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

Percentage Accurate: 90.4% → 95.1%

Time: 8.9s

Alternatives: 9

Speedup: 0.9×

• 49.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.4% 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: 95.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+256}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot -4\right) \cdot \left(z \cdot z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(z \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 4e+256)
   (fma x x (* (* y -4.0) (- (* z z) t)))
   (* z (* y (* z -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 4e+256) {
		tmp = fma(x, x, ((y * -4.0) * ((z * z) - t)));
	} else {
		tmp = z * (y * (z * -4.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 4e+256)
		tmp = fma(x, x, Float64(Float64(y * -4.0) * Float64(Float64(z * z) - t)));
	else
		tmp = Float64(z * Float64(y * Float64(z * -4.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 4e+256], N[(x * x + N[(N[(y * -4.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(y * N[(z * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.0000000000000001e256

   1. Initial program 98.8%

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

      1. fma-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   if 4.0000000000000001e256 < (*.f64 z z)

   1. Initial program 68.4%

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

      1. sub-neg 68.4%

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

      2. distribute-rgt-in 62.4%

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

   3. Applied egg-rr 62.4%

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

   4. Taylor expanded in z around inf 74.5%

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

      1. unpow2 74.5%

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

      2. *-commutative 74.5%

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

      3. associate-*l* 74.5%

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

      4. *-commutative 74.5%

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

      5. associate-*r* 74.5%

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

      6. associate-*l* 91.0%

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

   6. Simplified 91.0%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+256}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot -4\right) \cdot \left(z \cdot z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(z \cdot -4\right)\right)\\ \end{array} \]

Alternative 2: 83.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot x - 4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ t_2 := \left(y \cdot -4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{if}\;x \cdot x \leq 4.3 \cdot 10^{-157}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot x \leq 3.6 \cdot 10^{-63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot x \leq 1.4 \cdot 10^{+22}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot x \leq 2.75 \cdot 10^{+307}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x x) (* 4.0 (* (* z z) y))))
        (t_2 (* (* y -4.0) (- (* z z) t))))
   (if (<= (* x x) 4.3e-157)
     t_2
     (if (<= (* x x) 3.6e-63)
       t_1
       (if (<= (* x x) 1.4e+22)
         t_2
         (if (<= (* x x) 2.75e+307) t_1 (* x x)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) - (4.0 * ((z * z) * y));
	double t_2 = (y * -4.0) * ((z * z) - t);
	double tmp;
	if ((x * x) <= 4.3e-157) {
		tmp = t_2;
	} else if ((x * x) <= 3.6e-63) {
		tmp = t_1;
	} else if ((x * x) <= 1.4e+22) {
		tmp = t_2;
	} else if ((x * x) <= 2.75e+307) {
		tmp = t_1;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x * x) - (4.0d0 * ((z * z) * y))
    t_2 = (y * (-4.0d0)) * ((z * z) - t)
    if ((x * x) <= 4.3d-157) then
        tmp = t_2
    else if ((x * x) <= 3.6d-63) then
        tmp = t_1
    else if ((x * x) <= 1.4d+22) then
        tmp = t_2
    else if ((x * x) <= 2.75d+307) then
        tmp = t_1
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * x) - (4.0 * ((z * z) * y));
	double t_2 = (y * -4.0) * ((z * z) - t);
	double tmp;
	if ((x * x) <= 4.3e-157) {
		tmp = t_2;
	} else if ((x * x) <= 3.6e-63) {
		tmp = t_1;
	} else if ((x * x) <= 1.4e+22) {
		tmp = t_2;
	} else if ((x * x) <= 2.75e+307) {
		tmp = t_1;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * x) - (4.0 * ((z * z) * y))
	t_2 = (y * -4.0) * ((z * z) - t)
	tmp = 0
	if (x * x) <= 4.3e-157:
		tmp = t_2
	elif (x * x) <= 3.6e-63:
		tmp = t_1
	elif (x * x) <= 1.4e+22:
		tmp = t_2
	elif (x * x) <= 2.75e+307:
		tmp = t_1
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) - Float64(4.0 * Float64(Float64(z * z) * y)))
	t_2 = Float64(Float64(y * -4.0) * Float64(Float64(z * z) - t))
	tmp = 0.0
	if (Float64(x * x) <= 4.3e-157)
		tmp = t_2;
	elseif (Float64(x * x) <= 3.6e-63)
		tmp = t_1;
	elseif (Float64(x * x) <= 1.4e+22)
		tmp = t_2;
	elseif (Float64(x * x) <= 2.75e+307)
		tmp = t_1;
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * x) - (4.0 * ((z * z) * y));
	t_2 = (y * -4.0) * ((z * z) - t);
	tmp = 0.0;
	if ((x * x) <= 4.3e-157)
		tmp = t_2;
	elseif ((x * x) <= 3.6e-63)
		tmp = t_1;
	elseif ((x * x) <= 1.4e+22)
		tmp = t_2;
	elseif ((x * x) <= 2.75e+307)
		tmp = t_1;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] - N[(4.0 * N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * -4.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 4.3e-157], t$95$2, If[LessEqual[N[(x * x), $MachinePrecision], 3.6e-63], t$95$1, If[LessEqual[N[(x * x), $MachinePrecision], 1.4e+22], t$95$2, If[LessEqual[N[(x * x), $MachinePrecision], 2.75e+307], t$95$1, N[(x * x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 4.2999999999999998e-157 or 3.60000000000000008e-63 < (*.f64 x x) < 1.4e22

   1. Initial program 91.9%

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

   2. Taylor expanded in x around 0 90.3%

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

      1. associate-*r* 90.3%

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

      2. unpow2 90.3%

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

      3. *-commutative 90.3%

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

      4. *-commutative 90.3%

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

   4. Simplified 90.3%

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

   if 4.2999999999999998e-157 < (*.f64 x x) < 3.60000000000000008e-63 or 1.4e22 < (*.f64 x x) < 2.74999999999999993e307

   1. Initial program 90.0%

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

   2. Taylor expanded in z around inf 74.5%

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

      1. unpow2 74.5%

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

   4. Simplified 74.5%

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

   if 2.74999999999999993e307 < (*.f64 x x)

   1. Initial program 82.8%

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

   2. Taylor expanded in x around inf 89.1%

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

      1. unpow2 89.1%

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

   4. Simplified 89.1%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4.3 \cdot 10^{-157}:\\ \;\;\;\;\left(y \cdot -4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{elif}\;x \cdot x \leq 3.6 \cdot 10^{-63}:\\ \;\;\;\;x \cdot x - 4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \mathbf{elif}\;x \cdot x \leq 1.4 \cdot 10^{+22}:\\ \;\;\;\;\left(y \cdot -4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{elif}\;x \cdot x \leq 2.75 \cdot 10^{+307}:\\ \;\;\;\;x \cdot x - 4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 3: 59.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\left(z \cdot z\right) \cdot -4\right)\\ t_2 := y \cdot \left(t \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 8.4 \cdot 10^{-305}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot x \leq 3.3 \cdot 10^{-234}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot x \leq 3.5 \cdot 10^{-180}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot x \leq 1.4 \cdot 10^{+139}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (* (* z z) -4.0))) (t_2 (* y (* t 4.0))))
   (if (<= (* x x) 8.4e-305)
     t_2
     (if (<= (* x x) 3.3e-234)
       t_1
       (if (<= (* x x) 3.5e-180)
         t_2
         (if (<= (* x x) 1.4e+139) t_1 (* x x)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * ((z * z) * -4.0);
	double t_2 = y * (t * 4.0);
	double tmp;
	if ((x * x) <= 8.4e-305) {
		tmp = t_2;
	} else if ((x * x) <= 3.3e-234) {
		tmp = t_1;
	} else if ((x * x) <= 3.5e-180) {
		tmp = t_2;
	} else if ((x * x) <= 1.4e+139) {
		tmp = t_1;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = y * ((z * z) * (-4.0d0))
    t_2 = y * (t * 4.0d0)
    if ((x * x) <= 8.4d-305) then
        tmp = t_2
    else if ((x * x) <= 3.3d-234) then
        tmp = t_1
    else if ((x * x) <= 3.5d-180) then
        tmp = t_2
    else if ((x * x) <= 1.4d+139) then
        tmp = t_1
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = y * ((z * z) * -4.0);
	double t_2 = y * (t * 4.0);
	double tmp;
	if ((x * x) <= 8.4e-305) {
		tmp = t_2;
	} else if ((x * x) <= 3.3e-234) {
		tmp = t_1;
	} else if ((x * x) <= 3.5e-180) {
		tmp = t_2;
	} else if ((x * x) <= 1.4e+139) {
		tmp = t_1;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y * ((z * z) * -4.0)
	t_2 = y * (t * 4.0)
	tmp = 0
	if (x * x) <= 8.4e-305:
		tmp = t_2
	elif (x * x) <= 3.3e-234:
		tmp = t_1
	elif (x * x) <= 3.5e-180:
		tmp = t_2
	elif (x * x) <= 1.4e+139:
		tmp = t_1
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(Float64(z * z) * -4.0))
	t_2 = Float64(y * Float64(t * 4.0))
	tmp = 0.0
	if (Float64(x * x) <= 8.4e-305)
		tmp = t_2;
	elseif (Float64(x * x) <= 3.3e-234)
		tmp = t_1;
	elseif (Float64(x * x) <= 3.5e-180)
		tmp = t_2;
	elseif (Float64(x * x) <= 1.4e+139)
		tmp = t_1;
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y * ((z * z) * -4.0);
	t_2 = y * (t * 4.0);
	tmp = 0.0;
	if ((x * x) <= 8.4e-305)
		tmp = t_2;
	elseif ((x * x) <= 3.3e-234)
		tmp = t_1;
	elseif ((x * x) <= 3.5e-180)
		tmp = t_2;
	elseif ((x * x) <= 1.4e+139)
		tmp = t_1;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(N[(z * z), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(t * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 8.4e-305], t$95$2, If[LessEqual[N[(x * x), $MachinePrecision], 3.3e-234], t$95$1, If[LessEqual[N[(x * x), $MachinePrecision], 3.5e-180], t$95$2, If[LessEqual[N[(x * x), $MachinePrecision], 1.4e+139], t$95$1, N[(x * x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 8.3999999999999999e-305 or 3.30000000000000014e-234 < (*.f64 x x) < 3.5000000000000001e-180

   1. Initial program 91.6%

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

   2. Taylor expanded in t around inf 64.2%

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

      1. associate-*r* 64.2%

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

      2. *-commutative 64.2%

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

   4. Simplified 64.2%

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

   if 8.3999999999999999e-305 < (*.f64 x x) < 3.30000000000000014e-234 or 3.5000000000000001e-180 < (*.f64 x x) < 1.3999999999999999e139

   1. Initial program 90.9%

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

   2. Taylor expanded in z around inf 53.6%

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

      1. *-commutative 53.6%

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

      2. unpow2 53.6%

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

      3. associate-*l* 53.6%

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

   4. Simplified 53.6%

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

   if 1.3999999999999999e139 < (*.f64 x x)

   1. Initial program 85.4%

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

   2. Taylor expanded in x around inf 74.8%

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

      1. unpow2 74.8%

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

   4. Simplified 74.8%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 8.4 \cdot 10^{-305}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{elif}\;x \cdot x \leq 3.3 \cdot 10^{-234}:\\ \;\;\;\;y \cdot \left(\left(z \cdot z\right) \cdot -4\right)\\ \mathbf{elif}\;x \cdot x \leq 3.5 \cdot 10^{-180}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{elif}\;x \cdot x \leq 1.4 \cdot 10^{+139}:\\ \;\;\;\;y \cdot \left(\left(z \cdot z\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 4: 60.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(t \cdot 4\right)\\ \mathbf{if}\;z \cdot z \leq 10^{-174}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot z \leq 20000:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+82}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(z \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (* t 4.0))))
   (if (<= (* z z) 1e-174)
     t_1
     (if (<= (* z z) 20000.0)
       (* x x)
       (if (<= (* z z) 5e+82) t_1 (* z (* y (* z -4.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * (t * 4.0);
	double tmp;
	if ((z * z) <= 1e-174) {
		tmp = t_1;
	} else if ((z * z) <= 20000.0) {
		tmp = x * x;
	} else if ((z * z) <= 5e+82) {
		tmp = t_1;
	} else {
		tmp = z * (y * (z * -4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = y * (t * 4.0d0)
    if ((z * z) <= 1d-174) then
        tmp = t_1
    else if ((z * z) <= 20000.0d0) then
        tmp = x * x
    else if ((z * z) <= 5d+82) then
        tmp = t_1
    else
        tmp = z * (y * (z * (-4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = y * (t * 4.0);
	double tmp;
	if ((z * z) <= 1e-174) {
		tmp = t_1;
	} else if ((z * z) <= 20000.0) {
		tmp = x * x;
	} else if ((z * z) <= 5e+82) {
		tmp = t_1;
	} else {
		tmp = z * (y * (z * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y * (t * 4.0)
	tmp = 0
	if (z * z) <= 1e-174:
		tmp = t_1
	elif (z * z) <= 20000.0:
		tmp = x * x
	elif (z * z) <= 5e+82:
		tmp = t_1
	else:
		tmp = z * (y * (z * -4.0))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(t * 4.0))
	tmp = 0.0
	if (Float64(z * z) <= 1e-174)
		tmp = t_1;
	elseif (Float64(z * z) <= 20000.0)
		tmp = Float64(x * x);
	elseif (Float64(z * z) <= 5e+82)
		tmp = t_1;
	else
		tmp = Float64(z * Float64(y * Float64(z * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y * (t * 4.0);
	tmp = 0.0;
	if ((z * z) <= 1e-174)
		tmp = t_1;
	elseif ((z * z) <= 20000.0)
		tmp = x * x;
	elseif ((z * z) <= 5e+82)
		tmp = t_1;
	else
		tmp = z * (y * (z * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(t * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * z), $MachinePrecision], 1e-174], t$95$1, If[LessEqual[N[(z * z), $MachinePrecision], 20000.0], N[(x * x), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 5e+82], t$95$1, N[(z * N[(y * N[(z * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 1e-174 or 2e4 < (*.f64 z z) < 5.00000000000000015e82

   1. Initial program 99.1%

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

   2. Taylor expanded in t around inf 55.4%

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

      1. associate-*r* 55.4%

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

      2. *-commutative 55.4%

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

   4. Simplified 55.4%

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

   if 1e-174 < (*.f64 z z) < 2e4

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 50.3%

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

      1. unpow2 50.3%

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

   4. Simplified 50.3%

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

   if 5.00000000000000015e82 < (*.f64 z z)

   1. Initial program 76.2%

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

      1. sub-neg 76.2%

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

      2. distribute-rgt-in 71.8%

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

   3. Applied egg-rr 71.8%

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

   4. Taylor expanded in z around inf 72.4%

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

      1. unpow2 72.4%

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

      2. *-commutative 72.4%

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

      3. associate-*l* 72.4%

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

      4. *-commutative 72.4%

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

      5. associate-*r* 72.4%

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

      6. associate-*l* 84.3%

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

   6. Simplified 84.3%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-174}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{elif}\;z \cdot z \leq 20000:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+82}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(z \cdot -4\right)\right)\\ \end{array} \]

Alternative 5: 57.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4.5 \cdot 10^{-157} \lor \neg \left(x \cdot x \leq 1.35 \cdot 10^{-59}\right) \land x \cdot x \leq 2.6 \cdot 10^{+24}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* x x) 4.5e-157)
         (and (not (<= (* x x) 1.35e-59)) (<= (* x x) 2.6e+24)))
   (* y (* t 4.0))
   (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) <= 4.5e-157) || (!((x * x) <= 1.35e-59) && ((x * x) <= 2.6e+24))) {
		tmp = y * (t * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x * x) <= 4.5d-157) .or. (.not. ((x * x) <= 1.35d-59)) .and. ((x * x) <= 2.6d+24)) then
        tmp = y * (t * 4.0d0)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) <= 4.5e-157) || (!((x * x) <= 1.35e-59) && ((x * x) <= 2.6e+24))) {
		tmp = y * (t * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x * x) <= 4.5e-157) or (not ((x * x) <= 1.35e-59) and ((x * x) <= 2.6e+24)):
		tmp = y * (t * 4.0)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x * x) <= 4.5e-157) || (!(Float64(x * x) <= 1.35e-59) && (Float64(x * x) <= 2.6e+24)))
		tmp = Float64(y * Float64(t * 4.0));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x * x) <= 4.5e-157) || (~(((x * x) <= 1.35e-59)) && ((x * x) <= 2.6e+24)))
		tmp = y * (t * 4.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x * x), $MachinePrecision], 4.5e-157], And[N[Not[LessEqual[N[(x * x), $MachinePrecision], 1.35e-59]], $MachinePrecision], LessEqual[N[(x * x), $MachinePrecision], 2.6e+24]]], N[(y * N[(t * 4.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.49999999999999999e-157 or 1.3499999999999999e-59 < (*.f64 x x) < 2.5999999999999998e24

   1. Initial program 91.8%

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

   2. Taylor expanded in t around inf 55.8%

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

      1. associate-*r* 55.8%

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

      2. *-commutative 55.8%

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

   4. Simplified 55.8%

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

   if 4.49999999999999999e-157 < (*.f64 x x) < 1.3499999999999999e-59 or 2.5999999999999998e24 < (*.f64 x x)

   1. Initial program 86.9%

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

   2. Taylor expanded in x around inf 60.2%

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

      1. unpow2 60.2%

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

   4. Simplified 60.2%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4.5 \cdot 10^{-157} \lor \neg \left(x \cdot x \leq 1.35 \cdot 10^{-59}\right) \land x \cdot x \leq 2.6 \cdot 10^{+24}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 6: 94.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.9 \cdot 10^{+184}:\\ \;\;\;\;x \cdot x + 4 \cdot \left(t \cdot y - z \cdot \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(z \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.9e+184)
   (+ (* x x) (* 4.0 (- (* t y) (* z (* z y)))))
   (* z (* y (* z -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.9e+184) {
		tmp = (x * x) + (4.0 * ((t * y) - (z * (z * y))));
	} else {
		tmp = z * (y * (z * -4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 1.9d+184) then
        tmp = (x * x) + (4.0d0 * ((t * y) - (z * (z * y))))
    else
        tmp = z * (y * (z * (-4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.9e+184) {
		tmp = (x * x) + (4.0 * ((t * y) - (z * (z * y))));
	} else {
		tmp = z * (y * (z * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 1.9e+184:
		tmp = (x * x) + (4.0 * ((t * y) - (z * (z * y))))
	else:
		tmp = z * (y * (z * -4.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.9e+184)
		tmp = Float64(Float64(x * x) + Float64(4.0 * Float64(Float64(t * y) - Float64(z * Float64(z * y)))));
	else
		tmp = Float64(z * Float64(y * Float64(z * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 1.9e+184)
		tmp = (x * x) + (4.0 * ((t * y) - (z * (z * y))));
	else
		tmp = z * (y * (z * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 1.9e+184], N[(N[(x * x), $MachinePrecision] + N[(4.0 * N[(N[(t * y), $MachinePrecision] - N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(y * N[(z * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.9000000000000001e184

   1. Initial program 91.4%

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

      1. sub-neg 91.4%

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

      2. distribute-rgt-in 89.6%

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

   3. Applied egg-rr 89.6%

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

      1. distribute-rgt-out 91.4%

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

      2. sub-neg 91.4%

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

      3. add-sqr-sqrt 43.2%

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

      4. sqrt-unprod 61.5%

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

      5. sqr-neg 61.5%

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

      6. sqrt-unprod 27.9%

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

      7. add-sqr-sqrt 58.6%

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

      8. distribute-rgt-out-- 57.3%

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

      9. associate-*r* 57.3%

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

      10. *-commutative 57.3%

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

      11. associate-*r* 57.3%

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

      12. distribute-rgt-out-- 57.3%

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

      13. *-commutative 57.3%

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

      14. associate-*l* 62.5%

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

      15. *-commutative 62.5%

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

      16. add-sqr-sqrt 28.7%

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

      17. sqrt-unprod 64.3%

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

      18. sqr-neg 64.3%

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

      19. sqrt-unprod 44.8%

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

      20. add-sqr-sqrt 95.0%

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

   5. Applied egg-rr 95.0%

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

   if 1.9000000000000001e184 < z

   1. Initial program 72.5%

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

      1. sub-neg 72.5%

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

      2. distribute-rgt-in 66.0%

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

   3. Applied egg-rr 66.0%

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

   4. Taylor expanded in z around inf 82.2%

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

      1. unpow2 82.2%

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

      2. *-commutative 82.2%

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

      3. associate-*l* 82.2%

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

      4. *-commutative 82.2%

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

      5. associate-*r* 82.2%

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

      6. associate-*l* 96.9%

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

   6. Simplified 96.9%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.9 \cdot 10^{+184}:\\ \;\;\;\;x \cdot x + 4 \cdot \left(t \cdot y - z \cdot \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(z \cdot -4\right)\right)\\ \end{array} \]

Alternative 7: 92.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 6.4 \cdot 10^{+122}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(z \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 6.4e+122)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (* z (* y (* z -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 6.4e+122) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = z * (y * (z * -4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 6.4d+122) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = z * (y * (z * (-4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 6.4e+122) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = z * (y * (z * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 6.4e+122:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = z * (y * (z * -4.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 6.4e+122)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(z * Float64(y * Float64(z * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 6.4e+122)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = z * (y * (z * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 6.4e+122], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(y * N[(z * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 6.40000000000000024e122

   1. Initial program 91.9%

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

   if 6.40000000000000024e122 < z

   1. Initial program 74.4%

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

      1. sub-neg 74.4%

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

      2. distribute-rgt-in 69.5%

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

   3. Applied egg-rr 69.5%

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

   4. Taylor expanded in z around inf 81.9%

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

      1. unpow2 81.9%

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

      2. *-commutative 81.9%

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

      3. associate-*l* 81.9%

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

      4. *-commutative 81.9%

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

      5. associate-*r* 81.9%

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

      6. associate-*l* 95.3%

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

   6. Simplified 95.3%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.4 \cdot 10^{+122}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(z \cdot -4\right)\right)\\ \end{array} \]

Alternative 8: 70.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.8 \cdot 10^{+154}:\\ \;\;\;\;\left(y \cdot -4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 7.8e+154) (* (* y -4.0) (- (* z z) t)) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 7.8e+154) {
		tmp = (y * -4.0) * ((z * z) - 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 <= 7.8d+154) then
        tmp = (y * (-4.0d0)) * ((z * z) - 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 <= 7.8e+154) {
		tmp = (y * -4.0) * ((z * z) - t);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 7.8e+154:
		tmp = (y * -4.0) * ((z * z) - t)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 7.8e+154)
		tmp = Float64(Float64(y * -4.0) * Float64(Float64(z * z) - t));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 7.8e+154)
		tmp = (y * -4.0) * ((z * z) - t);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 7.8e+154], N[(N[(y * -4.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.8000000000000006e154

   1. Initial program 89.1%

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

   2. Taylor expanded in x around 0 71.1%

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

      1. associate-*r* 71.1%

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

      2. unpow2 71.1%

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

      3. *-commutative 71.1%

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

      4. *-commutative 71.1%

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

   4. Simplified 71.1%

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

   if 7.8000000000000006e154 < x

   1. Initial program 89.2%

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

   2. Taylor expanded in x around inf 94.6%

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

      1. unpow2 94.6%

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

   4. Simplified 94.6%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.8 \cdot 10^{+154}:\\ \;\;\;\;\left(y \cdot -4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 9: 41.2% 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 89.1%

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

2. Taylor expanded in x around inf 35.4%

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

   1. unpow2 35.4%

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

4. Simplified 35.4%

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

5. Final simplification 35.4%

   \[\leadsto x \cdot x \]

Developer target: 90.4% 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 2023290 
(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))))