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

Percentage Accurate: 91.2% → 94.9%

Time: 7.3s

Alternatives: 6

Speedup: 0.7×

• 50.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (* x x) (* (* y 4.0) (- (* z z) t))))

C

double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * x) - ((y * 4.0d0) * ((z * z) - t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}

Python

def code(x, y, z, t):
	return (x * x) - ((y * 4.0) * ((z * z) - t))

Julia

function code(x, y, z, t)
	return Float64(Float64(x * x) - Float64(Float64(y * 4.0) * Float64(Float64(z * z) - t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * x) - ((y * 4.0) * ((z * z) - t));
end

Wolfram

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

Initial Program: 91.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (* x x) (* (* y 4.0) (- (* z z) t))))

C

double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * x) - ((y * 4.0d0) * ((z * z) - t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}

Python

def code(x, y, z, t):
	return (x * x) - ((y * 4.0) * ((z * z) - t))

Julia

function code(x, y, z, t)
	return Float64(Float64(x * x) - Float64(Float64(y * 4.0) * Float64(Float64(z * z) - t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * x) - ((y * 4.0) * ((z * z) - t));
end

Wolfram

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

Alternative 1: 94.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+277}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-4 \cdot \left(z \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 4e+277)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (* z (* -4.0 (* z y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 4e+277) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = z * (-4.0 * (z * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z * z) <= 4d+277) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = z * ((-4.0d0) * (z * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 4e+277) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = z * (-4.0 * (z * y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 4e+277:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = z * (-4.0 * (z * y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 4e+277)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(z * Float64(-4.0 * Float64(z * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 4e+277)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = z * (-4.0 * (z * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 4e+277], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(-4.0 * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.00000000000000001e277

   1. Initial program 97.6%

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

   if 4.00000000000000001e277 < (*.f64 z z)

   1. Initial program 81.3%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 96.3

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

   5. Simplified 96.3%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+277}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-4 \cdot \left(z \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+172}:\\ \;\;\;\;\mathsf{fma}\left(4, y \cdot t, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-4 \cdot \left(z \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+172) (fma 4.0 (* y t) (* x x)) (* z (* -4.0 (* z y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+172) {
		tmp = fma(4.0, (y * t), (x * x));
	} else {
		tmp = z * (-4.0 * (z * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+172)
		tmp = fma(4.0, Float64(y * t), Float64(x * x));
	else
		tmp = Float64(z * Float64(-4.0 * Float64(z * y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+172], N[(4.0 * N[(y * t), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(z * N[(-4.0 * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.0000000000000001e172

   1. Initial program 97.9%

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

   3. Taylor expanded in z around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 88.8

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

   5. Simplified 88.8%

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

   if 1.0000000000000001e172 < (*.f64 z z)

   1. Initial program 84.1%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 92.0

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

   5. Simplified 92.0%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+172}:\\ \;\;\;\;\mathsf{fma}\left(4, y \cdot t, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-4 \cdot \left(z \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+172}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-4 \cdot \left(z \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+172) (* x x) (* z (* -4.0 (* z y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+172) {
		tmp = x * x;
	} else {
		tmp = z * (-4.0 * (z * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z * z) <= 1d+172) then
        tmp = x * x
    else
        tmp = z * ((-4.0d0) * (z * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+172) {
		tmp = x * x;
	} else {
		tmp = z * (-4.0 * (z * y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 1e+172:
		tmp = x * x
	else:
		tmp = z * (-4.0 * (z * y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+172)
		tmp = Float64(x * x);
	else
		tmp = Float64(z * Float64(-4.0 * Float64(z * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 1e+172)
		tmp = x * x;
	else
		tmp = z * (-4.0 * (z * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+172], N[(x * x), $MachinePrecision], N[(z * N[(-4.0 * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.0000000000000001e172

   1. Initial program 97.9%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 59.1

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

   5. Simplified 59.1%

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

   if 1.0000000000000001e172 < (*.f64 z z)

   1. Initial program 84.1%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 92.0

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

   5. Simplified 92.0%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+172}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-4 \cdot \left(z \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2.05 \cdot 10^{+172}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2.05e+172) (* x x) (* -4.0 (* (* z z) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2.05e+172) {
		tmp = x * x;
	} else {
		tmp = -4.0 * ((z * z) * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z * z) <= 2.05d+172) then
        tmp = x * x
    else
        tmp = (-4.0d0) * ((z * z) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2.05e+172) {
		tmp = x * x;
	} else {
		tmp = -4.0 * ((z * z) * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 2.05e+172:
		tmp = x * x
	else:
		tmp = -4.0 * ((z * z) * y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2.05e+172)
		tmp = Float64(x * x);
	else
		tmp = Float64(-4.0 * Float64(Float64(z * z) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 2.05e+172)
		tmp = x * x;
	else
		tmp = -4.0 * ((z * z) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2.05e+172], N[(x * x), $MachinePrecision], N[(-4.0 * N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.05e172

   1. Initial program 97.9%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 59.1

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

   5. Simplified 59.1%

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

   if 2.05e172 < (*.f64 z z)

   1. Initial program 84.1%

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

   3. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

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

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

      7. distribute-rgt-in N/A

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

   5. Simplified 83.1%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 83.9

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

   8. Simplified 83.9%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2.05 \cdot 10^{+172}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 45.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 400000:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 400000.0) (* 4.0 (* y t)) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 400000.0) {
		tmp = 4.0 * (y * t);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= 400000.0d0) then
        tmp = 4.0d0 * (y * t)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 400000.0) {
		tmp = 4.0 * (y * t);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 400000.0:
		tmp = 4.0 * (y * t)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 400000.0)
		tmp = Float64(4.0 * Float64(y * t));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 400000.0)
		tmp = 4.0 * (y * t);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 400000.0], N[(4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4e5

   1. Initial program 94.2%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 32.6

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

   5. Simplified 32.6%

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

   if 4e5 < x

   1. Initial program 87.6%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 77.0

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

   5. Simplified 77.0%

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

Alternative 6: 41.8% accurate, 4.5× 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. Add Preprocessing

3. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. lower-*.f64 44.9

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

5. Simplified 44.9%

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

Developer Target 1: 91.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (* x x) (* 4.0 (* y (- (* z z) t)))))

C

double code(double x, double y, double z, double t) {
	return (x * x) - (4.0 * (y * ((z * z) - t)));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * x) - (4.0d0 * (y * ((z * z) - t)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * x) - (4.0 * (y * ((z * z) - t)));
}

Python

def code(x, y, z, t):
	return (x * x) - (4.0 * (y * ((z * z) - t)))

Julia

function code(x, y, z, t)
	return Float64(Float64(x * x) - Float64(4.0 * Float64(y * Float64(Float64(z * z) - t))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * x) - (4.0 * (y * ((z * z) - t)));
end

Wolfram

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

Reproduce

herbie shell --seed 2024215 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

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

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