
	elseif (Float64(x * x) <= 8e+204)
		tmp = t_1;
	elseif (Float64(x * x) <= 3.2e+214)
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	elseif (Float64(x * x) <= 1.9e+274)
		tmp = t_1;
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, y, z, t)
	t_1 = (x * x) - (t * (y * -4.0));
	tmp = 0.0;
	if ((x * x) <= 5e-73)
		tmp = -4.0 * (((z * z) - t) * y);
	elseif ((x * x) <= 8e+204)
		tmp = t_1;
	elseif ((x * x) <= 3.2e+214)
		tmp = z * (z * (y * -4.0));
	elseif ((x * x) <= 1.9e+274)
		tmp = t_1;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] - N[(t * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 5e-73], N[(-4.0 * N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 8e+204], t$95$1, If[LessEqual[N[(x * x), $MachinePrecision], 3.2e+214], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 1.9e+274], t$95$1, N[(x * x), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x x) < 4.9999999999999998e-73

   1. Initial program 97.2%

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

   2. Taylor expanded in x around 0 92.6%

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

      1. *-commutative 92.6%

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

      2. unpow2 92.6%

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

   4. Simplified 92.6%

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

   if 4.9999999999999998e-73 < (*.f64 x x) < 7.99999999999999991e204 or 3.19999999999999995e214 < (*.f64 x x) < 1.8999999999999999e274

   1. Initial program 96.1%

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

   2. Taylor expanded in z around 0 79.8%

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

      1. associate-*r* 79.8%

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

   4. Simplified 79.8%

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

   if 7.99999999999999991e204 < (*.f64 x x) < 3.19999999999999995e214

   1. Initial program 99.7%

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

   2. Taylor expanded in z around inf 99.7%

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

      1. metadata-eval 99.7%

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

      2. distribute-lft-neg-in 99.7%

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

      3. *-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. *-commutative 99.7%

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

      6. associate-*r* 99.7%

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

      7. associate-*l* 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. *-commutative 100.0%

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

      11. distribute-lft-neg-in 100.0%

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

      12. metadata-eval 100.0%

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

   4. Simplified 100.0%

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

   if 1.8999999999999999e274 < (*.f64 x x)

   1. Initial program 80.3%

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

   2. Taylor expanded in x around inf 96.1%

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

      1. unpow2 96.1%

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

   4. Simplified 96.1%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-73}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z - t\right) \cdot y\right)\\ \mathbf{elif}\;x \cdot x \leq 8 \cdot 10^{+204}:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \mathbf{elif}\;x \cdot x \leq 3.2 \cdot 10^{+214}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \mathbf{elif}\;x \cdot x \leq 1.9 \cdot 10^{+274}:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 3: 60.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-192}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+216}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e-192)
   (* t (* y 4.0))
   (if (<= (* z z) 5e+216) (* x x) (* z (* z (* y -4.0))))))

C

x = abs(x);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e-192) {
		tmp = t * (y * 4.0);
	} else if ((z * z) <= 5e+216) {
		tmp = x * x;
	} else {
		tmp = z * (z * (y * -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) <= 5d-192) then
        tmp = t * (y * 4.0d0)
    else if ((z * z) <= 5d+216) then
        tmp = x * x
    else
        tmp = z * (z * (y * (-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) <= 5e-192) {
		tmp = t * (y * 4.0);
	} else if ((z * z) <= 5e+216) {
		tmp = x * x;
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 5e-192:
		tmp = t * (y * 4.0)
	elif (z * z) <= 5e+216:
		tmp = x * x
	else:
		tmp = z * (z * (y * -4.0))
	return tmp

Julia

x = abs(x)
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e-192)
		tmp = Float64(t * Float64(y * 4.0));
	elseif (Float64(z * z) <= 5e+216)
		tmp = Float64(x * x);
	else
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 5e-192)
		tmp = t * (y * 4.0);
	elseif ((z * z) <= 5e+216)
		tmp = x * x;
	else
		tmp = z * (z * (y * -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], 5e-192], N[(t * N[(y * 4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 5e+216], N[(x * x), $MachinePrecision], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 5.0000000000000001e-192

   1. Initial program 96.3%

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

   2. Taylor expanded in t around inf 61.0%

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

      1. associate-*r* 61.0%

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

      2. *-commutative 61.0%

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

   4. Simplified 61.0%

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

   if 5.0000000000000001e-192 < (*.f64 z z) < 4.9999999999999998e216

   1. Initial program 97.0%

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

   2. Taylor expanded in x around inf 57.8%

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

      1. unpow2 57.8%

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

   4. Simplified 57.8%

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

   if 4.9999999999999998e216 < (*.f64 z z)

   1. Initial program 80.2%

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

   2. Taylor expanded in z around inf 76.5%

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

      1. metadata-eval 76.5%

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

      2. distribute-lft-neg-in 76.5%

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

      3. *-commutative 76.5%

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

      4. unpow2 76.5%

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

      5. *-commutative 76.5%

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

      6. associate-*r* 76.5%

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

      7. associate-*l* 80.2%

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

      8. distribute-rgt-neg-in 80.2%

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

      9. distribute-rgt-neg-in 80.2%

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

      10. *-commutative 80.2%

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

      11. distribute-lft-neg-in 80.2%

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

      12. metadata-eval 80.2%

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

   4. Simplified 80.2%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-192}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+216}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternative 4: 92.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+137}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= x 1.35e+137) (+ (* 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 <= 1.35e+137) {
		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 <= 1.35d+137) 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 <= 1.35e+137) {
		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 <= 1.35e+137:
		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 (x <= 1.35e+137)
		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 <= 1.35e+137)
		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[x, 1.35e+137], 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 x < 1.35000000000000009e137

   1. Initial program 94.6%

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

   if 1.35000000000000009e137 < x

   1. Initial program 76.9%

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

   2. Taylor expanded in x around inf 94.9%

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

      1. unpow2 94.9%

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

   4. Simplified 94.9%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+137}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 5: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4.2 \cdot 10^{+215}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z - t\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 4.2e+215) (* -4.0 (* (- (* z z) t) y)) (* x x)))

C

x = abs(x);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 4.2e+215) {
		tmp = -4.0 * (((z * z) - t) * y);
	} 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) <= 4.2d+215) then
        tmp = (-4.0d0) * (((z * z) - t) * y)
    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) <= 4.2e+215) {
		tmp = -4.0 * (((z * z) - t) * y);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 4.2e+215:
		tmp = -4.0 * (((z * z) - t) * y)
	else:
		tmp = x * x
	return tmp

Julia

x = abs(x)
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 4.2e+215)
		tmp = Float64(-4.0 * Float64(Float64(Float64(z * z) - t) * y));
	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) <= 4.2e+215)
		tmp = -4.0 * (((z * z) - t) * y);
	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], 4.2e+215], N[(-4.0 * N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.2000000000000003e215

   1. Initial program 97.1%

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

   2. Taylor expanded in x around 0 81.4%

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

      1. *-commutative 81.4%

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

      2. unpow2 81.4%

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

   4. Simplified 81.4%

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

   if 4.2000000000000003e215 < (*.f64 x x)

   1. Initial program 82.1%

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

   2. Taylor expanded in x around inf 93.5%

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

      1. unpow2 93.5%

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

   4. Simplified 93.5%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4.2 \cdot 10^{+215}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z - t\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 6: 56.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.06 \cdot 10^{+184}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 1.06e+184) (* t (* y 4.0)) (* x x)))

C

x = abs(x);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 1.06e+184) {
		tmp = t * (y * 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) <= 1.06d+184) then
        tmp = t * (y * 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) <= 1.06e+184) {
		tmp = t * (y * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 1.06e+184:
		tmp = t * (y * 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) <= 1.06e+184)
		tmp = Float64(t * Float64(y * 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) <= 1.06e+184)
		tmp = t * (y * 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], 1.06e+184], N[(t * N[(y * 4.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.06e184

   1. Initial program 96.9%

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

   2. Taylor expanded in t around inf 48.2%

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

      1. associate-*r* 48.2%

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

      2. *-commutative 48.2%

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

   4. Simplified 48.2%

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

   if 1.06e184 < (*.f64 x x)

   1. Initial program 83.9%

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

   2. Taylor expanded in x around inf 87.5%

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

      1. unpow2 87.5%

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

   4. Simplified 87.5%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.06 \cdot 10^{+184}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 7: 41.1% 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 91.9%

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

2. Taylor expanded in x around inf 46.1%

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

   1. unpow2 46.1%

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

4. Simplified 46.1%

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

5. Final simplification 46.1%

   \[\leadsto x \cdot x \]

Developer target: 90.8% 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 2023257 
(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))))