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

Percentage Accurate: 91.2% → 95.6%

Time: 4.5s

Alternatives: 8

Speedup: 0.9×

• 48.8% 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: 91.2% 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: 95.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 9 \cdot 10^{+145}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= z 9e+145)
   (fma x x (* (- (* z z) t) (* y -4.0)))
   (* z (* z (* y -4.0)))))

C

z = abs(z);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 9e+145) {
		tmp = fma(x, x, (((z * z) - t) * (y * -4.0)));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Julia

z = abs(z)
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 9e+145)
		tmp = fma(x, x, Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0)));
	else
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	end
	return tmp
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[z, 9e+145], N[(x * x + N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 8.9999999999999996e145

   1. Initial program 93.6%

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

      1. fma-neg 96.0%

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

      2. *-commutative 96.0%

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

      3. distribute-rgt-neg-in 96.0%

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

      4. distribute-rgt-neg-in 96.0%

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

      5. metadata-eval 96.0%

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

   3. Simplified 96.0%

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

   if 8.9999999999999996e145 < z

   1. Initial program 68.1%

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

   2. Taylor expanded in z around inf 70.6%

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

      1. metadata-eval 70.6%

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

      2. distribute-lft-neg-in 70.6%

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

      3. *-commutative 70.6%

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

      4. unpow2 70.6%

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

      5. *-commutative 70.6%

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

      6. associate-*r* 70.6%

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

      7. associate-*l* 85.0%

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

      8. distribute-rgt-neg-in 85.0%

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

      9. distribute-rgt-neg-in 85.0%

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

      10. distribute-rgt-neg-in 85.0%

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

      11. metadata-eval 85.0%

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

   4. Simplified 85.0%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9 \cdot 10^{+145}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternative 2: 57.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} t_1 := t \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;z \leq 8.5 \cdot 10^{-224}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-168}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-78}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.16 \cdot 10^{+36}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{+83}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* t (* y 4.0))))
   (if (<= z 8.5e-224)
     t_1
     (if (<= z 8e-168)
       (* x x)
       (if (<= z 1.45e-78)
         t_1
         (if (<= z 1.16e+36)
           (* x x)
           (if (<= z 5.5e+56)
             t_1
             (if (<= z 4.4e+83) (* x x) (* -4.0 (* (* z z) y))))))))))

C

z = abs(z);
double code(double x, double y, double z, double t) {
	double t_1 = t * (y * 4.0);
	double tmp;
	if (z <= 8.5e-224) {
		tmp = t_1;
	} else if (z <= 8e-168) {
		tmp = x * x;
	} else if (z <= 1.45e-78) {
		tmp = t_1;
	} else if (z <= 1.16e+36) {
		tmp = x * x;
	} else if (z <= 5.5e+56) {
		tmp = t_1;
	} else if (z <= 4.4e+83) {
		tmp = x * x;
	} else {
		tmp = -4.0 * ((z * z) * y);
	}
	return tmp;
}

Fortran

NOTE: z 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 = t * (y * 4.0d0)
    if (z <= 8.5d-224) then
        tmp = t_1
    else if (z <= 8d-168) then
        tmp = x * x
    else if (z <= 1.45d-78) then
        tmp = t_1
    else if (z <= 1.16d+36) then
        tmp = x * x
    else if (z <= 5.5d+56) then
        tmp = t_1
    else if (z <= 4.4d+83) then
        tmp = x * x
    else
        tmp = (-4.0d0) * ((z * z) * y)
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z, double t) {
	double t_1 = t * (y * 4.0);
	double tmp;
	if (z <= 8.5e-224) {
		tmp = t_1;
	} else if (z <= 8e-168) {
		tmp = x * x;
	} else if (z <= 1.45e-78) {
		tmp = t_1;
	} else if (z <= 1.16e+36) {
		tmp = x * x;
	} else if (z <= 5.5e+56) {
		tmp = t_1;
	} else if (z <= 4.4e+83) {
		tmp = x * x;
	} else {
		tmp = -4.0 * ((z * z) * y);
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z, t):
	t_1 = t * (y * 4.0)
	tmp = 0
	if z <= 8.5e-224:
		tmp = t_1
	elif z <= 8e-168:
		tmp = x * x
	elif z <= 1.45e-78:
		tmp = t_1
	elif z <= 1.16e+36:
		tmp = x * x
	elif z <= 5.5e+56:
		tmp = t_1
	elif z <= 4.4e+83:
		tmp = x * x
	else:
		tmp = -4.0 * ((z * z) * y)
	return tmp

Julia

z = abs(z)
function code(x, y, z, t)
	t_1 = Float64(t * Float64(y * 4.0))
	tmp = 0.0
	if (z <= 8.5e-224)
		tmp = t_1;
	elseif (z <= 8e-168)
		tmp = Float64(x * x);
	elseif (z <= 1.45e-78)
		tmp = t_1;
	elseif (z <= 1.16e+36)
		tmp = Float64(x * x);
	elseif (z <= 5.5e+56)
		tmp = t_1;
	elseif (z <= 4.4e+83)
		tmp = Float64(x * x);
	else
		tmp = Float64(-4.0 * Float64(Float64(z * z) * y));
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z, t)
	t_1 = t * (y * 4.0);
	tmp = 0.0;
	if (z <= 8.5e-224)
		tmp = t_1;
	elseif (z <= 8e-168)
		tmp = x * x;
	elseif (z <= 1.45e-78)
		tmp = t_1;
	elseif (z <= 1.16e+36)
		tmp = x * x;
	elseif (z <= 5.5e+56)
		tmp = t_1;
	elseif (z <= 4.4e+83)
		tmp = x * x;
	else
		tmp = -4.0 * ((z * z) * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(t * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 8.5e-224], t$95$1, If[LessEqual[z, 8e-168], N[(x * x), $MachinePrecision], If[LessEqual[z, 1.45e-78], t$95$1, If[LessEqual[z, 1.16e+36], N[(x * x), $MachinePrecision], If[LessEqual[z, 5.5e+56], t$95$1, If[LessEqual[z, 4.4e+83], N[(x * x), $MachinePrecision], N[(-4.0 * N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 8.4999999999999996e-224 or 8.0000000000000004e-168 < z < 1.45e-78 or 1.15999999999999998e36 < z < 5.5000000000000002e56

   1. Initial program 92.0%

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

   2. Taylor expanded in t around inf 46.0%

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

      1. associate-*r* 46.0%

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

   4. Simplified 46.0%

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

   if 8.4999999999999996e-224 < z < 8.0000000000000004e-168 or 1.45e-78 < z < 1.15999999999999998e36 or 5.5000000000000002e56 < z < 4.39999999999999997e83

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 71.0%

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

      1. unpow2 71.0%

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

   4. Simplified 71.0%

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

   if 4.39999999999999997e83 < z

   1. Initial program 76.3%

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

   2. Taylor expanded in z around inf 69.7%

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

      1. unpow2 69.7%

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

   4. Simplified 69.7%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8.5 \cdot 10^{-224}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-168}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-78}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;z \leq 1.16 \cdot 10^{+36}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+56}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{+83}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \end{array} \]

Alternative 3: 60.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} t_1 := t \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;z \leq 3.35 \cdot 10^{-223}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{-168}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{-78}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+35}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+83}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* t (* y 4.0))))
   (if (<= z 3.35e-223)
     t_1
     (if (<= z 4.1e-168)
       (* x x)
       (if (<= z 1.02e-78)
         t_1
         (if (<= z 5e+35)
           (* x x)
           (if (<= z 8e+62)
             t_1
             (if (<= z 4.2e+83) (* x x) (* z (* z (* y -4.0)))))))))))

C

z = abs(z);
double code(double x, double y, double z, double t) {
	double t_1 = t * (y * 4.0);
	double tmp;
	if (z <= 3.35e-223) {
		tmp = t_1;
	} else if (z <= 4.1e-168) {
		tmp = x * x;
	} else if (z <= 1.02e-78) {
		tmp = t_1;
	} else if (z <= 5e+35) {
		tmp = x * x;
	} else if (z <= 8e+62) {
		tmp = t_1;
	} else if (z <= 4.2e+83) {
		tmp = x * x;
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Fortran

NOTE: z 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 = t * (y * 4.0d0)
    if (z <= 3.35d-223) then
        tmp = t_1
    else if (z <= 4.1d-168) then
        tmp = x * x
    else if (z <= 1.02d-78) then
        tmp = t_1
    else if (z <= 5d+35) then
        tmp = x * x
    else if (z <= 8d+62) then
        tmp = t_1
    else if (z <= 4.2d+83) then
        tmp = x * x
    else
        tmp = z * (z * (y * (-4.0d0)))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z, double t) {
	double t_1 = t * (y * 4.0);
	double tmp;
	if (z <= 3.35e-223) {
		tmp = t_1;
	} else if (z <= 4.1e-168) {
		tmp = x * x;
	} else if (z <= 1.02e-78) {
		tmp = t_1;
	} else if (z <= 5e+35) {
		tmp = x * x;
	} else if (z <= 8e+62) {
		tmp = t_1;
	} else if (z <= 4.2e+83) {
		tmp = x * x;
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z, t):
	t_1 = t * (y * 4.0)
	tmp = 0
	if z <= 3.35e-223:
		tmp = t_1
	elif z <= 4.1e-168:
		tmp = x * x
	elif z <= 1.02e-78:
		tmp = t_1
	elif z <= 5e+35:
		tmp = x * x
	elif z <= 8e+62:
		tmp = t_1
	elif z <= 4.2e+83:
		tmp = x * x
	else:
		tmp = z * (z * (y * -4.0))
	return tmp

Julia

z = abs(z)
function code(x, y, z, t)
	t_1 = Float64(t * Float64(y * 4.0))
	tmp = 0.0
	if (z <= 3.35e-223)
		tmp = t_1;
	elseif (z <= 4.1e-168)
		tmp = Float64(x * x);
	elseif (z <= 1.02e-78)
		tmp = t_1;
	elseif (z <= 5e+35)
		tmp = Float64(x * x);
	elseif (z <= 8e+62)
		tmp = t_1;
	elseif (z <= 4.2e+83)
		tmp = Float64(x * x);
	else
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z, t)
	t_1 = t * (y * 4.0);
	tmp = 0.0;
	if (z <= 3.35e-223)
		tmp = t_1;
	elseif (z <= 4.1e-168)
		tmp = x * x;
	elseif (z <= 1.02e-78)
		tmp = t_1;
	elseif (z <= 5e+35)
		tmp = x * x;
	elseif (z <= 8e+62)
		tmp = t_1;
	elseif (z <= 4.2e+83)
		tmp = x * x;
	else
		tmp = z * (z * (y * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(t * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 3.35e-223], t$95$1, If[LessEqual[z, 4.1e-168], N[(x * x), $MachinePrecision], If[LessEqual[z, 1.02e-78], t$95$1, If[LessEqual[z, 5e+35], N[(x * x), $MachinePrecision], If[LessEqual[z, 8e+62], t$95$1, If[LessEqual[z, 4.2e+83], N[(x * x), $MachinePrecision], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 3.3500000000000001e-223 or 4.0999999999999998e-168 < z < 1.02e-78 or 5.00000000000000021e35 < z < 8.00000000000000028e62

   1. Initial program 92.0%

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

   2. Taylor expanded in t around inf 46.0%

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

      1. associate-*r* 46.0%

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

   4. Simplified 46.0%

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

   if 3.3500000000000001e-223 < z < 4.0999999999999998e-168 or 1.02e-78 < z < 5.00000000000000021e35 or 8.00000000000000028e62 < z < 4.20000000000000005e83

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 71.0%

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

      1. unpow2 71.0%

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

   4. Simplified 71.0%

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

   if 4.20000000000000005e83 < z

   1. Initial program 76.3%

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

   2. Taylor expanded in z around inf 69.7%

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

      1. metadata-eval 69.7%

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

      2. distribute-lft-neg-in 69.7%

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

      3. *-commutative 69.7%

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

      4. unpow2 69.7%

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

      5. *-commutative 69.7%

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

      6. associate-*r* 69.7%

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

      7. associate-*l* 79.5%

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

      8. distribute-rgt-neg-in 79.5%

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

      9. distribute-rgt-neg-in 79.5%

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

      10. distribute-rgt-neg-in 79.5%

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

      11. metadata-eval 79.5%

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

   4. Simplified 79.5%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.35 \cdot 10^{-223}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{-168}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{-78}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+35}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+62}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+83}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternative 4: 94.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 3.1 \cdot 10^{+138}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= z 3.1e+138)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (* z (* z (* y -4.0)))))

C

z = abs(z);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3.1e+138) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Fortran

NOTE: z 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 <= 3.1d+138) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = z * (z * (y * (-4.0d0)))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3.1e+138) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z, t):
	tmp = 0
	if z <= 3.1e+138:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = z * (z * (y * -4.0))
	return tmp

Julia

z = abs(z)
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 3.1e+138)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 3.1e+138)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = z * (z * (y * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[z, 3.1e+138], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.0999999999999998e138

   1. Initial program 93.5%

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

   if 3.0999999999999998e138 < z

   1. Initial program 71.0%

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

   2. Taylor expanded in z around inf 73.4%

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

      1. metadata-eval 73.4%

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

      2. distribute-lft-neg-in 73.4%

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

      3. *-commutative 73.4%

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

      4. unpow2 73.4%

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

      5. *-commutative 73.4%

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

      6. associate-*r* 73.4%

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

      7. associate-*l* 86.4%

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

      8. distribute-rgt-neg-in 86.4%

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

      9. distribute-rgt-neg-in 86.4%

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

      10. distribute-rgt-neg-in 86.4%

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

      11. metadata-eval 86.4%

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

   4. Simplified 86.4%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.1 \cdot 10^{+138}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternative 5: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 3 \cdot 10^{+213}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 3e+213) (* (- (* z z) t) (* y -4.0)) (* x x)))

C

z = abs(z);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 3e+213) {
		tmp = ((z * z) - t) * (y * -4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

NOTE: z 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) <= 3d+213) then
        tmp = ((z * z) - t) * (y * (-4.0d0))
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 3e+213) {
		tmp = ((z * z) - t) * (y * -4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 3e+213:
		tmp = ((z * z) - t) * (y * -4.0)
	else:
		tmp = x * x
	return tmp

Julia

z = abs(z)
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 3e+213)
		tmp = Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 3e+213)
		tmp = ((z * z) - t) * (y * -4.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 3e+213], N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 3.0000000000000001e213

   1. Initial program 93.9%

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

   2. Taylor expanded in x around 0 83.4%

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

      1. *-commutative 83.4%

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

      2. *-commutative 83.4%

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

      3. unpow2 83.4%

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

      4. *-commutative 83.4%

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

      5. associate-*l* 83.4%

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

   4. Simplified 83.4%

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

   if 3.0000000000000001e213 < (*.f64 x x)

   1. Initial program 81.0%

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

   2. Taylor expanded in x around inf 84.5%

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

      1. unpow2 84.5%

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

   4. Simplified 84.5%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 3 \cdot 10^{+213}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 6: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+167}:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+167) (- (* x x) (* t (* y -4.0))) (* z (* z (* y -4.0)))))

C

z = abs(z);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+167) {
		tmp = (x * x) - (t * (y * -4.0));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Fortran

NOTE: z 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) <= 2d+167) then
        tmp = (x * x) - (t * (y * (-4.0d0)))
    else
        tmp = z * (z * (y * (-4.0d0)))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+167) {
		tmp = (x * x) - (t * (y * -4.0));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 2e+167:
		tmp = (x * x) - (t * (y * -4.0))
	else:
		tmp = z * (z * (y * -4.0))
	return tmp

Julia

z = abs(z)
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+167)
		tmp = Float64(Float64(x * x) - Float64(t * Float64(y * -4.0)));
	else
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 2e+167)
		tmp = (x * x) - (t * (y * -4.0));
	else
		tmp = z * (z * (y * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+167], N[(N[(x * x), $MachinePrecision] - N[(t * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000001e167

   1. Initial program 98.6%

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

   2. Taylor expanded in z around 0 91.0%

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

      1. *-commutative 91.0%

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

      2. *-commutative 91.0%

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

      3. associate-*l* 91.0%

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

   4. Simplified 91.0%

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

   if 2.0000000000000001e167 < (*.f64 z z)

   1. Initial program 77.6%

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

   2. Taylor expanded in z around inf 74.5%

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

      1. metadata-eval 74.5%

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

      2. distribute-lft-neg-in 74.5%

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

      3. *-commutative 74.5%

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

      4. unpow2 74.5%

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

      5. *-commutative 74.5%

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

      6. associate-*r* 74.5%

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

      7. associate-*l* 82.2%

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

      8. distribute-rgt-neg-in 82.2%

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

      9. distribute-rgt-neg-in 82.2%

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

      10. distribute-rgt-neg-in 82.2%

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

      11. metadata-eval 82.2%

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

   4. Simplified 82.2%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+167}:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternative 7: 59.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 6.6 \cdot 10^{+37}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 6.6e+37) (* t (* y 4.0)) (* x x)))

C

z = abs(z);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 6.6e+37) {
		tmp = t * (y * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

NOTE: z 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) <= 6.6d+37) then
        tmp = t * (y * 4.0d0)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 6.6e+37) {
		tmp = t * (y * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 6.6e+37:
		tmp = t * (y * 4.0)
	else:
		tmp = x * x
	return tmp

Julia

z = abs(z)
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 6.6e+37)
		tmp = Float64(t * Float64(y * 4.0));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 6.6e+37)
		tmp = t * (y * 4.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 6.6e+37], N[(t * N[(y * 4.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 6.6000000000000002e37

   1. Initial program 93.5%

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

   2. Taylor expanded in t around inf 50.5%

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

      1. associate-*r* 50.5%

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

   4. Simplified 50.5%

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

   if 6.6000000000000002e37 < (*.f64 x x)

   1. Initial program 84.8%

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

   2. Taylor expanded in x around inf 71.5%

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

      1. unpow2 71.5%

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

   4. Simplified 71.5%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 6.6 \cdot 10^{+37}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 8: 41.7% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: z 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

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.7%

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

2. Taylor expanded in x around inf 36.5%

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

   1. unpow2 36.5%

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

4. Simplified 36.5%

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

5. Final simplification 36.5%

   \[\leadsto x \cdot x \]

Developer target: 91.2% 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 2023221 
(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))))