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

Percentage Accurate: 90.7% → 95.3%

Time: 8.6s

Alternatives: 8

Speedup: 0.6×

• 48.5% 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: 95.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+307}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e+307)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (* z (* z (* y -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e+307) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	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) <= 5d+307) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = z * (z * (y * (-4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e+307) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 5e+307:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = z * (z * (y * -4.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e+307)
		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

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 5e+307)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = z * (z * (y * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+307], 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 (*.f64 z z) < 5e307

   1. Initial program 97.4%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   if 5e307 < (*.f64 z z)

   1. Initial program 67.2%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \color{blue}{\left(y \cdot {z}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \color{blue}{\left({z}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \left(z \cdot \color{blue}{z}\right)\right)\right) \]

      4. *-lowering-*.f64 75.4%

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

   5. Simplified 75.4%

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(4\right)\right) \cdot y\right) \cdot \left(z \cdot z\right) \]

      3. distribute-lft-neg-in N/A

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

      4. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z\right) \]

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot z\right), \color{blue}{z}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{neg}\left(y \cdot 4\right)\right), z\right), z\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot \left(\mathsf{neg}\left(4\right)\right)\right), z\right), z\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot -4\right), z\right), z\right) \]

      10. *-lowering-*.f64 93.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, -4\right), z\right), z\right) \]

   7. Applied egg-rr 93.5%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+307}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-13}:\\ \;\;\;\;x \cdot x - -4 \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+307}:\\ \;\;\;\;x \cdot x - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e-13)
   (- (* x x) (* -4.0 (* y t)))
   (if (<= (* z z) 5e+307)
     (- (* x x) (* (* z z) (* y 4.0)))
     (* z (* z (* y -4.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e-13) {
		tmp = (x * x) - (-4.0 * (y * t));
	} else if ((z * z) <= 5e+307) {
		tmp = (x * x) - ((z * z) * (y * 4.0));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	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-13) then
        tmp = (x * x) - ((-4.0d0) * (y * t))
    else if ((z * z) <= 5d+307) then
        tmp = (x * x) - ((z * z) * (y * 4.0d0))
    else
        tmp = z * (z * (y * (-4.0d0)))
    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-13) {
		tmp = (x * x) - (-4.0 * (y * t));
	} else if ((z * z) <= 5e+307) {
		tmp = (x * x) - ((z * z) * (y * 4.0));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 2e-13:
		tmp = (x * x) - (-4.0 * (y * t))
	elif (z * z) <= 5e+307:
		tmp = (x * x) - ((z * z) * (y * 4.0))
	else:
		tmp = z * (z * (y * -4.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e-13)
		tmp = Float64(Float64(x * x) - Float64(-4.0 * Float64(y * t)));
	elseif (Float64(z * z) <= 5e+307)
		tmp = Float64(Float64(x * x) - Float64(Float64(z * z) * Float64(y * 4.0)));
	else
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 2e-13)
		tmp = (x * x) - (-4.0 * (y * t));
	elseif ((z * z) <= 5e+307)
		tmp = (x * x) - ((z * z) * (y * 4.0));
	else
		tmp = z * (z * (y * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e-13], N[(N[(x * x), $MachinePrecision] - N[(-4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 5e+307], N[(N[(x * x), $MachinePrecision] - N[(N[(z * z), $MachinePrecision] * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 2.0000000000000001e-13

   1. Initial program 97.0%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \color{blue}{\left(-4 \cdot \left(t \cdot y\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(-4, \color{blue}{\left(t \cdot y\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(-4, \left(y \cdot \color{blue}{t}\right)\right)\right) \]

      3. *-lowering-*.f64 94.6%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \color{blue}{t}\right)\right)\right) \]

   5. Simplified 94.6%

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

   if 2.0000000000000001e-13 < (*.f64 z z) < 5e307

   1. Initial program 98.3%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), \color{blue}{\left({z}^{2}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), \left(z \cdot \color{blue}{z}\right)\right)\right) \]

      2. *-lowering-*.f64 83.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

   5. Simplified 83.2%

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

   if 5e307 < (*.f64 z z)

   1. Initial program 67.2%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \color{blue}{\left(y \cdot {z}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \color{blue}{\left({z}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \left(z \cdot \color{blue}{z}\right)\right)\right) \]

      4. *-lowering-*.f64 75.4%

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

   5. Simplified 75.4%

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(4\right)\right) \cdot y\right) \cdot \left(z \cdot z\right) \]

      3. distribute-lft-neg-in N/A

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

      4. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z\right) \]

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot z\right), \color{blue}{z}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{neg}\left(y \cdot 4\right)\right), z\right), z\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot \left(\mathsf{neg}\left(4\right)\right)\right), z\right), z\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot -4\right), z\right), z\right) \]

      10. *-lowering-*.f64 93.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, -4\right), z\right), z\right) \]

   7. Applied egg-rr 93.5%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-13}:\\ \;\;\;\;x \cdot x - -4 \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+307}:\\ \;\;\;\;x \cdot x - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+155}:\\ \;\;\;\;x \cdot x - -4 \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e+155) (- (* x x) (* -4.0 (* y t))) (* z (* z (* y -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e+155) {
		tmp = (x * x) - (-4.0 * (y * t));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	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) <= 5d+155) then
        tmp = (x * x) - ((-4.0d0) * (y * t))
    else
        tmp = z * (z * (y * (-4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e+155) {
		tmp = (x * x) - (-4.0 * (y * t));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 5e+155:
		tmp = (x * x) - (-4.0 * (y * t))
	else:
		tmp = z * (z * (y * -4.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e+155)
		tmp = Float64(Float64(x * x) - Float64(-4.0 * Float64(y * t)));
	else
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 5e+155)
		tmp = (x * x) - (-4.0 * (y * t));
	else
		tmp = z * (z * (y * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+155], N[(N[(x * x), $MachinePrecision] - N[(-4.0 * N[(y * t), $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) < 4.9999999999999999e155

   1. Initial program 97.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \color{blue}{\left(-4 \cdot \left(t \cdot y\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(-4, \color{blue}{\left(t \cdot y\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(-4, \left(y \cdot \color{blue}{t}\right)\right)\right) \]

      3. *-lowering-*.f64 88.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \color{blue}{t}\right)\right)\right) \]

   5. Simplified 88.9%

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

   if 4.9999999999999999e155 < (*.f64 z z)

   1. Initial program 76.4%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \color{blue}{\left(y \cdot {z}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \color{blue}{\left({z}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \left(z \cdot \color{blue}{z}\right)\right)\right) \]

      4. *-lowering-*.f64 73.4%

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

   5. Simplified 73.4%

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(4\right)\right) \cdot y\right) \cdot \left(z \cdot z\right) \]

      3. distribute-lft-neg-in N/A

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

      4. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z\right) \]

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot z\right), \color{blue}{z}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{neg}\left(y \cdot 4\right)\right), z\right), z\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot \left(\mathsf{neg}\left(4\right)\right)\right), z\right), z\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot -4\right), z\right), z\right) \]

      10. *-lowering-*.f64 85.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, -4\right), z\right), z\right) \]

   7. Applied egg-rr 85.8%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+155}:\\ \;\;\;\;x \cdot x - -4 \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4.6 \cdot 10^{+47}:\\ \;\;\;\;-4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 4.6e+47) (* -4.0 (* y (- (* z z) t))) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 4.6e+47) {
		tmp = -4.0 * (y * ((z * z) - t));
	} 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 * x) <= 4.6d+47) then
        tmp = (-4.0d0) * (y * ((z * z) - t))
    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 * x) <= 4.6e+47) {
		tmp = -4.0 * (y * ((z * z) - t));
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 4.6e+47:
		tmp = -4.0 * (y * ((z * z) - t))
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 4.6e+47)
		tmp = Float64(-4.0 * Float64(y * Float64(Float64(z * z) - t)));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 4.6e+47)
		tmp = -4.0 * (y * ((z * z) - t));
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 4.6e+47], N[(-4.0 * N[(y * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.5999999999999997e47

   1. Initial program 91.5%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \color{blue}{\left(y \cdot \left({z}^{2} - t\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \color{blue}{\left({z}^{2} - t\right)}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left({z}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(z \cdot z\right), t\right)\right)\right) \]

      5. *-lowering-*.f64 82.9%

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), t\right)\right)\right) \]

   5. Simplified 82.9%

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

   if 4.5999999999999997e47 < (*.f64 x x)

   1. Initial program 88.7%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 73.6%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

   5. Simplified 73.6%

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

Alternative 5: 61.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+155}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e+155) (* x x) (* z (* z (* y -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e+155) {
		tmp = x * x;
	} else {
		tmp = z * (z * (y * -4.0));
	}
	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) <= 5d+155) then
        tmp = x * x
    else
        tmp = z * (z * (y * (-4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e+155) {
		tmp = x * x;
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 5e+155:
		tmp = x * x
	else:
		tmp = z * (z * (y * -4.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e+155)
		tmp = Float64(x * x);
	else
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 5e+155)
		tmp = x * x;
	else
		tmp = z * (z * (y * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+155], N[(x * x), $MachinePrecision], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.9999999999999999e155

   1. Initial program 97.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 53.8%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

   5. Simplified 53.8%

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

   if 4.9999999999999999e155 < (*.f64 z z)

   1. Initial program 76.4%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \color{blue}{\left(y \cdot {z}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \color{blue}{\left({z}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \left(z \cdot \color{blue}{z}\right)\right)\right) \]

      4. *-lowering-*.f64 73.4%

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

   5. Simplified 73.4%

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(4\right)\right) \cdot y\right) \cdot \left(z \cdot z\right) \]

      3. distribute-lft-neg-in N/A

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

      4. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z\right) \]

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot z\right), \color{blue}{z}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{neg}\left(y \cdot 4\right)\right), z\right), z\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot \left(\mathsf{neg}\left(4\right)\right)\right), z\right), z\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot -4\right), z\right), z\right) \]

      10. *-lowering-*.f64 85.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, -4\right), z\right), z\right) \]

   7. Applied egg-rr 85.8%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+155}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+156}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+156) (* x x) (* -4.0 (* (* z z) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+156) {
		tmp = x * x;
	} 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+156) then
        tmp = x * x
    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+156) {
		tmp = x * x;
	} else {
		tmp = -4.0 * ((z * z) * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 2e+156:
		tmp = x * x
	else:
		tmp = -4.0 * ((z * z) * y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+156)
		tmp = Float64(x * x);
	else
		tmp = Float64(-4.0 * Float64(Float64(z * z) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 2e+156)
		tmp = x * x;
	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+156], N[(x * x), $MachinePrecision], N[(-4.0 * N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2e156

   1. Initial program 97.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 53.8%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

   5. Simplified 53.8%

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

   if 2e156 < (*.f64 z z)

   1. Initial program 76.4%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \color{blue}{\left(y \cdot {z}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \color{blue}{\left({z}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \left(z \cdot \color{blue}{z}\right)\right)\right) \]

      4. *-lowering-*.f64 73.4%

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

   5. Simplified 73.4%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+156}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 45.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{+18}:\\ \;\;\;\;\left(y \cdot t\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 2.6e+18) (* (* y t) 4.0) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 2.6e+18) {
		tmp = (y * t) * 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 <= 2.6d+18) then
        tmp = (y * t) * 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 <= 2.6e+18) {
		tmp = (y * t) * 4.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 2.6e+18:
		tmp = (y * t) * 4.0
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 2.6e+18)
		tmp = Float64(Float64(y * t) * 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 <= 2.6e+18)
		tmp = (y * t) * 4.0;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 2.6e+18], N[(N[(y * t), $MachinePrecision] * 4.0), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.6e18

   1. Initial program 91.2%

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

      1. flip-- N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(y \cdot 4\right) \cdot \frac{\left(z \cdot z\right) \cdot \left(z \cdot z\right) - t \cdot t}{\color{blue}{z \cdot z + t}}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(y \cdot 4\right) \cdot \frac{1}{\color{blue}{\frac{z \cdot z + t}{\left(z \cdot z\right) \cdot \left(z \cdot z\right) - t \cdot t}}}\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{y \cdot 4}{\color{blue}{\frac{z \cdot z + t}{\left(z \cdot z\right) \cdot \left(z \cdot z\right) - t \cdot t}}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\left(y \cdot 4\right), \color{blue}{\left(\frac{z \cdot z + t}{\left(z \cdot z\right) \cdot \left(z \cdot z\right) - t \cdot t}\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \left(\frac{\color{blue}{z \cdot z + t}}{\left(z \cdot z\right) \cdot \left(z \cdot z\right) - t \cdot t}\right)\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \left(\frac{1}{\color{blue}{\frac{\left(z \cdot z\right) \cdot \left(z \cdot z\right) - t \cdot t}{z \cdot z + t}}}\right)\right)\right) \]

      7. flip-- N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \left(\frac{1}{z \cdot z - \color{blue}{t}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{/.f64}\left(1, \color{blue}{\left(z \cdot z - t\right)}\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(z \cdot z\right), \color{blue}{t}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 91.1%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), t\right)\right)\right)\right) \]

   4. Applied egg-rr 91.1%

      \[\leadsto x \cdot x - \color{blue}{\frac{y \cdot 4}{\frac{1}{z \cdot z - t}}} \]

   5. Taylor expanded in t around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(4, \color{blue}{\left(t \cdot y\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(4, \left(y \cdot \color{blue}{t}\right)\right) \]

      3. *-lowering-*.f64 37.9%

         \[\leadsto \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(y, \color{blue}{t}\right)\right) \]

   7. Simplified 37.9%

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

   if 2.6e18 < x

   1. Initial program 87.0%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 69.7%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

   5. Simplified 69.7%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{+18}:\\ \;\;\;\;\left(y \cdot t\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 41.1% 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 90.2%

   \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 42.6%

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

5. Simplified 42.6%

   \[\leadsto \color{blue}{x \cdot x} \]
6. Add Preprocessing

Developer Target 1: 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 2024152 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* x x) (* 4 (* y (- (* z z) t)))))

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