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

Percentage Accurate: 90.2% → 97.5%

Time: 13.6s

Alternatives: 10

Speedup: 0.9×

• 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: 90.2% 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: 97.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+195}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+195)
   (fma x x (* (- (* z z) t) (* y -4.0)))
   (fma x x (* z (* z (* y -4.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+195) {
		tmp = fma(x, x, (((z * z) - t) * (y * -4.0)));
	} else {
		tmp = fma(x, x, (z * (z * (y * -4.0))));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+195)
		tmp = fma(x, x, Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0)));
	else
		tmp = fma(x, x, Float64(z * Float64(z * Float64(y * -4.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+195], N[(x * x + N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999995e195

   1. Initial program 98.8%

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

      1. fma-neg 98.8%

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

      2. *-commutative 98.8%

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

      3. distribute-rgt-neg-in 98.8%

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

      4. distribute-rgt-neg-in 98.8%

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

      5. metadata-eval 98.8%

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

   3. Simplified 98.8%

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

   if 1.99999999999999995e195 < (*.f64 z z)

   1. Initial program 68.3%

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

   2. Taylor expanded in t around 0 68.3%

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

      1. unpow2 68.3%

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

      2. fma-neg 76.1%

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

      3. *-commutative 76.1%

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

      4. unpow2 76.1%

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

      5. *-commutative 76.1%

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

      6. associate-*r* 76.1%

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

      7. associate-*l* 96.5%

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

      8. distribute-rgt-neg-in 96.5%

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

      9. distribute-rgt-neg-in 96.5%

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

      10. distribute-rgt-neg-in 96.5%

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

      11. metadata-eval 96.5%

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

   4. Simplified 96.5%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+195}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\right)\\ \end{array} \]

Alternative 2: 97.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+195}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+195)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (fma x x (* z (* z (* y -4.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+195) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = fma(x, x, (z * (z * (y * -4.0))));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+195)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = fma(x, x, Float64(z * Float64(z * Float64(y * -4.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+195], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999995e195

   1. Initial program 98.8%

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

   if 1.99999999999999995e195 < (*.f64 z z)

   1. Initial program 68.3%

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

   2. Taylor expanded in t around 0 68.3%

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

      1. unpow2 68.3%

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

      2. fma-neg 76.1%

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

      3. *-commutative 76.1%

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

      4. unpow2 76.1%

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

      5. *-commutative 76.1%

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

      6. associate-*r* 76.1%

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

      7. associate-*l* 96.5%

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

      8. distribute-rgt-neg-in 96.5%

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

      9. distribute-rgt-neg-in 96.5%

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

      10. distribute-rgt-neg-in 96.5%

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

      11. metadata-eval 96.5%

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

   4. Simplified 96.5%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+195}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\right)\\ \end{array} \]

Alternative 3: 77.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.7 \cdot 10^{+35}:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+213}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 2.7e+35)
   (- (* x x) (* t (* y -4.0)))
   (if (<= z 5e+213)
     (- (* 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 <= 2.7e+35) {
		tmp = (x * x) - (t * (y * -4.0));
	} else if (z <= 5e+213) {
		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 <= 2.7d+35) then
        tmp = (x * x) - (t * (y * (-4.0d0)))
    else if (z <= 5d+213) 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 <= 2.7e+35) {
		tmp = (x * x) - (t * (y * -4.0));
	} else if (z <= 5e+213) {
		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 <= 2.7e+35:
		tmp = (x * x) - (t * (y * -4.0))
	elif z <= 5e+213:
		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 (z <= 2.7e+35)
		tmp = Float64(Float64(x * x) - Float64(t * Float64(y * -4.0)));
	elseif (z <= 5e+213)
		tmp = Float64(Float64(x * x) - Float64(z * Float64(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 <= 2.7e+35)
		tmp = (x * x) - (t * (y * -4.0));
	elseif (z <= 5e+213)
		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[z, 2.7e+35], N[(N[(x * x), $MachinePrecision] - N[(t * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5e+213], N[(N[(x * x), $MachinePrecision] - N[(z * N[(z * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 2.70000000000000003e35

   1. Initial program 90.8%

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

   2. Taylor expanded in z around 0 79.1%

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

      1. *-commutative 79.1%

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

      2. *-commutative 79.1%

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

      3. associate-*l* 79.1%

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

   4. Simplified 79.1%

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

   if 2.70000000000000003e35 < z < 4.9999999999999998e213

   1. Initial program 79.1%

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

   2. Taylor expanded in z around inf 76.0%

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

      1. unpow2 76.0%

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

      2. associate-*r* 76.0%

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

      3. *-commutative 76.0%

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

      4. associate-*r* 93.5%

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

      5. *-commutative 93.5%

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

   4. Simplified 93.5%

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

   if 4.9999999999999998e213 < 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 91.3%

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

      1. metadata-eval 91.3%

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

      2. distribute-lft-neg-in 91.3%

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

      3. *-commutative 91.3%

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

      4. unpow2 91.3%

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

      5. *-commutative 91.3%

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

      6. associate-*r* 91.3%

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

      7. associate-*l* 95.4%

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

      8. distribute-rgt-neg-in 95.4%

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

      9. distribute-rgt-neg-in 95.4%

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

      10. distribute-rgt-neg-in 95.4%

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

      11. metadata-eval 95.4%

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

   4. Simplified 95.4%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.7 \cdot 10^{+35}:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+213}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternative 4: 93.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.46 \cdot 10^{+103}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+214}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.46e+103)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (if (<= z 2.6e+214)
     (- (* 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 <= 1.46e+103) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else if (z <= 2.6e+214) {
		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 <= 1.46d+103) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else if (z <= 2.6d+214) 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 <= 1.46e+103) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else if (z <= 2.6e+214) {
		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 <= 1.46e+103:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	elif z <= 2.6e+214:
		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 (z <= 1.46e+103)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	elseif (z <= 2.6e+214)
		tmp = Float64(Float64(x * x) - Float64(z * Float64(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 <= 1.46e+103)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	elseif (z <= 2.6e+214)
		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[z, 1.46e+103], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.6e+214], N[(N[(x * x), $MachinePrecision] - N[(z * N[(z * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 1.45999999999999998e103

   1. Initial program 91.2%

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

   if 1.45999999999999998e103 < z < 2.59999999999999993e214

   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 71.0%

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

      1. unpow2 71.0%

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

      2. associate-*r* 71.0%

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

      3. *-commutative 71.0%

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

      4. associate-*r* 95.4%

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

      5. *-commutative 95.4%

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

   4. Simplified 95.4%

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

   if 2.59999999999999993e214 < 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 91.3%

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

      1. metadata-eval 91.3%

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

      2. distribute-lft-neg-in 91.3%

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

      3. *-commutative 91.3%

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

      4. unpow2 91.3%

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

      5. *-commutative 91.3%

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

      6. associate-*r* 91.3%

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

      7. associate-*l* 95.4%

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

      8. distribute-rgt-neg-in 95.4%

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

      9. distribute-rgt-neg-in 95.4%

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

      10. distribute-rgt-neg-in 95.4%

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

      11. metadata-eval 95.4%

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

   4. Simplified 95.4%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.46 \cdot 10^{+103}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+214}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternative 5: 50.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 4 \cdot \left(t \cdot y\right)\\ t_2 := z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \mathbf{if}\;x \leq 3.2 \cdot 10^{-201}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-83}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 13:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* 4.0 (* t y))) (t_2 (* z (* z (* y -4.0)))))
   (if (<= x 3.2e-201)
     t_2
     (if (<= x 2.5e-121)
       t_1
       (if (<= x 5.4e-83) t_2 (if (<= x 13.0) t_1 (* x x)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 4.0 * (t * y);
	double t_2 = z * (z * (y * -4.0));
	double tmp;
	if (x <= 3.2e-201) {
		tmp = t_2;
	} else if (x <= 2.5e-121) {
		tmp = t_1;
	} else if (x <= 5.4e-83) {
		tmp = t_2;
	} else if (x <= 13.0) {
		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) :: t_2
    real(8) :: tmp
    t_1 = 4.0d0 * (t * y)
    t_2 = z * (z * (y * (-4.0d0)))
    if (x <= 3.2d-201) then
        tmp = t_2
    else if (x <= 2.5d-121) then
        tmp = t_1
    else if (x <= 5.4d-83) then
        tmp = t_2
    else if (x <= 13.0d0) 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 * (t * y);
	double t_2 = z * (z * (y * -4.0));
	double tmp;
	if (x <= 3.2e-201) {
		tmp = t_2;
	} else if (x <= 2.5e-121) {
		tmp = t_1;
	} else if (x <= 5.4e-83) {
		tmp = t_2;
	} else if (x <= 13.0) {
		tmp = t_1;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 4.0 * (t * y)
	t_2 = z * (z * (y * -4.0))
	tmp = 0
	if x <= 3.2e-201:
		tmp = t_2
	elif x <= 2.5e-121:
		tmp = t_1
	elif x <= 5.4e-83:
		tmp = t_2
	elif x <= 13.0:
		tmp = t_1
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(4.0 * Float64(t * y))
	t_2 = Float64(z * Float64(z * Float64(y * -4.0)))
	tmp = 0.0
	if (x <= 3.2e-201)
		tmp = t_2;
	elseif (x <= 2.5e-121)
		tmp = t_1;
	elseif (x <= 5.4e-83)
		tmp = t_2;
	elseif (x <= 13.0)
		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 * (t * y);
	t_2 = z * (z * (y * -4.0));
	tmp = 0.0;
	if (x <= 3.2e-201)
		tmp = t_2;
	elseif (x <= 2.5e-121)
		tmp = t_1;
	elseif (x <= 5.4e-83)
		tmp = t_2;
	elseif (x <= 13.0)
		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[(t * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 3.2e-201], t$95$2, If[LessEqual[x, 2.5e-121], t$95$1, If[LessEqual[x, 5.4e-83], t$95$2, If[LessEqual[x, 13.0], t$95$1, N[(x * x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.2000000000000001e-201 or 2.49999999999999995e-121 < x < 5.39999999999999982e-83

   1. Initial program 89.2%

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

   2. Taylor expanded in z around inf 39.2%

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

      1. metadata-eval 39.2%

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

      2. distribute-lft-neg-in 39.2%

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

      3. *-commutative 39.2%

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

      4. unpow2 39.2%

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

      5. *-commutative 39.2%

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

      6. associate-*r* 39.2%

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

      7. associate-*l* 41.9%

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

      8. distribute-rgt-neg-in 41.9%

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

      9. distribute-rgt-neg-in 41.9%

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

      10. distribute-rgt-neg-in 41.9%

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

      11. metadata-eval 41.9%

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

   4. Simplified 41.9%

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

   if 3.2000000000000001e-201 < x < 2.49999999999999995e-121 or 5.39999999999999982e-83 < x < 13

   1. Initial program 96.1%

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

   2. Taylor expanded in t around inf 42.1%

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

   if 13 < x

   1. Initial program 81.8%

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

   2. Taylor expanded in x around inf 80.2%

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

      1. unpow2 80.2%

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

   4. Simplified 80.2%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{-201}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-121}:\\ \;\;\;\;4 \cdot \left(t \cdot y\right)\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-83}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \mathbf{elif}\;x \leq 13:\\ \;\;\;\;4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 6: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.52 \cdot 10^{+115}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 1.52e+115) (* (- (* z z) t) (* y -4.0)) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 1.52e+115) {
		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 * x) <= 1.52d+115) 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 * x) <= 1.52e+115) {
		tmp = ((z * z) - t) * (y * -4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 1.52e+115:
		tmp = ((z * z) - t) * (y * -4.0)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 1.52e+115)
		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 * x) <= 1.52e+115)
		tmp = ((z * z) - t) * (y * -4.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 1.52e+115], 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) < 1.52e115

   1. Initial program 94.7%

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

   2. Taylor expanded in x around 0 81.6%

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

      1. *-commutative 81.6%

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

      2. *-commutative 81.6%

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

      3. unpow2 81.6%

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

      4. *-commutative 81.6%

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

      5. associate-*l* 81.6%

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

   4. Simplified 81.6%

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

   if 1.52e115 < (*.f64 x x)

   1. Initial program 79.7%

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

   2. Taylor expanded in x around inf 79.1%

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

      1. unpow2 79.1%

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

   4. Simplified 79.1%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.52 \cdot 10^{+115}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 7: 74.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.2 \cdot 10^{+55}:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.2e+55) (- (* x x) (* t (* y -4.0))) (* z (* z (* y -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.2e+55) {
		tmp = (x * x) - (t * (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 <= 1.2d+55) then
        tmp = (x * x) - (t * (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 <= 1.2e+55) {
		tmp = (x * x) - (t * (y * -4.0));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 1.2e+55:
		tmp = (x * x) - (t * (y * -4.0))
	else:
		tmp = z * (z * (y * -4.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.2e+55)
		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

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 1.2e+55)
		tmp = (x * x) - (t * (y * -4.0));
	else
		tmp = z * (z * (y * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 1.2e+55], 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 z < 1.2e55

   1. Initial program 90.8%

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

   2. Taylor expanded in z around 0 79.1%

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

      1. *-commutative 79.1%

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

      2. *-commutative 79.1%

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

      3. associate-*l* 79.1%

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

   4. Simplified 79.1%

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

   if 1.2e55 < z

   1. Initial program 78.5%

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

   2. Taylor expanded in z around inf 76.8%

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

      1. metadata-eval 76.8%

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

      2. distribute-lft-neg-in 76.8%

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

      3. *-commutative 76.8%

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

      4. unpow2 76.8%

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

      5. *-commutative 76.8%

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

      6. associate-*r* 76.8%

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

      7. associate-*l* 83.8%

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

      8. distribute-rgt-neg-in 83.8%

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

      9. distribute-rgt-neg-in 83.8%

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

      10. distribute-rgt-neg-in 83.8%

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

      11. metadata-eval 83.8%

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

   4. Simplified 83.8%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.2 \cdot 10^{+55}:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternative 8: 48.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{-202}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \mathbf{elif}\;x \leq 14.2:\\ \;\;\;\;4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 1.35e-202)
   (* -4.0 (* (* z z) y))
   (if (<= x 14.2) (* 4.0 (* t y)) (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.35e-202) {
		tmp = -4.0 * ((z * z) * y);
	} else if (x <= 14.2) {
		tmp = 4.0 * (t * y);
	} 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 <= 1.35d-202) then
        tmp = (-4.0d0) * ((z * z) * y)
    else if (x <= 14.2d0) then
        tmp = 4.0d0 * (t * y)
    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 <= 1.35e-202) {
		tmp = -4.0 * ((z * z) * y);
	} else if (x <= 14.2) {
		tmp = 4.0 * (t * y);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 1.35e-202:
		tmp = -4.0 * ((z * z) * y)
	elif x <= 14.2:
		tmp = 4.0 * (t * y)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 1.35e-202)
		tmp = Float64(-4.0 * Float64(Float64(z * z) * y));
	elseif (x <= 14.2)
		tmp = Float64(4.0 * Float64(t * y));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 1.35e-202)
		tmp = -4.0 * ((z * z) * y);
	elseif (x <= 14.2)
		tmp = 4.0 * (t * y);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 1.35e-202], N[(-4.0 * N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 14.2], N[(4.0 * N[(t * y), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.3499999999999999e-202

   1. Initial program 89.3%

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

   2. Taylor expanded in z around inf 38.8%

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

      1. unpow2 38.8%

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

   4. Simplified 38.8%

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

   if 1.3499999999999999e-202 < x < 14.199999999999999

   1. Initial program 94.0%

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

   2. Taylor expanded in t around inf 36.5%

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

   if 14.199999999999999 < x

   1. Initial program 81.8%

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

   2. Taylor expanded in x around inf 80.2%

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

      1. unpow2 80.2%

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

   4. Simplified 80.2%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{-202}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \mathbf{elif}\;x \leq 14.2:\\ \;\;\;\;4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 9: 45.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 13.5:\\ \;\;\;\;4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= x 13.5) (* 4.0 (* t y)) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 13.5) {
		tmp = 4.0 * (t * y);
	} 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 <= 13.5d0) then
        tmp = 4.0d0 * (t * y)
    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 <= 13.5) {
		tmp = 4.0 * (t * y);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 13.5:
		tmp = 4.0 * (t * y)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 13.5)
		tmp = Float64(4.0 * Float64(t * y));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 13.5)
		tmp = 4.0 * (t * y);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 13.5], N[(4.0 * N[(t * y), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 13.5

   1. Initial program 90.1%

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

   2. Taylor expanded in t around inf 36.1%

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

   if 13.5 < x

   1. Initial program 81.8%

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

   2. Taylor expanded in x around inf 80.2%

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

      1. unpow2 80.2%

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

   4. Simplified 80.2%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 13.5:\\ \;\;\;\;4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 10: 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 88.2%

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

2. Taylor expanded in x around inf 44.2%

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

   1. unpow2 44.2%

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

4. Simplified 44.2%

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

5. Final simplification 44.2%

   \[\leadsto x \cdot x \]

Developer target: 90.2% 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 2023278 
(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))))