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

Percentage Accurate: 90.9% → 97.9%

Time: 7.9s

Alternatives: 9

Speedup: 0.8×

• 49.7% 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: 90.9% 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: 97.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+247}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e+247)
   (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) <= 5e+247) {
		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) <= 5e+247)
		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], 5e+247], 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) < 5.00000000000000023e247

   1. Initial program 98.9%

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

      1. fma-neg 98.9%

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

      2. *-commutative 98.9%

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

      3. distribute-rgt-neg-in 98.9%

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

      4. distribute-rgt-neg-in 98.9%

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

      5. metadata-eval 98.9%

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

   3. Simplified 98.9%

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

   if 5.00000000000000023e247 < (*.f64 z z)

   1. Initial program 74.8%

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

   2. Taylor expanded in t around 0 74.8%

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

      1. unpow2 74.8%

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

      2. fma-neg 86.2%

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

      3. *-commutative 86.2%

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

      4. unpow2 86.2%

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

      5. *-commutative 86.2%

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

      6. associate-*r* 86.2%

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

      7. associate-*l* 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. distribute-rgt-neg-in 100.0%

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

      11. metadata-eval 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+247}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\right)\\ \end{array} \]

Alternative 2: 97.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+247}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e+247)
   (+ (* 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) <= 5e+247) {
		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) <= 5e+247)
		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], 5e+247], 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) < 5.00000000000000023e247

   1. Initial program 98.9%

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

   if 5.00000000000000023e247 < (*.f64 z z)

   1. Initial program 74.8%

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

   2. Taylor expanded in t around 0 74.8%

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

      1. unpow2 74.8%

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

      2. fma-neg 86.2%

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

      3. *-commutative 86.2%

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

      4. unpow2 86.2%

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

      5. *-commutative 86.2%

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

      6. associate-*r* 86.2%

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

      7. associate-*l* 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. distribute-rgt-neg-in 100.0%

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

      11. metadata-eval 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+247}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\right)\\ \end{array} \]

Alternative 3: 88.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-41}:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+295}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e-41)
   (- (* x x) (* t (* y -4.0)))
   (if (<= (* z z) 5e+295)
     (- (* 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) <= 5e-41) {
		tmp = (x * x) - (t * (y * -4.0));
	} else if ((z * z) <= 5e+295) {
		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) <= 5d-41) then
        tmp = (x * x) - (t * (y * (-4.0d0)))
    else if ((z * z) <= 5d+295) 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) <= 5e-41) {
		tmp = (x * x) - (t * (y * -4.0));
	} else if ((z * z) <= 5e+295) {
		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) <= 5e-41:
		tmp = (x * x) - (t * (y * -4.0))
	elif (z * z) <= 5e+295:
		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) <= 5e-41)
		tmp = Float64(Float64(x * x) - Float64(t * Float64(y * -4.0)));
	elseif (Float64(z * z) <= 5e+295)
		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 * z) <= 5e-41)
		tmp = (x * x) - (t * (y * -4.0));
	elseif ((z * z) <= 5e+295)
		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], 5e-41], N[(N[(x * x), $MachinePrecision] - N[(t * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 5e+295], 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 (*.f64 z z) < 4.9999999999999996e-41

   1. Initial program 99.2%

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

   2. Taylor expanded in z around 0 96.3%

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

      1. *-commutative 96.3%

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

      2. *-commutative 96.3%

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

      3. associate-*l* 96.3%

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

   4. Simplified 96.3%

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

   if 4.9999999999999996e-41 < (*.f64 z z) < 4.99999999999999991e295

   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 inf 85.3%

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

      1. unpow2 85.3%

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

      2. associate-*r* 85.3%

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

      3. *-commutative 85.3%

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

      4. associate-*r* 85.3%

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

      5. *-commutative 85.3%

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

   4. Simplified 85.3%

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

   if 4.99999999999999991e295 < (*.f64 z z)

   1. Initial program 70.1%

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

   2. Taylor expanded in z around inf 83.7%

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

      1. metadata-eval 83.7%

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

      2. distribute-lft-neg-in 83.7%

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

      3. *-commutative 83.7%

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

      4. unpow2 83.7%

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

      5. *-commutative 83.7%

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

      6. associate-*r* 83.7%

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

      7. associate-*l* 96.7%

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

      8. distribute-rgt-neg-in 96.7%

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

      9. distribute-rgt-neg-in 96.7%

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

      10. distribute-rgt-neg-in 96.7%

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

      11. metadata-eval 96.7%

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

   4. Simplified 96.7%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-41}:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+295}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternative 4: 58.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4.1 \cdot 10^{+20} \lor \neg \left(x \cdot x \leq 1.45 \cdot 10^{+54}\right) \land x \cdot x \leq 1.75 \cdot 10^{+118}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* x x) 4.1e+20)
         (and (not (<= (* x x) 1.45e+54)) (<= (* x x) 1.75e+118)))
   (* t (* y 4.0))
   (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) <= 4.1e+20) || (!((x * x) <= 1.45e+54) && ((x * x) <= 1.75e+118))) {
		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 * x) <= 4.1d+20) .or. (.not. ((x * x) <= 1.45d+54)) .and. ((x * x) <= 1.75d+118)) 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 * x) <= 4.1e+20) || (!((x * x) <= 1.45e+54) && ((x * x) <= 1.75e+118))) {
		tmp = t * (y * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x * x) <= 4.1e+20) or (not ((x * x) <= 1.45e+54) and ((x * x) <= 1.75e+118)):
		tmp = t * (y * 4.0)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x * x) <= 4.1e+20) || (!(Float64(x * x) <= 1.45e+54) && (Float64(x * x) <= 1.75e+118)))
		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 * x) <= 4.1e+20) || (~(((x * x) <= 1.45e+54)) && ((x * x) <= 1.75e+118)))
		tmp = t * (y * 4.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x * x), $MachinePrecision], 4.1e+20], And[N[Not[LessEqual[N[(x * x), $MachinePrecision], 1.45e+54]], $MachinePrecision], LessEqual[N[(x * x), $MachinePrecision], 1.75e+118]]], N[(t * N[(y * 4.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.1e20 or 1.4499999999999999e54 < (*.f64 x x) < 1.75000000000000008e118

   1. Initial program 93.6%

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

   2. Taylor expanded in t around inf 55.4%

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

      1. associate-*r* 55.4%

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

   4. Simplified 55.4%

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

   if 4.1e20 < (*.f64 x x) < 1.4499999999999999e54 or 1.75000000000000008e118 < (*.f64 x x)

   1. Initial program 90.7%

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

   2. Taylor expanded in x around inf 73.5%

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

      1. unpow2 73.5%

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

   4. Simplified 73.5%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4.1 \cdot 10^{+20} \lor \neg \left(x \cdot x \leq 1.45 \cdot 10^{+54}\right) \land x \cdot x \leq 1.75 \cdot 10^{+118}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 5: 95.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+295}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e+295)
   (+ (* 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+295) {
		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+295) 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+295) {
		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+295:
		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+295)
		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+295)
		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+295], 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) < 4.99999999999999991e295

   1. Initial program 98.9%

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

   if 4.99999999999999991e295 < (*.f64 z z)

   1. Initial program 70.1%

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

   2. Taylor expanded in z around inf 83.7%

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

      1. metadata-eval 83.7%

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

      2. distribute-lft-neg-in 83.7%

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

      3. *-commutative 83.7%

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

      4. unpow2 83.7%

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

      5. *-commutative 83.7%

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

      6. associate-*r* 83.7%

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

      7. associate-*l* 96.7%

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

      8. distribute-rgt-neg-in 96.7%

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

      9. distribute-rgt-neg-in 96.7%

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

      10. distribute-rgt-neg-in 96.7%

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

      11. metadata-eval 96.7%

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

   4. Simplified 96.7%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+295}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternative 6: 50.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;z \leq 1.22 \cdot 10^{-224}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-185}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-71}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{+20}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* t (* y 4.0))))
   (if (<= z 1.22e-224)
     (* x x)
     (if (<= z 1.35e-185)
       t_1
       (if (<= z 6.5e-71)
         (* x x)
         (if (<= z 2.15e+20) t_1 (* -4.0 (* (* z z) y))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = t * (y * 4.0);
	double tmp;
	if (z <= 1.22e-224) {
		tmp = x * x;
	} else if (z <= 1.35e-185) {
		tmp = t_1;
	} else if (z <= 6.5e-71) {
		tmp = x * x;
	} else if (z <= 2.15e+20) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = t * (y * 4.0d0)
    if (z <= 1.22d-224) then
        tmp = x * x
    else if (z <= 1.35d-185) then
        tmp = t_1
    else if (z <= 6.5d-71) then
        tmp = x * x
    else if (z <= 2.15d+20) then
        tmp = t_1
    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 t_1 = t * (y * 4.0);
	double tmp;
	if (z <= 1.22e-224) {
		tmp = x * x;
	} else if (z <= 1.35e-185) {
		tmp = t_1;
	} else if (z <= 6.5e-71) {
		tmp = x * x;
	} else if (z <= 2.15e+20) {
		tmp = t_1;
	} else {
		tmp = -4.0 * ((z * z) * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = t * (y * 4.0)
	tmp = 0
	if z <= 1.22e-224:
		tmp = x * x
	elif z <= 1.35e-185:
		tmp = t_1
	elif z <= 6.5e-71:
		tmp = x * x
	elif z <= 2.15e+20:
		tmp = t_1
	else:
		tmp = -4.0 * ((z * z) * y)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(t * Float64(y * 4.0))
	tmp = 0.0
	if (z <= 1.22e-224)
		tmp = Float64(x * x);
	elseif (z <= 1.35e-185)
		tmp = t_1;
	elseif (z <= 6.5e-71)
		tmp = Float64(x * x);
	elseif (z <= 2.15e+20)
		tmp = t_1;
	else
		tmp = Float64(-4.0 * Float64(Float64(z * z) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = t * (y * 4.0);
	tmp = 0.0;
	if (z <= 1.22e-224)
		tmp = x * x;
	elseif (z <= 1.35e-185)
		tmp = t_1;
	elseif (z <= 6.5e-71)
		tmp = x * x;
	elseif (z <= 2.15e+20)
		tmp = t_1;
	else
		tmp = -4.0 * ((z * z) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(t * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 1.22e-224], N[(x * x), $MachinePrecision], If[LessEqual[z, 1.35e-185], t$95$1, If[LessEqual[z, 6.5e-71], N[(x * x), $MachinePrecision], If[LessEqual[z, 2.15e+20], t$95$1, N[(-4.0 * N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 1.22000000000000003e-224 or 1.34999999999999994e-185 < z < 6.50000000000000005e-71

   1. Initial program 93.3%

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

   2. Taylor expanded in x around inf 44.3%

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

      1. unpow2 44.3%

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

   4. Simplified 44.3%

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

   if 1.22000000000000003e-224 < z < 1.34999999999999994e-185 or 6.50000000000000005e-71 < z < 2.15e20

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 66.1%

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

      1. associate-*r* 66.1%

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

   4. Simplified 66.1%

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

   if 2.15e20 < z

   1. Initial program 83.2%

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

   2. Taylor expanded in z around inf 64.3%

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

      1. unpow2 64.3%

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

   4. Simplified 64.3%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.22 \cdot 10^{-224}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-185}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-71}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{+20}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \end{array} \]

Alternative 7: 51.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;z \leq 7.2 \cdot 10^{-225}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-184}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-70}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+20}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* t (* y 4.0))))
   (if (<= z 7.2e-225)
     (* x x)
     (if (<= z 3.7e-184)
       t_1
       (if (<= z 1.6e-70)
         (* x x)
         (if (<= z 2.8e+20) t_1 (* z (* z (* y -4.0)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = t * (y * 4.0);
	double tmp;
	if (z <= 7.2e-225) {
		tmp = x * x;
	} else if (z <= 3.7e-184) {
		tmp = t_1;
	} else if (z <= 1.6e-70) {
		tmp = x * x;
	} else if (z <= 2.8e+20) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = t * (y * 4.0d0)
    if (z <= 7.2d-225) then
        tmp = x * x
    else if (z <= 3.7d-184) then
        tmp = t_1
    else if (z <= 1.6d-70) then
        tmp = x * x
    else if (z <= 2.8d+20) then
        tmp = t_1
    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 t_1 = t * (y * 4.0);
	double tmp;
	if (z <= 7.2e-225) {
		tmp = x * x;
	} else if (z <= 3.7e-184) {
		tmp = t_1;
	} else if (z <= 1.6e-70) {
		tmp = x * x;
	} else if (z <= 2.8e+20) {
		tmp = t_1;
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = t * (y * 4.0)
	tmp = 0
	if z <= 7.2e-225:
		tmp = x * x
	elif z <= 3.7e-184:
		tmp = t_1
	elif z <= 1.6e-70:
		tmp = x * x
	elif z <= 2.8e+20:
		tmp = t_1
	else:
		tmp = z * (z * (y * -4.0))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(t * Float64(y * 4.0))
	tmp = 0.0
	if (z <= 7.2e-225)
		tmp = Float64(x * x);
	elseif (z <= 3.7e-184)
		tmp = t_1;
	elseif (z <= 1.6e-70)
		tmp = Float64(x * x);
	elseif (z <= 2.8e+20)
		tmp = t_1;
	else
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = t * (y * 4.0);
	tmp = 0.0;
	if (z <= 7.2e-225)
		tmp = x * x;
	elseif (z <= 3.7e-184)
		tmp = t_1;
	elseif (z <= 1.6e-70)
		tmp = x * x;
	elseif (z <= 2.8e+20)
		tmp = t_1;
	else
		tmp = z * (z * (y * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(t * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 7.2e-225], N[(x * x), $MachinePrecision], If[LessEqual[z, 3.7e-184], t$95$1, If[LessEqual[z, 1.6e-70], N[(x * x), $MachinePrecision], If[LessEqual[z, 2.8e+20], t$95$1, N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 7.20000000000000018e-225 or 3.6999999999999999e-184 < z < 1.5999999999999999e-70

   1. Initial program 93.3%

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

   2. Taylor expanded in x around inf 44.3%

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

      1. unpow2 44.3%

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

   4. Simplified 44.3%

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

   if 7.20000000000000018e-225 < z < 3.6999999999999999e-184 or 1.5999999999999999e-70 < z < 2.8e20

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 66.1%

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

      1. associate-*r* 66.1%

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

   4. Simplified 66.1%

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

   if 2.8e20 < z

   1. Initial program 83.2%

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

   2. Taylor expanded in z around inf 64.3%

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

      1. metadata-eval 64.3%

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

      2. distribute-lft-neg-in 64.3%

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

      3. *-commutative 64.3%

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

      4. unpow2 64.3%

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

      5. *-commutative 64.3%

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

      6. associate-*r* 64.3%

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

      7. associate-*l* 76.7%

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

      8. distribute-rgt-neg-in 76.7%

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

      9. distribute-rgt-neg-in 76.7%

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

      10. distribute-rgt-neg-in 76.7%

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

      11. metadata-eval 76.7%

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

   4. Simplified 76.7%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7.2 \cdot 10^{-225}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-184}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-70}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+20}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternative 8: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+199}:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+199) (- (* 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 * z) <= 1e+199) {
		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 * z) <= 1d+199) 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 * z) <= 1e+199) {
		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 * z) <= 1e+199:
		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 (Float64(z * z) <= 1e+199)
		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 * z) <= 1e+199)
		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[N[(z * z), $MachinePrecision], 1e+199], N[(N[(x * x), $MachinePrecision] - N[(t * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.0000000000000001e199

   1. Initial program 98.8%

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

   2. Taylor expanded in z around 0 86.7%

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

      1. *-commutative 86.7%

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

      2. *-commutative 86.7%

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

      3. associate-*l* 86.7%

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

   4. Simplified 86.7%

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

   if 1.0000000000000001e199 < (*.f64 z z)

   1. Initial program 78.2%

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

   2. Taylor expanded in z around inf 80.9%

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

      1. metadata-eval 80.9%

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

      2. distribute-lft-neg-in 80.9%

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

      3. *-commutative 80.9%

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

      4. unpow2 80.9%

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

      5. *-commutative 80.9%

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

      6. associate-*r* 80.9%

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

      7. associate-*l* 90.3%

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

      8. distribute-rgt-neg-in 90.3%

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

      9. distribute-rgt-neg-in 90.3%

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

      10. distribute-rgt-neg-in 90.3%

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

      11. metadata-eval 90.3%

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

   4. Simplified 90.3%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+199}:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternative 9: 41.8% 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 92.3%

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

2. Taylor expanded in x around inf 39.3%

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

   1. unpow2 39.3%

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

4. Simplified 39.3%

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

5. Final simplification 39.3%

   \[\leadsto x \cdot x \]

Developer target: 90.9% 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 2023229 
(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))))