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

Percentage Accurate: 91.5% → 94.3%

Time: 9.8s

Alternatives: 7

Speedup: 0.5×

• 49.6% 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: 91.5% 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: 94.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+163}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= x 2e+163) (fma x x (* (- (* z z) t) (* y -4.0))) (pow x 2.0)))

C

x = abs(x);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 2e+163) {
		tmp = fma(x, x, (((z * z) - t) * (y * -4.0)));
	} else {
		tmp = pow(x, 2.0);
	}
	return tmp;
}

Julia

x = abs(x)
function code(x, y, z, t)
	tmp = 0.0
	if (x <= 2e+163)
		tmp = fma(x, x, Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0)));
	else
		tmp = x ^ 2.0;
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[x, 2e+163], N[(x * x + N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[x, 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.9999999999999999e163

   1. Initial program 90.4%

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

      1. fma-neg 91.8%

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

      2. distribute-lft-neg-in 91.8%

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

      3. *-commutative 91.8%

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

      4. distribute-rgt-neg-in 91.8%

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

      5. metadata-eval 91.8%

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

   3. Simplified 91.8%

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

   if 1.9999999999999999e163 < x

   1. Initial program 83.3%

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

      1. associate-*l* 83.3%

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

   3. Simplified 83.3%

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

   4. Taylor expanded in x around inf 97.2%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+163}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{2}\\ \end{array} \]

Alternative 2: 93.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{+149}:\\ \;\;\;\;x \cdot x + y \cdot \left(4 \cdot \left(t - z \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= x 5.5e+149) (+ (* x x) (* y (* 4.0 (- t (* z z))))) (pow x 2.0)))

C

x = abs(x);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 5.5e+149) {
		tmp = (x * x) + (y * (4.0 * (t - (z * z))));
	} else {
		tmp = pow(x, 2.0);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
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 <= 5.5d+149) then
        tmp = (x * x) + (y * (4.0d0 * (t - (z * z))))
    else
        tmp = x ** 2.0d0
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 5.5e+149) {
		tmp = (x * x) + (y * (4.0 * (t - (z * z))));
	} else {
		tmp = Math.pow(x, 2.0);
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y, z, t):
	tmp = 0
	if x <= 5.5e+149:
		tmp = (x * x) + (y * (4.0 * (t - (z * z))))
	else:
		tmp = math.pow(x, 2.0)
	return tmp

Julia

x = abs(x)
function code(x, y, z, t)
	tmp = 0.0
	if (x <= 5.5e+149)
		tmp = Float64(Float64(x * x) + Float64(y * Float64(4.0 * Float64(t - Float64(z * z)))));
	else
		tmp = x ^ 2.0;
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 5.5e+149)
		tmp = (x * x) + (y * (4.0 * (t - (z * z))));
	else
		tmp = x ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[x, 5.5e+149], N[(N[(x * x), $MachinePrecision] + N[(y * N[(4.0 * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[x, 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.49999999999999999e149

   1. Initial program 90.6%

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

      1. associate-*l* 91.1%

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

   3. Simplified 91.1%

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

   if 5.49999999999999999e149 < x

   1. Initial program 83.3%

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

      1. associate-*l* 83.3%

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

   3. Simplified 83.3%

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

   4. Taylor expanded in x around inf 95.2%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{+149}:\\ \;\;\;\;x \cdot x + y \cdot \left(4 \cdot \left(t - z \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{2}\\ \end{array} \]

Alternative 3: 93.5% accurate, 0.5× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= (+ (* x x) (* (* y 4.0) (- t (* z z)))) INFINITY)
   (- (* x x) (* y (* (- (* z z) t) 4.0)))
   (- (* x x) (* y (* t -4.0)))))

C

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

Java

x = Math.abs(x);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) + ((y * 4.0) * (t - (z * z)))) <= Double.POSITIVE_INFINITY) {
		tmp = (x * x) - (y * (((z * z) - t) * 4.0));
	} else {
		tmp = (x * x) - (y * (t * -4.0));
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y, z, t):
	tmp = 0
	if ((x * x) + ((y * 4.0) * (t - (z * z)))) <= math.inf:
		tmp = (x * x) - (y * (((z * z) - t) * 4.0))
	else:
		tmp = (x * x) - (y * (t * -4.0))
	return tmp

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(x * x), $MachinePrecision] - N[(y * N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x x) (*.f64 (*.f64 y 4) (-.f64 (*.f64 z z) t))) < +inf.0

   1. Initial program 94.6%

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

      1. associate-*l* 95.0%

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

   3. Simplified 95.0%

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

   if +inf.0 < (-.f64 (*.f64 x x) (*.f64 (*.f64 y 4) (-.f64 (*.f64 z z) t)))

   1. Initial program 0.0%

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

      1. associate-*l* 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in z around 0 42.9%

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

      1. *-commutative 42.9%

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

      2. *-commutative 42.9%

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

      3. associate-*l* 42.9%

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

   6. Simplified 42.9%

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

4. Final simplification 92.1%

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

Alternative 4: 74.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 9 \cdot 10^{+129}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -4\right) \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= z 9e+129) (- (* x x) (* y (* t -4.0))) (* (* y -4.0) (* z z))))

C

x = abs(x);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 9e+129) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = (y * -4.0) * (z * z);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
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 <= 9d+129) then
        tmp = (x * x) - (y * (t * (-4.0d0)))
    else
        tmp = (y * (-4.0d0)) * (z * z)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 9e+129) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = (y * -4.0) * (z * z);
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y, z, t):
	tmp = 0
	if z <= 9e+129:
		tmp = (x * x) - (y * (t * -4.0))
	else:
		tmp = (y * -4.0) * (z * z)
	return tmp

Julia

x = abs(x)
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 9e+129)
		tmp = Float64(Float64(x * x) - Float64(y * Float64(t * -4.0)));
	else
		tmp = Float64(Float64(y * -4.0) * Float64(z * z));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 9e+129)
		tmp = (x * x) - (y * (t * -4.0));
	else
		tmp = (y * -4.0) * (z * z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[z, 9e+129], N[(N[(x * x), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * -4.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 9.0000000000000003e129

   1. Initial program 92.6%

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

      1. associate-*l* 93.1%

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

   3. Simplified 93.1%

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

   4. Taylor expanded in z around 0 77.6%

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

      1. *-commutative 77.6%

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

      2. *-commutative 77.6%

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

      3. associate-*l* 77.6%

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

   6. Simplified 77.6%

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

   if 9.0000000000000003e129 < z

   1. Initial program 73.2%

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

      1. associate-*l* 73.2%

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

   3. Simplified 73.2%

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

      1. add-sqr-sqrt 41.1%

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

      2. difference-of-squares 41.1%

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

   5. Applied egg-rr 41.1%

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

   6. Taylor expanded in z around inf 41.3%

      \[\leadsto \color{blue}{-4 \cdot \left(y \cdot \left(z \cdot \left(\sqrt{t} + -1 \cdot \sqrt{t}\right)\right)\right) + -4 \cdot \left(y \cdot {z}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative 41.3%

         \[\leadsto \color{blue}{-4 \cdot \left(y \cdot {z}^{2}\right) + -4 \cdot \left(y \cdot \left(z \cdot \left(\sqrt{t} + -1 \cdot \sqrt{t}\right)\right)\right)} \]

      2. associate-*r* 41.3%

         \[\leadsto \color{blue}{\left(-4 \cdot y\right) \cdot {z}^{2}} + -4 \cdot \left(y \cdot \left(z \cdot \left(\sqrt{t} + -1 \cdot \sqrt{t}\right)\right)\right) \]

      3. associate-*r* 41.3%

         \[\leadsto \left(-4 \cdot y\right) \cdot {z}^{2} + \color{blue}{\left(-4 \cdot y\right) \cdot \left(z \cdot \left(\sqrt{t} + -1 \cdot \sqrt{t}\right)\right)} \]

      4. distribute-lft-out 41.3%

         \[\leadsto \color{blue}{\left(-4 \cdot y\right) \cdot \left({z}^{2} + z \cdot \left(\sqrt{t} + -1 \cdot \sqrt{t}\right)\right)} \]

      5. unpow2 41.3%

         \[\leadsto \left(-4 \cdot y\right) \cdot \left(\color{blue}{z \cdot z} + z \cdot \left(\sqrt{t} + -1 \cdot \sqrt{t}\right)\right) \]

      6. *-commutative 41.3%

         \[\leadsto \left(-4 \cdot y\right) \cdot \left(z \cdot z + \color{blue}{\left(\sqrt{t} + -1 \cdot \sqrt{t}\right) \cdot z}\right) \]

      7. distribute-rgt1-in 41.3%

         \[\leadsto \left(-4 \cdot y\right) \cdot \left(z \cdot z + \color{blue}{\left(\left(-1 + 1\right) \cdot \sqrt{t}\right)} \cdot z\right) \]

      8. metadata-eval 41.3%

         \[\leadsto \left(-4 \cdot y\right) \cdot \left(z \cdot z + \left(\color{blue}{0} \cdot \sqrt{t}\right) \cdot z\right) \]

      9. mul0-lft 73.3%

         \[\leadsto \left(-4 \cdot y\right) \cdot \left(z \cdot z + \color{blue}{0} \cdot z\right) \]

      10. distribute-rgt-out 73.3%

          \[\leadsto \left(-4 \cdot y\right) \cdot \color{blue}{\left(z \cdot \left(z + 0\right)\right)} \]

   8. Simplified 73.3%

      \[\leadsto \color{blue}{\left(-4 \cdot y\right) \cdot \left(z \cdot \left(z + 0\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9 \cdot 10^{+129}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -4\right) \cdot \left(z \cdot z\right)\\ \end{array} \]

Alternative 5: 44.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 4.6 \cdot 10^{+16}:\\ \;\;\;\;4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -4\right) \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= z 4.6e+16) (* 4.0 (* t y)) (* (* y -4.0) (* z z))))

C

x = abs(x);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 4.6e+16) {
		tmp = 4.0 * (t * y);
	} else {
		tmp = (y * -4.0) * (z * z);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
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 <= 4.6d+16) then
        tmp = 4.0d0 * (t * y)
    else
        tmp = (y * (-4.0d0)) * (z * z)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 4.6e+16) {
		tmp = 4.0 * (t * y);
	} else {
		tmp = (y * -4.0) * (z * z);
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y, z, t):
	tmp = 0
	if z <= 4.6e+16:
		tmp = 4.0 * (t * y)
	else:
		tmp = (y * -4.0) * (z * z)
	return tmp

Julia

x = abs(x)
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 4.6e+16)
		tmp = Float64(4.0 * Float64(t * y));
	else
		tmp = Float64(Float64(y * -4.0) * Float64(z * z));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 4.6e+16)
		tmp = 4.0 * (t * y);
	else
		tmp = (y * -4.0) * (z * z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[z, 4.6e+16], N[(4.0 * N[(t * y), $MachinePrecision]), $MachinePrecision], N[(N[(y * -4.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.6e16

   1. Initial program 93.1%

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

      1. associate-*l* 93.5%

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

   3. Simplified 93.5%

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

   4. Taylor expanded in t around inf 38.1%

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

      1. *-commutative 38.1%

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

   6. Simplified 38.1%

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

   if 4.6e16 < z

   1. Initial program 77.0%

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

      1. associate-*l* 77.0%

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

   3. Simplified 77.0%

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

      1. add-sqr-sqrt 46.9%

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

      2. difference-of-squares 46.9%

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

   5. Applied egg-rr 46.9%

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

   6. Taylor expanded in z around inf 38.6%

      \[\leadsto \color{blue}{-4 \cdot \left(y \cdot \left(z \cdot \left(\sqrt{t} + -1 \cdot \sqrt{t}\right)\right)\right) + -4 \cdot \left(y \cdot {z}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative 38.6%

         \[\leadsto \color{blue}{-4 \cdot \left(y \cdot {z}^{2}\right) + -4 \cdot \left(y \cdot \left(z \cdot \left(\sqrt{t} + -1 \cdot \sqrt{t}\right)\right)\right)} \]

      2. associate-*r* 38.6%

         \[\leadsto \color{blue}{\left(-4 \cdot y\right) \cdot {z}^{2}} + -4 \cdot \left(y \cdot \left(z \cdot \left(\sqrt{t} + -1 \cdot \sqrt{t}\right)\right)\right) \]

      3. associate-*r* 38.6%

         \[\leadsto \left(-4 \cdot y\right) \cdot {z}^{2} + \color{blue}{\left(-4 \cdot y\right) \cdot \left(z \cdot \left(\sqrt{t} + -1 \cdot \sqrt{t}\right)\right)} \]

      4. distribute-lft-out 38.6%

         \[\leadsto \color{blue}{\left(-4 \cdot y\right) \cdot \left({z}^{2} + z \cdot \left(\sqrt{t} + -1 \cdot \sqrt{t}\right)\right)} \]

      5. unpow2 38.6%

         \[\leadsto \left(-4 \cdot y\right) \cdot \left(\color{blue}{z \cdot z} + z \cdot \left(\sqrt{t} + -1 \cdot \sqrt{t}\right)\right) \]

      6. *-commutative 38.6%

         \[\leadsto \left(-4 \cdot y\right) \cdot \left(z \cdot z + \color{blue}{\left(\sqrt{t} + -1 \cdot \sqrt{t}\right) \cdot z}\right) \]

      7. distribute-rgt1-in 38.6%

         \[\leadsto \left(-4 \cdot y\right) \cdot \left(z \cdot z + \color{blue}{\left(\left(-1 + 1\right) \cdot \sqrt{t}\right)} \cdot z\right) \]

      8. metadata-eval 38.6%

         \[\leadsto \left(-4 \cdot y\right) \cdot \left(z \cdot z + \left(\color{blue}{0} \cdot \sqrt{t}\right) \cdot z\right) \]

      9. mul0-lft 65.2%

         \[\leadsto \left(-4 \cdot y\right) \cdot \left(z \cdot z + \color{blue}{0} \cdot z\right) \]

      10. distribute-rgt-out 65.2%

          \[\leadsto \left(-4 \cdot y\right) \cdot \color{blue}{\left(z \cdot \left(z + 0\right)\right)} \]

   8. Simplified 65.2%

      \[\leadsto \color{blue}{\left(-4 \cdot y\right) \cdot \left(z \cdot \left(z + 0\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.6 \cdot 10^{+16}:\\ \;\;\;\;4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -4\right) \cdot \left(z \cdot z\right)\\ \end{array} \]

Alternative 6: 31.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.65 \cdot 10^{+236}:\\ \;\;\;\;4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(4 \cdot \left(-y\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= x 2.65e+236) (* 4.0 (* t y)) (* t (* 4.0 (- y)))))

C

x = abs(x);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 2.65e+236) {
		tmp = 4.0 * (t * y);
	} else {
		tmp = t * (4.0 * -y);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= 2.65d+236) then
        tmp = 4.0d0 * (t * y)
    else
        tmp = t * (4.0d0 * -y)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 2.65e+236) {
		tmp = 4.0 * (t * y);
	} else {
		tmp = t * (4.0 * -y);
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y, z, t):
	tmp = 0
	if x <= 2.65e+236:
		tmp = 4.0 * (t * y)
	else:
		tmp = t * (4.0 * -y)
	return tmp

Julia

x = abs(x)
function code(x, y, z, t)
	tmp = 0.0
	if (x <= 2.65e+236)
		tmp = Float64(4.0 * Float64(t * y));
	else
		tmp = Float64(t * Float64(4.0 * Float64(-y)));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 2.65e+236)
		tmp = 4.0 * (t * y);
	else
		tmp = t * (4.0 * -y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[x, 2.65e+236], N[(4.0 * N[(t * y), $MachinePrecision]), $MachinePrecision], N[(t * N[(4.0 * (-y)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.64999999999999989e236

   1. Initial program 90.4%

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

      1. associate-*l* 90.8%

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

   3. Simplified 90.8%

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

   4. Taylor expanded in t around inf 33.4%

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

      1. *-commutative 33.4%

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

   6. Simplified 33.4%

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

   if 2.64999999999999989e236 < x

   1. Initial program 73.3%

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

      1. associate-*l* 73.3%

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

   3. Simplified 73.3%

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

   4. Taylor expanded in t around inf 1.1%

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

      1. *-commutative 1.1%

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

   6. Simplified 1.1%

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

      1. *-commutative 1.1%

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

      2. metadata-eval 1.1%

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

      3. distribute-rgt-neg-in 1.1%

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

      4. associate-*r* 1.1%

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

      5. add-sqr-sqrt 0.2%

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

      6. sqrt-unprod 0.8%

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

      7. associate-*r* 0.8%

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

      8. associate-*r* 0.8%

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

      9. swap-sqr 0.8%

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

      10. metadata-eval 0.8%

          \[\leadsto -\sqrt{\left(\left(y \cdot t\right) \cdot \left(y \cdot t\right)\right) \cdot \color{blue}{16}} \]

      11. metadata-eval 0.8%

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

      12. swap-sqr 0.8%

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

      13. *-commutative 0.8%

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

      14. *-commutative 0.8%

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

      15. sqrt-unprod 0.6%

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

      16. add-sqr-sqrt 15.6%

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

      17. associate-*r* 15.6%

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

      18. *-commutative 15.6%

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

   8. Applied egg-rr 15.6%

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

4. Final simplification 32.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.65 \cdot 10^{+236}:\\ \;\;\;\;4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(4 \cdot \left(-y\right)\right)\\ \end{array} \]

Alternative 7: 31.7% accurate, 2.6× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z t) :precision binary64 (* 4.0 (* t y)))

C

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

Fortran

NOTE: x should be positive before calling this function
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 = 4.0d0 * (t * y)
end function

Java

x = Math.abs(x);
public static double code(double x, double y, double z, double t) {
	return 4.0 * (t * y);
}

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_, t_] := N[(4.0 * N[(t * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.4%

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

   1. associate-*l* 89.8%

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

3. Simplified 89.8%

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

4. Taylor expanded in t around inf 31.5%

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

   1. *-commutative 31.5%

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

6. Simplified 31.5%

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

7. Final simplification 31.5%

   \[\leadsto 4 \cdot \left(t \cdot y\right) \]

Developer target: 91.6% 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 2023336 
(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))))