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

Percentage Accurate: 91.3% → 95.6%

Time: 7.5s

Alternatives: 8

Speedup: 0.8×

• 48.3% 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: 91.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+254}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(z \cdot \left(z \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+254)
   (fma x x (* (- (* z z) t) (* y -4.0)))
   (* -4.0 (* z (* z y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+254) {
		tmp = fma(x, x, (((z * z) - t) * (y * -4.0)));
	} else {
		tmp = -4.0 * (z * (z * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+254)
		tmp = fma(x, x, Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0)));
	else
		tmp = Float64(-4.0 * Float64(z * Float64(z * y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+254], N[(x * x + N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.9999999999999994e253

   1. Initial program 96.5%

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

      1. fma-neg 97.6%

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

      2. *-commutative 97.6%

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

      3. distribute-rgt-neg-in 97.6%

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

      4. distribute-rgt-neg-in 97.6%

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

      5. metadata-eval 97.6%

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

   3. Simplified 97.6%

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

   if 9.9999999999999994e253 < (*.f64 z z)

   1. Initial program 69.9%

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

   2. Taylor expanded in z around inf 78.3%

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

      1. unpow2 78.3%

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

      2. *-commutative 78.3%

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

      3. associate-*l* 83.6%

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

   4. Simplified 83.6%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+254}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(z \cdot \left(z \cdot y\right)\right)\\ \end{array} \]

Alternative 2: 95.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+254}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(z \cdot \left(z \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+254)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (* -4.0 (* z (* z y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+254) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = -4.0 * (z * (z * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z * z) <= 1d+254) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = (-4.0d0) * (z * (z * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+254) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = -4.0 * (z * (z * y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 1e+254:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = -4.0 * (z * (z * y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+254)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(-4.0 * Float64(z * Float64(z * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 1e+254)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = -4.0 * (z * (z * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+254], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.9999999999999994e253

   1. Initial program 96.5%

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

   if 9.9999999999999994e253 < (*.f64 z z)

   1. Initial program 69.9%

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

   2. Taylor expanded in z around inf 78.3%

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

      1. unpow2 78.3%

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

      2. *-commutative 78.3%

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

      3. associate-*l* 83.6%

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

   4. Simplified 83.6%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+254}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(z \cdot \left(z \cdot y\right)\right)\\ \end{array} \]

Alternative 3: 45.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;z \leq 2.75 \cdot 10^{-269}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-177}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+27}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* t (* y 4.0))))
   (if (<= z 2.75e-269)
     t_1
     (if (<= z 1.55e-177)
       (* x x)
       (if (<= z 6.2e-26)
         t_1
         (if (<= z 1.6e+27) (* x x) (* -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 <= 2.75e-269) {
		tmp = t_1;
	} else if (z <= 1.55e-177) {
		tmp = x * x;
	} else if (z <= 6.2e-26) {
		tmp = t_1;
	} else if (z <= 1.6e+27) {
		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) :: t_1
    real(8) :: tmp
    t_1 = t * (y * 4.0d0)
    if (z <= 2.75d-269) then
        tmp = t_1
    else if (z <= 1.55d-177) then
        tmp = x * x
    else if (z <= 6.2d-26) then
        tmp = t_1
    else if (z <= 1.6d+27) 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 t_1 = t * (y * 4.0);
	double tmp;
	if (z <= 2.75e-269) {
		tmp = t_1;
	} else if (z <= 1.55e-177) {
		tmp = x * x;
	} else if (z <= 6.2e-26) {
		tmp = t_1;
	} else if (z <= 1.6e+27) {
		tmp = x * x;
	} 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 <= 2.75e-269:
		tmp = t_1
	elif z <= 1.55e-177:
		tmp = x * x
	elif z <= 6.2e-26:
		tmp = t_1
	elif z <= 1.6e+27:
		tmp = x * x
	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 <= 2.75e-269)
		tmp = t_1;
	elseif (z <= 1.55e-177)
		tmp = Float64(x * x);
	elseif (z <= 6.2e-26)
		tmp = t_1;
	elseif (z <= 1.6e+27)
		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)
	t_1 = t * (y * 4.0);
	tmp = 0.0;
	if (z <= 2.75e-269)
		tmp = t_1;
	elseif (z <= 1.55e-177)
		tmp = x * x;
	elseif (z <= 6.2e-26)
		tmp = t_1;
	elseif (z <= 1.6e+27)
		tmp = x * x;
	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, 2.75e-269], t$95$1, If[LessEqual[z, 1.55e-177], N[(x * x), $MachinePrecision], If[LessEqual[z, 6.2e-26], t$95$1, If[LessEqual[z, 1.6e+27], N[(x * x), $MachinePrecision], N[(-4.0 * N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 2.75000000000000005e-269 or 1.55000000000000009e-177 < z < 6.19999999999999966e-26

   1. Initial program 91.0%

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

   2. Taylor expanded in t around inf 42.5%

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

      1. associate-*r* 42.5%

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

   4. Simplified 42.5%

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

   if 2.75000000000000005e-269 < z < 1.55000000000000009e-177 or 6.19999999999999966e-26 < z < 1.60000000000000008e27

   1. Initial program 95.2%

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

   2. Taylor expanded in x around inf 67.5%

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

      1. unpow2 67.5%

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

   4. Simplified 67.5%

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

   if 1.60000000000000008e27 < z

   1. Initial program 78.8%

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

   2. Taylor expanded in z around inf 68.7%

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

      1. unpow2 68.7%

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

   4. Simplified 68.7%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.75 \cdot 10^{-269}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-177}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-26}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+27}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \end{array} \]

Alternative 4: 47.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;z \leq 2.42 \cdot 10^{-269}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-177}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{+26}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(z \cdot \left(z \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* t (* y 4.0))))
   (if (<= z 2.42e-269)
     t_1
     (if (<= z 7.8e-177)
       (* x x)
       (if (<= z 4.8e-25)
         t_1
         (if (<= z 1.75e+26) (* x x) (* -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 <= 2.42e-269) {
		tmp = t_1;
	} else if (z <= 7.8e-177) {
		tmp = x * x;
	} else if (z <= 4.8e-25) {
		tmp = t_1;
	} else if (z <= 1.75e+26) {
		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) :: t_1
    real(8) :: tmp
    t_1 = t * (y * 4.0d0)
    if (z <= 2.42d-269) then
        tmp = t_1
    else if (z <= 7.8d-177) then
        tmp = x * x
    else if (z <= 4.8d-25) then
        tmp = t_1
    else if (z <= 1.75d+26) 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 t_1 = t * (y * 4.0);
	double tmp;
	if (z <= 2.42e-269) {
		tmp = t_1;
	} else if (z <= 7.8e-177) {
		tmp = x * x;
	} else if (z <= 4.8e-25) {
		tmp = t_1;
	} else if (z <= 1.75e+26) {
		tmp = x * x;
	} 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 <= 2.42e-269:
		tmp = t_1
	elif z <= 7.8e-177:
		tmp = x * x
	elif z <= 4.8e-25:
		tmp = t_1
	elif z <= 1.75e+26:
		tmp = x * x
	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 <= 2.42e-269)
		tmp = t_1;
	elseif (z <= 7.8e-177)
		tmp = Float64(x * x);
	elseif (z <= 4.8e-25)
		tmp = t_1;
	elseif (z <= 1.75e+26)
		tmp = Float64(x * x);
	else
		tmp = Float64(-4.0 * Float64(z * Float64(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 <= 2.42e-269)
		tmp = t_1;
	elseif (z <= 7.8e-177)
		tmp = x * x;
	elseif (z <= 4.8e-25)
		tmp = t_1;
	elseif (z <= 1.75e+26)
		tmp = x * x;
	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, 2.42e-269], t$95$1, If[LessEqual[z, 7.8e-177], N[(x * x), $MachinePrecision], If[LessEqual[z, 4.8e-25], t$95$1, If[LessEqual[z, 1.75e+26], N[(x * x), $MachinePrecision], N[(-4.0 * N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 2.42e-269 or 7.80000000000000028e-177 < z < 4.80000000000000018e-25

   1. Initial program 91.0%

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

   2. Taylor expanded in t around inf 42.5%

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

      1. associate-*r* 42.5%

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

   4. Simplified 42.5%

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

   if 2.42e-269 < z < 7.80000000000000028e-177 or 4.80000000000000018e-25 < z < 1.75e26

   1. Initial program 95.2%

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

   2. Taylor expanded in x around inf 67.5%

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

      1. unpow2 67.5%

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

   4. Simplified 67.5%

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

   if 1.75e26 < z

   1. Initial program 78.8%

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

   2. Taylor expanded in z around inf 68.7%

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

      1. unpow2 68.7%

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

      2. *-commutative 68.7%

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

      3. associate-*l* 71.0%

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

   4. Simplified 71.0%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.42 \cdot 10^{-269}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-177}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-25}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{+26}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(z \cdot \left(z \cdot y\right)\right)\\ \end{array} \]

Alternative 5: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2.3 \cdot 10^{+118}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 2.3e+118) (* (- (* z z) t) (* y -4.0)) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 2.3e+118) {
		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) <= 2.3d+118) 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) <= 2.3e+118) {
		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) <= 2.3e+118:
		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) <= 2.3e+118)
		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) <= 2.3e+118)
		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], 2.3e+118], 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) < 2.30000000000000016e118

   1. Initial program 96.7%

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

   2. Taylor expanded in x around 0 88.8%

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

      1. *-commutative 88.8%

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

      2. *-commutative 88.8%

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

      3. unpow2 88.8%

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

      4. *-commutative 88.8%

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

      5. associate-*l* 88.8%

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

   4. Simplified 88.8%

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

   if 2.30000000000000016e118 < (*.f64 x x)

   1. Initial program 71.6%

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

   2. Taylor expanded in x around inf 78.4%

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

      1. unpow2 78.4%

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

   4. Simplified 78.4%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2.3 \cdot 10^{+118}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 6: 82.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 6 \cdot 10^{+103}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 6e+103)
   (* (- (* z z) t) (* y -4.0))
   (- (* x x) (* t (* y -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 6e+103) {
		tmp = ((z * z) - t) * (y * -4.0);
	} else {
		tmp = (x * x) - (t * (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 ((x * x) <= 6d+103) then
        tmp = ((z * z) - t) * (y * (-4.0d0))
    else
        tmp = (x * x) - (t * (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 ((x * x) <= 6e+103) {
		tmp = ((z * z) - t) * (y * -4.0);
	} else {
		tmp = (x * x) - (t * (y * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 6e+103:
		tmp = ((z * z) - t) * (y * -4.0)
	else:
		tmp = (x * x) - (t * (y * -4.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 6e+103)
		tmp = Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0));
	else
		tmp = Float64(Float64(x * x) - Float64(t * Float64(y * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 6e+103)
		tmp = ((z * z) - t) * (y * -4.0);
	else
		tmp = (x * x) - (t * (y * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 6e+103], N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(t * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 6e103

   1. Initial program 96.5%

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

   2. Taylor expanded in x around 0 89.6%

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

      1. *-commutative 89.6%

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

      2. *-commutative 89.6%

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

      3. unpow2 89.6%

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

      4. *-commutative 89.6%

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

      5. associate-*l* 89.6%

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

   4. Simplified 89.6%

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

   if 6e103 < (*.f64 x x)

   1. Initial program 73.9%

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

   2. Taylor expanded in z around 0 80.9%

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

      1. associate-*r* 80.9%

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

   4. Simplified 80.9%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 6 \cdot 10^{+103}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \end{array} \]

Alternative 7: 44.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.42 \cdot 10^{+59}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 1.42e+59) (* t (* y 4.0)) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.42e+59) {
		tmp = t * (y * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= 1.42d+59) then
        tmp = t * (y * 4.0d0)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.42e+59) {
		tmp = t * (y * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 1.42e+59:
		tmp = t * (y * 4.0)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 1.42e+59)
		tmp = Float64(t * Float64(y * 4.0));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 1.42e+59)
		tmp = t * (y * 4.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 1.42e+59], N[(t * N[(y * 4.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.42000000000000005e59

   1. Initial program 90.6%

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

   2. Taylor expanded in t around inf 38.6%

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

      1. associate-*r* 38.6%

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

   4. Simplified 38.6%

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

   if 1.42000000000000005e59 < x

   1. Initial program 73.6%

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

   2. Taylor expanded in x around inf 83.5%

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

      1. unpow2 83.5%

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

   4. Simplified 83.5%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.42 \cdot 10^{+59}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 8: 41.4% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

2. Taylor expanded in x around inf 34.9%

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

   1. unpow2 34.9%

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

4. Simplified 34.9%

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

5. Final simplification 34.9%

   \[\leadsto x \cdot x \]

Developer target: 91.4% 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 2023258 
(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))))