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

Percentage Accurate: 90.7% → 96.2%

Time: 9.1s

Alternatives: 9

Speedup: 0.8×

• 49.5% 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: 96.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(z \cdot \left(z \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+302)
   (fma x x (* (- (* z z) t) (* y -4.0)))
   (* -4.0 (* z (* z y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+302) {
		tmp = fma(x, x, (((z * z) - t) * (y * -4.0)));
	} else {
		tmp = -4.0 * (z * (z * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+302)
		tmp = fma(x, x, Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0)));
	else
		tmp = Float64(-4.0 * Float64(z * Float64(z * y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+302], N[(x * x + N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.0000000000000001e302

   1. Initial program 96.3%

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

      1. fma-neg 97.9%

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

      2. *-commutative 97.9%

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

      3. distribute-rgt-neg-in 97.9%

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

      4. distribute-rgt-neg-in 97.9%

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

      5. metadata-eval 97.9%

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

   3. Simplified 97.9%

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

   if 1.0000000000000001e302 < (*.f64 z z)

   1. Initial program 62.2%

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

   2. Taylor expanded in z around inf 74.7%

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

      1. unpow2 74.7%

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

      2. *-commutative 74.7%

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

      3. associate-*l* 84.7%

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

   4. Simplified 84.7%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(z \cdot \left(z \cdot y\right)\right)\\ \end{array} \]

Alternative 2: 89.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 500000000000:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \mathbf{elif}\;z \cdot z \leq 4 \cdot 10^{+276}:\\ \;\;\;\;x \cdot x - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(z \cdot \left(z \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 500000000000.0)
   (- (* x x) (* t (* y -4.0)))
   (if (<= (* z z) 4e+276)
     (- (* x x) (* (* z z) (* y 4.0)))
     (* -4.0 (* z (* z y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 500000000000.0) {
		tmp = (x * x) - (t * (y * -4.0));
	} else if ((z * z) <= 4e+276) {
		tmp = (x * x) - ((z * z) * (y * 4.0));
	} else {
		tmp = -4.0 * (z * (z * y));
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((z * z) <= 500000000000.0d0) then
        tmp = (x * x) - (t * (y * (-4.0d0)))
    else if ((z * z) <= 4d+276) then
        tmp = (x * x) - ((z * z) * (y * 4.0d0))
    else
        tmp = (-4.0d0) * (z * (z * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 500000000000.0) {
		tmp = (x * x) - (t * (y * -4.0));
	} else if ((z * z) <= 4e+276) {
		tmp = (x * x) - ((z * z) * (y * 4.0));
	} else {
		tmp = -4.0 * (z * (z * y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 500000000000.0:
		tmp = (x * x) - (t * (y * -4.0))
	elif (z * z) <= 4e+276:
		tmp = (x * x) - ((z * z) * (y * 4.0))
	else:
		tmp = -4.0 * (z * (z * y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 500000000000.0)
		tmp = Float64(Float64(x * x) - Float64(t * Float64(y * -4.0)));
	elseif (Float64(z * z) <= 4e+276)
		tmp = Float64(Float64(x * x) - Float64(Float64(z * z) * Float64(y * 4.0)));
	else
		tmp = Float64(-4.0 * Float64(z * Float64(z * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 500000000000.0)
		tmp = (x * x) - (t * (y * -4.0));
	elseif ((z * z) <= 4e+276)
		tmp = (x * x) - ((z * z) * (y * 4.0));
	else
		tmp = -4.0 * (z * (z * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 500000000000.0], N[(N[(x * x), $MachinePrecision] - N[(t * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 4e+276], N[(N[(x * x), $MachinePrecision] - N[(N[(z * z), $MachinePrecision] * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 5e11

   1. Initial program 98.3%

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

   2. Taylor expanded in z around 0 95.3%

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

      1. associate-*r* 95.3%

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

   4. Simplified 95.3%

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

   if 5e11 < (*.f64 z z) < 4.0000000000000002e276

   1. Initial program 94.8%

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

   2. Taylor expanded in z around inf 84.0%

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

      1. *-commutative 84.0%

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

      2. unpow2 84.0%

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

      3. *-commutative 84.0%

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

      4. associate-*r* 84.0%

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

   4. Simplified 84.0%

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

   if 4.0000000000000002e276 < (*.f64 z z)

   1. Initial program 64.6%

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

   2. Taylor expanded in z around inf 75.5%

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

      1. unpow2 75.5%

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

      2. *-commutative 75.5%

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

      3. associate-*l* 84.2%

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

   4. Simplified 84.2%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 500000000000:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \mathbf{elif}\;z \cdot z \leq 4 \cdot 10^{+276}:\\ \;\;\;\;x \cdot x - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(z \cdot \left(z \cdot y\right)\right)\\ \end{array} \]

Alternative 3: 95.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+276}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(z \cdot \left(z \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 4e+276)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (* -4.0 (* z (* z y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 4e+276) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = -4.0 * (z * (z * y));
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((z * z) <= 4d+276) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = (-4.0d0) * (z * (z * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 4e+276) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = -4.0 * (z * (z * y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 4e+276:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = -4.0 * (z * (z * y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 4e+276)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(-4.0 * Float64(z * Float64(z * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 4e+276)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = -4.0 * (z * (z * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 4e+276], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.0000000000000002e276

   1. Initial program 97.2%

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

   if 4.0000000000000002e276 < (*.f64 z z)

   1. Initial program 64.6%

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

   2. Taylor expanded in z around inf 75.5%

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

      1. unpow2 75.5%

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

      2. *-commutative 75.5%

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

      3. associate-*l* 84.2%

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

   4. Simplified 84.2%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+276}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(z \cdot \left(z \cdot y\right)\right)\\ \end{array} \]

Alternative 4: 59.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -4 \cdot \left(z \cdot \left(z \cdot y\right)\right)\\ \mathbf{if}\;x \leq -2.35 \cdot 10^{+82}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-181}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-162}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;x \leq 2400000000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* -4.0 (* z (* z y)))))
   (if (<= x -2.35e+82)
     (* x x)
     (if (<= x -1.5e-181)
       t_1
       (if (<= x 3.1e-162)
         (* t (* y 4.0))
         (if (<= x 2400000000000.0) t_1 (* x x)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -4.0 * (z * (z * y));
	double tmp;
	if (x <= -2.35e+82) {
		tmp = x * x;
	} else if (x <= -1.5e-181) {
		tmp = t_1;
	} else if (x <= 3.1e-162) {
		tmp = t * (y * 4.0);
	} else if (x <= 2400000000000.0) {
		tmp = t_1;
	} else {
		tmp = x * x;
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-4.0d0) * (z * (z * y))
    if (x <= (-2.35d+82)) then
        tmp = x * x
    else if (x <= (-1.5d-181)) then
        tmp = t_1
    else if (x <= 3.1d-162) then
        tmp = t * (y * 4.0d0)
    else if (x <= 2400000000000.0d0) then
        tmp = t_1
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = -4.0 * (z * (z * y));
	double tmp;
	if (x <= -2.35e+82) {
		tmp = x * x;
	} else if (x <= -1.5e-181) {
		tmp = t_1;
	} else if (x <= 3.1e-162) {
		tmp = t * (y * 4.0);
	} else if (x <= 2400000000000.0) {
		tmp = t_1;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = -4.0 * (z * (z * y))
	tmp = 0
	if x <= -2.35e+82:
		tmp = x * x
	elif x <= -1.5e-181:
		tmp = t_1
	elif x <= 3.1e-162:
		tmp = t * (y * 4.0)
	elif x <= 2400000000000.0:
		tmp = t_1
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(-4.0 * Float64(z * Float64(z * y)))
	tmp = 0.0
	if (x <= -2.35e+82)
		tmp = Float64(x * x);
	elseif (x <= -1.5e-181)
		tmp = t_1;
	elseif (x <= 3.1e-162)
		tmp = Float64(t * Float64(y * 4.0));
	elseif (x <= 2400000000000.0)
		tmp = t_1;
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = -4.0 * (z * (z * y));
	tmp = 0.0;
	if (x <= -2.35e+82)
		tmp = x * x;
	elseif (x <= -1.5e-181)
		tmp = t_1;
	elseif (x <= 3.1e-162)
		tmp = t * (y * 4.0);
	elseif (x <= 2400000000000.0)
		tmp = t_1;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(-4.0 * N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.35e+82], N[(x * x), $MachinePrecision], If[LessEqual[x, -1.5e-181], t$95$1, If[LessEqual[x, 3.1e-162], N[(t * N[(y * 4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2400000000000.0], t$95$1, N[(x * x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.35e82 or 2.4e12 < x

   1. Initial program 80.9%

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

   2. Taylor expanded in x around inf 81.9%

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

      1. unpow2 81.9%

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

   4. Simplified 81.9%

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

   if -2.35e82 < x < -1.49999999999999987e-181 or 3.0999999999999999e-162 < x < 2.4e12

   1. Initial program 92.6%

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

   2. Taylor expanded in z around inf 52.3%

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

      1. unpow2 52.3%

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

      2. *-commutative 52.3%

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

      3. associate-*l* 59.6%

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

   4. Simplified 59.6%

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

   if -1.49999999999999987e-181 < x < 3.0999999999999999e-162

   1. Initial program 97.7%

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

   2. Taylor expanded in t around inf 77.0%

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

      1. associate-*r* 77.0%

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

      2. *-commutative 77.0%

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

   4. Simplified 77.0%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{+82}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-181}:\\ \;\;\;\;-4 \cdot \left(z \cdot \left(z \cdot y\right)\right)\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-162}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;x \leq 2400000000000:\\ \;\;\;\;-4 \cdot \left(z \cdot \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 5: 90.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 500000000000:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \frac{y}{\frac{0.25}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 500000000000.0)
   (- (* x x) (* t (* y -4.0)))
   (- (* x x) (* z (/ y (/ 0.25 z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 500000000000.0) {
		tmp = (x * x) - (t * (y * -4.0));
	} else {
		tmp = (x * x) - (z * (y / (0.25 / z)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((z * z) <= 500000000000.0d0) then
        tmp = (x * x) - (t * (y * (-4.0d0)))
    else
        tmp = (x * x) - (z * (y / (0.25d0 / z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 500000000000.0) {
		tmp = (x * x) - (t * (y * -4.0));
	} else {
		tmp = (x * x) - (z * (y / (0.25 / z)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 500000000000.0:
		tmp = (x * x) - (t * (y * -4.0))
	else:
		tmp = (x * x) - (z * (y / (0.25 / z)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 500000000000.0)
		tmp = Float64(Float64(x * x) - Float64(t * Float64(y * -4.0)));
	else
		tmp = Float64(Float64(x * x) - Float64(z * Float64(y / Float64(0.25 / z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 500000000000.0)
		tmp = (x * x) - (t * (y * -4.0));
	else
		tmp = (x * x) - (z * (y / (0.25 / z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 500000000000.0], N[(N[(x * x), $MachinePrecision] - N[(t * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(z * N[(y / N[(0.25 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5e11

   1. Initial program 98.3%

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

   2. Taylor expanded in z around 0 95.3%

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

      1. associate-*r* 95.3%

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

   4. Simplified 95.3%

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

   if 5e11 < (*.f64 z z)

   1. Initial program 78.0%

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

   2. Taylor expanded in z around 0 78.0%

      \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \color{blue}{\left({z}^{2} + -1 \cdot t\right)} \]
   3. Step-by-step derivation

      1. unpow2 78.0%

         \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \left(\color{blue}{z \cdot z} + -1 \cdot t\right) \]

      2. neg-mul-1 78.0%

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

      3. fma-def 78.0%

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

   4. Simplified 78.0%

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

      1. fma-neg 78.0%

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

      2. flip-- 26.7%

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

      3. associate-*r/ 21.0%

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

      4. difference-of-squares 25.7%

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

      5. fma-def 25.7%

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

      6. add-sqr-sqrt 11.6%

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

      7. sqrt-unprod 23.3%

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

      8. sqr-neg 23.3%

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

      9. sqrt-unprod 11.8%

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

      10. add-sqr-sqrt 21.1%

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

      11. fma-neg 21.1%

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

      12. pow2 21.1%

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

      13. add-sqr-sqrt 11.8%

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

      14. sqrt-unprod 21.1%

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

      15. sqr-neg 21.1%

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

      16. sqrt-unprod 9.3%

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

      17. add-sqr-sqrt 21.1%

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

      18. fma-def 21.1%

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

   6. Applied egg-rr 21.1%

      \[\leadsto x \cdot x - \color{blue}{\frac{\left(y \cdot 4\right) \cdot {\left(\mathsf{fma}\left(z, z, t\right)\right)}^{2}}{\mathsf{fma}\left(z, z, t\right)}} \]
   7. Step-by-step derivation

      1. associate-/l* 26.5%

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

      2. associate-/l* 26.5%

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

      3. unpow2 26.5%

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

      4. associate-/r* 41.6%

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

      5. *-inverses 71.5%

         \[\leadsto x \cdot x - \frac{y}{\frac{\frac{\color{blue}{1}}{\mathsf{fma}\left(z, z, t\right)}}{4}} \]

   8. Simplified 71.5%

      \[\leadsto x \cdot x - \color{blue}{\frac{y}{\frac{\frac{1}{\mathsf{fma}\left(z, z, t\right)}}{4}}} \]

   9. Taylor expanded in z around inf 73.2%

      \[\leadsto x \cdot x - \frac{y}{\color{blue}{\frac{0.25}{{z}^{2}}}} \]
   10. Step-by-step derivation

       1. unpow2 73.2%

          \[\leadsto x \cdot x - \frac{y}{\frac{0.25}{\color{blue}{z \cdot z}}} \]

   11. Simplified 73.2%

       \[\leadsto x \cdot x - \frac{y}{\color{blue}{\frac{0.25}{z \cdot z}}} \]
   12. Step-by-step derivation

       1. associate-/r* 74.3%

          \[\leadsto x \cdot x - \frac{y}{\color{blue}{\frac{\frac{0.25}{z}}{z}}} \]

       2. associate-/r/ 83.1%

          \[\leadsto x \cdot x - \color{blue}{\frac{y}{\frac{0.25}{z}} \cdot z} \]

   13. Applied egg-rr 83.1%

       \[\leadsto x \cdot x - \color{blue}{\frac{y}{\frac{0.25}{z}} \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 500000000000:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \frac{y}{\frac{0.25}{z}}\\ \end{array} \]

Alternative 6: 78.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+83}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-28}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -4.2e+83)
   (* x x)
   (if (<= x 5.8e-28) (* (- (* z z) t) (* y -4.0)) (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -4.2e+83) {
		tmp = x * x;
	} else if (x <= 5.8e-28) {
		tmp = ((z * z) - t) * (y * -4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (x <= (-4.2d+83)) then
        tmp = x * x
    else if (x <= 5.8d-28) then
        tmp = ((z * z) - t) * (y * (-4.0d0))
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -4.2e+83) {
		tmp = x * x;
	} else if (x <= 5.8e-28) {
		tmp = ((z * z) - t) * (y * -4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -4.2e+83:
		tmp = x * x
	elif x <= 5.8e-28:
		tmp = ((z * z) - t) * (y * -4.0)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -4.2e+83)
		tmp = Float64(x * x);
	elseif (x <= 5.8e-28)
		tmp = Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -4.2e+83)
		tmp = x * x;
	elseif (x <= 5.8e-28)
		tmp = ((z * z) - t) * (y * -4.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -4.2e+83], N[(x * x), $MachinePrecision], If[LessEqual[x, 5.8e-28], 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 x < -4.20000000000000005e83 or 5.80000000000000026e-28 < x

   1. Initial program 80.5%

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

   2. Taylor expanded in x around inf 79.8%

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

      1. unpow2 79.8%

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

   4. Simplified 79.8%

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

   if -4.20000000000000005e83 < x < 5.80000000000000026e-28

   1. Initial program 95.4%

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

   2. Taylor expanded in x around 0 89.5%

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

      1. associate-*r* 88.8%

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

      2. unpow2 88.8%

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

      3. *-commutative 88.8%

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

      4. associate-*l* 89.5%

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

   4. Simplified 89.5%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+83}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-28}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 7: 84.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+225}:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(z \cdot \left(z \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+225) (- (* x x) (* t (* y -4.0))) (* -4.0 (* z (* z y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+225) {
		tmp = (x * x) - (t * (y * -4.0));
	} else {
		tmp = -4.0 * (z * (z * y));
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((z * z) <= 2d+225) then
        tmp = (x * x) - (t * (y * (-4.0d0)))
    else
        tmp = (-4.0d0) * (z * (z * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+225) {
		tmp = (x * x) - (t * (y * -4.0));
	} else {
		tmp = -4.0 * (z * (z * y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 2e+225:
		tmp = (x * x) - (t * (y * -4.0))
	else:
		tmp = -4.0 * (z * (z * y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+225)
		tmp = Float64(Float64(x * x) - Float64(t * Float64(y * -4.0)));
	else
		tmp = Float64(-4.0 * Float64(z * Float64(z * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 2e+225)
		tmp = (x * x) - (t * (y * -4.0));
	else
		tmp = -4.0 * (z * (z * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+225], N[(N[(x * x), $MachinePrecision] - N[(t * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999986e225

   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 87.6%

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

      1. associate-*r* 87.6%

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

   4. Simplified 87.6%

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

   if 1.99999999999999986e225 < (*.f64 z z)

   1. Initial program 67.2%

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

   2. Taylor expanded in z around inf 73.2%

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

      1. unpow2 73.2%

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

      2. *-commutative 73.2%

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

      3. associate-*l* 80.7%

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

   4. Simplified 80.7%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+225}:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(z \cdot \left(z \cdot y\right)\right)\\ \end{array} \]

Alternative 8: 56.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+80}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{-59}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -7.5e+80) (* x x) (if (<= x 6.4e-59) (* t (* y 4.0)) (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -7.5e+80) {
		tmp = x * x;
	} else if (x <= 6.4e-59) {
		tmp = t * (y * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (x <= (-7.5d+80)) then
        tmp = x * x
    else if (x <= 6.4d-59) then
        tmp = t * (y * 4.0d0)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -7.5e+80) {
		tmp = x * x;
	} else if (x <= 6.4e-59) {
		tmp = t * (y * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -7.5e+80:
		tmp = x * x
	elif x <= 6.4e-59:
		tmp = t * (y * 4.0)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -7.5e+80)
		tmp = Float64(x * x);
	elseif (x <= 6.4e-59)
		tmp = Float64(t * Float64(y * 4.0));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -7.5e+80)
		tmp = x * x;
	elseif (x <= 6.4e-59)
		tmp = t * (y * 4.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -7.5e+80], N[(x * x), $MachinePrecision], If[LessEqual[x, 6.4e-59], N[(t * N[(y * 4.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.49999999999999994e80 or 6.3999999999999998e-59 < x

   1. Initial program 81.5%

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

   2. Taylor expanded in x around inf 77.2%

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

      1. unpow2 77.2%

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

   4. Simplified 77.2%

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

   if -7.49999999999999994e80 < x < 6.3999999999999998e-59

   1. Initial program 95.1%

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

   2. Taylor expanded in t around inf 53.4%

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

      1. associate-*r* 53.4%

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

      2. *-commutative 53.4%

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

   4. Simplified 53.4%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+80}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{-59}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 9: 40.2% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(x * x), $MachinePrecision]

Derivation

1. Initial program 87.8%

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

2. Taylor expanded in x around inf 45.5%

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

   1. unpow2 45.5%

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

4. Simplified 45.5%

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

5. Final simplification 45.5%

   \[\leadsto x \cdot x \]

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 2023182 
(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))))