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

Percentage Accurate: 90.9% → 95.5%

Time: 17.1s

Alternatives: 11

Speedup: 0.9×

• 48.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 9.2 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= z 9.2e+167)
   (fma x x (* (- (* z z) t) (* y -4.0)))
   (* z (* z (* y -4.0)))))

C

z = abs(z);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 9.2e+167) {
		tmp = fma(x, x, (((z * z) - t) * (y * -4.0)));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Julia

z = abs(z)
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 9.2e+167)
		tmp = fma(x, x, Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0)));
	else
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	end
	return tmp
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[z, 9.2e+167], N[(x * x + N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 9.19999999999999952e167

   1. Initial program 91.6%

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

      1. fma-neg 95.0%

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

      2. *-commutative 95.0%

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

      3. distribute-rgt-neg-in 95.0%

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

      4. distribute-rgt-neg-in 95.0%

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

      5. metadata-eval 95.0%

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

   3. Simplified 95.0%

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

   if 9.19999999999999952e167 < z

   1. Initial program 78.9%

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

   2. Taylor expanded in z around inf 78.9%

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

      1. metadata-eval 78.9%

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

      2. distribute-lft-neg-in 78.9%

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

      3. *-commutative 78.9%

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

      4. unpow2 78.9%

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

      5. *-commutative 78.9%

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

      6. associate-*r* 78.9%

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

      7. associate-*l* 95.7%

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

      8. distribute-rgt-neg-in 95.7%

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

      9. distribute-rgt-neg-in 95.7%

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

      10. distribute-rgt-neg-in 95.7%

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

      11. metadata-eval 95.7%

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

   4. Simplified 95.7%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9.2 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternative 2: 59.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} t_1 := -4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ t_2 := t \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 1.28 \cdot 10^{-267}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot x \leq 1.72 \cdot 10^{-239}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot x \leq 1.55 \cdot 10^{-173}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot x \leq 1.65 \cdot 10^{+19}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* -4.0 (* (* z z) y))) (t_2 (* t (* y 4.0))))
   (if (<= (* x x) 1.28e-267)
     t_2
     (if (<= (* x x) 1.72e-239)
       t_1
       (if (<= (* x x) 1.55e-173)
         t_2
         (if (<= (* x x) 1.65e+19) t_1 (* x x)))))))

C

z = abs(z);
double code(double x, double y, double z, double t) {
	double t_1 = -4.0 * ((z * z) * y);
	double t_2 = t * (y * 4.0);
	double tmp;
	if ((x * x) <= 1.28e-267) {
		tmp = t_2;
	} else if ((x * x) <= 1.72e-239) {
		tmp = t_1;
	} else if ((x * x) <= 1.55e-173) {
		tmp = t_2;
	} else if ((x * x) <= 1.65e+19) {
		tmp = t_1;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

NOTE: z 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (-4.0d0) * ((z * z) * y)
    t_2 = t * (y * 4.0d0)
    if ((x * x) <= 1.28d-267) then
        tmp = t_2
    else if ((x * x) <= 1.72d-239) then
        tmp = t_1
    else if ((x * x) <= 1.55d-173) then
        tmp = t_2
    else if ((x * x) <= 1.65d+19) then
        tmp = t_1
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z, double t) {
	double t_1 = -4.0 * ((z * z) * y);
	double t_2 = t * (y * 4.0);
	double tmp;
	if ((x * x) <= 1.28e-267) {
		tmp = t_2;
	} else if ((x * x) <= 1.72e-239) {
		tmp = t_1;
	} else if ((x * x) <= 1.55e-173) {
		tmp = t_2;
	} else if ((x * x) <= 1.65e+19) {
		tmp = t_1;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z, t):
	t_1 = -4.0 * ((z * z) * y)
	t_2 = t * (y * 4.0)
	tmp = 0
	if (x * x) <= 1.28e-267:
		tmp = t_2
	elif (x * x) <= 1.72e-239:
		tmp = t_1
	elif (x * x) <= 1.55e-173:
		tmp = t_2
	elif (x * x) <= 1.65e+19:
		tmp = t_1
	else:
		tmp = x * x
	return tmp

Julia

z = abs(z)
function code(x, y, z, t)
	t_1 = Float64(-4.0 * Float64(Float64(z * z) * y))
	t_2 = Float64(t * Float64(y * 4.0))
	tmp = 0.0
	if (Float64(x * x) <= 1.28e-267)
		tmp = t_2;
	elseif (Float64(x * x) <= 1.72e-239)
		tmp = t_1;
	elseif (Float64(x * x) <= 1.55e-173)
		tmp = t_2;
	elseif (Float64(x * x) <= 1.65e+19)
		tmp = t_1;
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z, t)
	t_1 = -4.0 * ((z * z) * y);
	t_2 = t * (y * 4.0);
	tmp = 0.0;
	if ((x * x) <= 1.28e-267)
		tmp = t_2;
	elseif ((x * x) <= 1.72e-239)
		tmp = t_1;
	elseif ((x * x) <= 1.55e-173)
		tmp = t_2;
	elseif ((x * x) <= 1.65e+19)
		tmp = t_1;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(-4.0 * N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 1.28e-267], t$95$2, If[LessEqual[N[(x * x), $MachinePrecision], 1.72e-239], t$95$1, If[LessEqual[N[(x * x), $MachinePrecision], 1.55e-173], t$95$2, If[LessEqual[N[(x * x), $MachinePrecision], 1.65e+19], t$95$1, N[(x * x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 1.2799999999999999e-267 or 1.72000000000000005e-239 < (*.f64 x x) < 1.55000000000000003e-173

   1. Initial program 95.9%

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

   2. Taylor expanded in t around inf 61.8%

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

      1. associate-*r* 61.8%

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

   4. Simplified 61.8%

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

   if 1.2799999999999999e-267 < (*.f64 x x) < 1.72000000000000005e-239 or 1.55000000000000003e-173 < (*.f64 x x) < 1.65e19

   1. Initial program 93.9%

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

   2. Taylor expanded in z around inf 51.6%

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

      1. unpow2 51.6%

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

   4. Simplified 51.6%

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

   if 1.65e19 < (*.f64 x x)

   1. Initial program 85.1%

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

   2. Taylor expanded in x around inf 78.6%

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

      1. unpow2 78.6%

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

   4. Simplified 78.6%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.28 \cdot 10^{-267}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;x \cdot x \leq 1.72 \cdot 10^{-239}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \mathbf{elif}\;x \cdot x \leq 1.55 \cdot 10^{-173}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;x \cdot x \leq 1.65 \cdot 10^{+19}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 3: 49.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} t_1 := t \cdot \left(y \cdot 4\right)\\ t_2 := z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \mathbf{if}\;x \leq 5 \cdot 10^{-201}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-135}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{-120}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-87}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1000000000000:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* t (* y 4.0))) (t_2 (* z (* z (* y -4.0)))))
   (if (<= x 5e-201)
     t_2
     (if (<= x 2.8e-135)
       t_1
       (if (<= x 2.75e-120)
         (* -4.0 (* (* z z) y))
         (if (<= x 2.8e-87) t_1 (if (<= x 1000000000000.0) t_2 (* x x))))))))

C

z = abs(z);
double code(double x, double y, double z, double t) {
	double t_1 = t * (y * 4.0);
	double t_2 = z * (z * (y * -4.0));
	double tmp;
	if (x <= 5e-201) {
		tmp = t_2;
	} else if (x <= 2.8e-135) {
		tmp = t_1;
	} else if (x <= 2.75e-120) {
		tmp = -4.0 * ((z * z) * y);
	} else if (x <= 2.8e-87) {
		tmp = t_1;
	} else if (x <= 1000000000000.0) {
		tmp = t_2;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

NOTE: z 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = t * (y * 4.0d0)
    t_2 = z * (z * (y * (-4.0d0)))
    if (x <= 5d-201) then
        tmp = t_2
    else if (x <= 2.8d-135) then
        tmp = t_1
    else if (x <= 2.75d-120) then
        tmp = (-4.0d0) * ((z * z) * y)
    else if (x <= 2.8d-87) then
        tmp = t_1
    else if (x <= 1000000000000.0d0) then
        tmp = t_2
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z, double t) {
	double t_1 = t * (y * 4.0);
	double t_2 = z * (z * (y * -4.0));
	double tmp;
	if (x <= 5e-201) {
		tmp = t_2;
	} else if (x <= 2.8e-135) {
		tmp = t_1;
	} else if (x <= 2.75e-120) {
		tmp = -4.0 * ((z * z) * y);
	} else if (x <= 2.8e-87) {
		tmp = t_1;
	} else if (x <= 1000000000000.0) {
		tmp = t_2;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z, t):
	t_1 = t * (y * 4.0)
	t_2 = z * (z * (y * -4.0))
	tmp = 0
	if x <= 5e-201:
		tmp = t_2
	elif x <= 2.8e-135:
		tmp = t_1
	elif x <= 2.75e-120:
		tmp = -4.0 * ((z * z) * y)
	elif x <= 2.8e-87:
		tmp = t_1
	elif x <= 1000000000000.0:
		tmp = t_2
	else:
		tmp = x * x
	return tmp

Julia

z = abs(z)
function code(x, y, z, t)
	t_1 = Float64(t * Float64(y * 4.0))
	t_2 = Float64(z * Float64(z * Float64(y * -4.0)))
	tmp = 0.0
	if (x <= 5e-201)
		tmp = t_2;
	elseif (x <= 2.8e-135)
		tmp = t_1;
	elseif (x <= 2.75e-120)
		tmp = Float64(-4.0 * Float64(Float64(z * z) * y));
	elseif (x <= 2.8e-87)
		tmp = t_1;
	elseif (x <= 1000000000000.0)
		tmp = t_2;
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z, t)
	t_1 = t * (y * 4.0);
	t_2 = z * (z * (y * -4.0));
	tmp = 0.0;
	if (x <= 5e-201)
		tmp = t_2;
	elseif (x <= 2.8e-135)
		tmp = t_1;
	elseif (x <= 2.75e-120)
		tmp = -4.0 * ((z * z) * y);
	elseif (x <= 2.8e-87)
		tmp = t_1;
	elseif (x <= 1000000000000.0)
		tmp = t_2;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(t * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5e-201], t$95$2, If[LessEqual[x, 2.8e-135], t$95$1, If[LessEqual[x, 2.75e-120], N[(-4.0 * N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.8e-87], t$95$1, If[LessEqual[x, 1000000000000.0], t$95$2, N[(x * x), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 4.9999999999999999e-201 or 2.8000000000000001e-87 < x < 1e12

   1. Initial program 91.8%

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

   2. Taylor expanded in z around inf 39.7%

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

      1. metadata-eval 39.7%

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

      2. distribute-lft-neg-in 39.7%

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

      3. *-commutative 39.7%

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

      4. unpow2 39.7%

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

      5. *-commutative 39.7%

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

      6. associate-*r* 39.7%

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

      7. associate-*l* 42.7%

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

      8. distribute-rgt-neg-in 42.7%

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

      9. distribute-rgt-neg-in 42.7%

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

      10. distribute-rgt-neg-in 42.7%

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

      11. metadata-eval 42.7%

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

   4. Simplified 42.7%

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

   if 4.9999999999999999e-201 < x < 2.80000000000000023e-135 or 2.7500000000000001e-120 < x < 2.8000000000000001e-87

   1. Initial program 99.9%

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

   2. Taylor expanded in t around inf 81.5%

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

      1. associate-*r* 81.5%

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

   4. Simplified 81.5%

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

   if 2.80000000000000023e-135 < x < 2.7500000000000001e-120

   1. Initial program 99.7%

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

   2. Taylor expanded in z around inf 99.7%

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

      1. unpow2 99.7%

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

   4. Simplified 99.7%

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

   if 1e12 < x

   1. Initial program 81.3%

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

   2. Taylor expanded in x around inf 79.1%

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

      1. unpow2 79.1%

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

   4. Simplified 79.1%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-201}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-135}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{-120}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-87}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;x \leq 1000000000000:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 4: 49.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} t_1 := t \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \leq 1.3 \cdot 10^{-200}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \mathbf{elif}\;x \leq 1.88 \cdot 10^{-134}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-120}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-87}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 980000000000:\\ \;\;\;\;z \cdot \left(-4 \cdot \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* t (* y 4.0))))
   (if (<= x 1.3e-200)
     (* z (* z (* y -4.0)))
     (if (<= x 1.88e-134)
       t_1
       (if (<= x 2.7e-120)
         (* -4.0 (* (* z z) y))
         (if (<= x 2.6e-87)
           t_1
           (if (<= x 980000000000.0) (* z (* -4.0 (* z y))) (* x x))))))))

C

z = abs(z);
double code(double x, double y, double z, double t) {
	double t_1 = t * (y * 4.0);
	double tmp;
	if (x <= 1.3e-200) {
		tmp = z * (z * (y * -4.0));
	} else if (x <= 1.88e-134) {
		tmp = t_1;
	} else if (x <= 2.7e-120) {
		tmp = -4.0 * ((z * z) * y);
	} else if (x <= 2.6e-87) {
		tmp = t_1;
	} else if (x <= 980000000000.0) {
		tmp = z * (-4.0 * (z * y));
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

NOTE: z 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) :: t_1
    real(8) :: tmp
    t_1 = t * (y * 4.0d0)
    if (x <= 1.3d-200) then
        tmp = z * (z * (y * (-4.0d0)))
    else if (x <= 1.88d-134) then
        tmp = t_1
    else if (x <= 2.7d-120) then
        tmp = (-4.0d0) * ((z * z) * y)
    else if (x <= 2.6d-87) then
        tmp = t_1
    else if (x <= 980000000000.0d0) then
        tmp = z * ((-4.0d0) * (z * y))
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z, double t) {
	double t_1 = t * (y * 4.0);
	double tmp;
	if (x <= 1.3e-200) {
		tmp = z * (z * (y * -4.0));
	} else if (x <= 1.88e-134) {
		tmp = t_1;
	} else if (x <= 2.7e-120) {
		tmp = -4.0 * ((z * z) * y);
	} else if (x <= 2.6e-87) {
		tmp = t_1;
	} else if (x <= 980000000000.0) {
		tmp = z * (-4.0 * (z * y));
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z, t):
	t_1 = t * (y * 4.0)
	tmp = 0
	if x <= 1.3e-200:
		tmp = z * (z * (y * -4.0))
	elif x <= 1.88e-134:
		tmp = t_1
	elif x <= 2.7e-120:
		tmp = -4.0 * ((z * z) * y)
	elif x <= 2.6e-87:
		tmp = t_1
	elif x <= 980000000000.0:
		tmp = z * (-4.0 * (z * y))
	else:
		tmp = x * x
	return tmp

Julia

z = abs(z)
function code(x, y, z, t)
	t_1 = Float64(t * Float64(y * 4.0))
	tmp = 0.0
	if (x <= 1.3e-200)
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	elseif (x <= 1.88e-134)
		tmp = t_1;
	elseif (x <= 2.7e-120)
		tmp = Float64(-4.0 * Float64(Float64(z * z) * y));
	elseif (x <= 2.6e-87)
		tmp = t_1;
	elseif (x <= 980000000000.0)
		tmp = Float64(z * Float64(-4.0 * Float64(z * y)));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z, t)
	t_1 = t * (y * 4.0);
	tmp = 0.0;
	if (x <= 1.3e-200)
		tmp = z * (z * (y * -4.0));
	elseif (x <= 1.88e-134)
		tmp = t_1;
	elseif (x <= 2.7e-120)
		tmp = -4.0 * ((z * z) * y);
	elseif (x <= 2.6e-87)
		tmp = t_1;
	elseif (x <= 980000000000.0)
		tmp = z * (-4.0 * (z * y));
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(t * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.3e-200], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.88e-134], t$95$1, If[LessEqual[x, 2.7e-120], N[(-4.0 * N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.6e-87], t$95$1, If[LessEqual[x, 980000000000.0], N[(z * N[(-4.0 * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 1.29999999999999995e-200

   1. Initial program 92.0%

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

   2. Taylor expanded in z around inf 39.7%

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

      1. metadata-eval 39.7%

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

      2. distribute-lft-neg-in 39.7%

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

      3. *-commutative 39.7%

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

      4. unpow2 39.7%

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

      5. *-commutative 39.7%

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

      6. associate-*r* 39.7%

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

      7. associate-*l* 42.0%

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

      8. distribute-rgt-neg-in 42.0%

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

      9. distribute-rgt-neg-in 42.0%

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

      10. distribute-rgt-neg-in 42.0%

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

      11. metadata-eval 42.0%

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

   4. Simplified 42.0%

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

   if 1.29999999999999995e-200 < x < 1.8799999999999999e-134 or 2.6999999999999999e-120 < x < 2.60000000000000002e-87

   1. Initial program 99.9%

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

   2. Taylor expanded in t around inf 81.5%

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

      1. associate-*r* 81.5%

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

   4. Simplified 81.5%

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

   if 1.8799999999999999e-134 < x < 2.6999999999999999e-120

   1. Initial program 99.7%

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

   2. Taylor expanded in z around inf 99.7%

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

      1. unpow2 99.7%

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

   4. Simplified 99.7%

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

   if 2.60000000000000002e-87 < x < 9.8e11

   1. Initial program 90.3%

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

   2. Taylor expanded in z around inf 39.3%

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

      1. unpow2 39.3%

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

   4. Simplified 39.3%

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

      1. add-sqr-sqrt 22.9%

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

      2. sqrt-unprod 23.2%

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

      3. *-commutative 23.2%

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

      4. *-commutative 23.2%

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

      5. swap-sqr 17.7%

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

      6. pow2 17.7%

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

      7. pow2 17.7%

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

      8. pow-prod-up 17.7%

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

      9. metadata-eval 17.7%

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

   6. Applied egg-rr 17.7%

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

   7. Taylor expanded in z around 0 39.3%

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

      1. *-commutative 39.3%

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

      2. *-commutative 39.3%

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

      3. unpow2 39.3%

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

      4. associate-*l* 48.8%

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

      5. associate-*l* 48.8%

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

   9. Simplified 48.8%

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

   if 9.8e11 < x

   1. Initial program 81.3%

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

   2. Taylor expanded in x around inf 79.1%

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

      1. unpow2 79.1%

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

   4. Simplified 79.1%

      \[\leadsto \color{blue}{x \cdot x} \]
3. Recombined 5 regimes into one program.

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{-200}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \mathbf{elif}\;x \leq 1.88 \cdot 10^{-134}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-120}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-87}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;x \leq 980000000000:\\ \;\;\;\;z \cdot \left(-4 \cdot \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 5: 83.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 7.2 \cdot 10^{-32}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \mathbf{elif}\;x \cdot x \leq 3.8 \cdot 10^{+239}:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 7.2e-32)
   (* (- (* z z) t) (* y -4.0))
   (if (<= (* x x) 3.8e+239) (- (* x x) (* t (* y -4.0))) (* x x))))

C

z = abs(z);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 7.2e-32) {
		tmp = ((z * z) - t) * (y * -4.0);
	} else if ((x * x) <= 3.8e+239) {
		tmp = (x * x) - (t * (y * -4.0));
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

NOTE: z 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 * x) <= 7.2d-32) then
        tmp = ((z * z) - t) * (y * (-4.0d0))
    else if ((x * x) <= 3.8d+239) then
        tmp = (x * x) - (t * (y * (-4.0d0)))
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 7.2e-32) {
		tmp = ((z * z) - t) * (y * -4.0);
	} else if ((x * x) <= 3.8e+239) {
		tmp = (x * x) - (t * (y * -4.0));
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 7.2e-32:
		tmp = ((z * z) - t) * (y * -4.0)
	elif (x * x) <= 3.8e+239:
		tmp = (x * x) - (t * (y * -4.0))
	else:
		tmp = x * x
	return tmp

Julia

z = abs(z)
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 7.2e-32)
		tmp = Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0));
	elseif (Float64(x * x) <= 3.8e+239)
		tmp = Float64(Float64(x * x) - Float64(t * Float64(y * -4.0)));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 7.2e-32)
		tmp = ((z * z) - t) * (y * -4.0);
	elseif ((x * x) <= 3.8e+239)
		tmp = (x * x) - (t * (y * -4.0));
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 7.2e-32], N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 3.8e+239], N[(N[(x * x), $MachinePrecision] - N[(t * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 7.19999999999999986e-32

   1. Initial program 94.7%

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

   2. Taylor expanded in x around 0 90.0%

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

      1. *-commutative 90.0%

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

      2. *-commutative 90.0%

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

      3. unpow2 90.0%

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

      4. *-commutative 90.0%

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

      5. associate-*l* 90.0%

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

   4. Simplified 90.0%

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

   if 7.19999999999999986e-32 < (*.f64 x x) < 3.8000000000000001e239

   1. Initial program 93.0%

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

   2. Taylor expanded in z around 0 74.8%

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

      1. *-commutative 74.8%

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

      2. *-commutative 74.8%

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

      3. associate-*l* 74.8%

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

   4. Simplified 74.8%

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

   if 3.8000000000000001e239 < (*.f64 x x)

   1. Initial program 82.1%

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

   2. Taylor expanded in x around inf 88.6%

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

      1. unpow2 88.6%

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

   4. Simplified 88.6%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 7.2 \cdot 10^{-32}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \mathbf{elif}\;x \cdot x \leq 3.8 \cdot 10^{+239}:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 6: 89.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 3.8 \cdot 10^{+21}:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{+157}:\\ \;\;\;\;x \cdot x + y \cdot \left(\left(z \cdot z\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= z 3.8e+21)
   (- (* x x) (* t (* y -4.0)))
   (if (<= z 2.45e+157)
     (+ (* x x) (* y (* (* z z) -4.0)))
     (* z (* z (* y -4.0))))))

C

z = abs(z);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3.8e+21) {
		tmp = (x * x) - (t * (y * -4.0));
	} else if (z <= 2.45e+157) {
		tmp = (x * x) + (y * ((z * z) * -4.0));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Fortran

NOTE: z 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 <= 3.8d+21) then
        tmp = (x * x) - (t * (y * (-4.0d0)))
    else if (z <= 2.45d+157) then
        tmp = (x * x) + (y * ((z * z) * (-4.0d0)))
    else
        tmp = z * (z * (y * (-4.0d0)))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3.8e+21) {
		tmp = (x * x) - (t * (y * -4.0));
	} else if (z <= 2.45e+157) {
		tmp = (x * x) + (y * ((z * z) * -4.0));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z, t):
	tmp = 0
	if z <= 3.8e+21:
		tmp = (x * x) - (t * (y * -4.0))
	elif z <= 2.45e+157:
		tmp = (x * x) + (y * ((z * z) * -4.0))
	else:
		tmp = z * (z * (y * -4.0))
	return tmp

Julia

z = abs(z)
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 3.8e+21)
		tmp = Float64(Float64(x * x) - Float64(t * Float64(y * -4.0)));
	elseif (z <= 2.45e+157)
		tmp = Float64(Float64(x * x) + Float64(y * Float64(Float64(z * z) * -4.0)));
	else
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 3.8e+21)
		tmp = (x * x) - (t * (y * -4.0));
	elseif (z <= 2.45e+157)
		tmp = (x * x) + (y * ((z * z) * -4.0));
	else
		tmp = z * (z * (y * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[z, 3.8e+21], N[(N[(x * x), $MachinePrecision] - N[(t * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.45e+157], N[(N[(x * x), $MachinePrecision] + N[(y * N[(N[(z * z), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 3.8e21

   1. Initial program 91.1%

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

   2. Taylor expanded in z around 0 73.9%

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

      1. *-commutative 73.9%

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

      2. *-commutative 73.9%

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

      3. associate-*l* 73.9%

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

   4. Simplified 73.9%

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

   if 3.8e21 < z < 2.4500000000000001e157

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 100.0%

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

      1. unpow2 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 100.0%

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

      5. *-commutative 100.0%

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

   4. Simplified 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-neg-in 100.0%

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

      6. metadata-eval 100.0%

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

      7. associate-*r* 100.0%

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

      8. associate-*r* 100.0%

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

      9. *-commutative 100.0%

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

      10. associate-*l* 100.0%

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

   6. Applied egg-rr 100.0%

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

   if 2.4500000000000001e157 < z

   1. Initial program 77.4%

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

   2. Taylor expanded in z around inf 81.4%

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

      1. metadata-eval 81.4%

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

      2. distribute-lft-neg-in 81.4%

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

      3. *-commutative 81.4%

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

      4. unpow2 81.4%

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

      5. *-commutative 81.4%

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

      6. associate-*r* 81.4%

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

      7. associate-*l* 92.4%

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

      8. distribute-rgt-neg-in 92.4%

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

      9. distribute-rgt-neg-in 92.4%

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

      10. distribute-rgt-neg-in 92.4%

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

      11. metadata-eval 92.4%

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

   4. Simplified 92.4%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.8 \cdot 10^{+21}:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{+157}:\\ \;\;\;\;x \cdot x + y \cdot \left(\left(z \cdot z\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternative 7: 90.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+28}:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e+28)
   (- (* x x) (* t (* y -4.0)))
   (- (* x x) (* z (* z (* y 4.0))))))

C

z = abs(z);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e+28) {
		tmp = (x * x) - (t * (y * -4.0));
	} else {
		tmp = (x * x) - (z * (z * (y * 4.0)));
	}
	return tmp;
}

Fortran

NOTE: z 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 * z) <= 5d+28) then
        tmp = (x * x) - (t * (y * (-4.0d0)))
    else
        tmp = (x * x) - (z * (z * (y * 4.0d0)))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e+28) {
		tmp = (x * x) - (t * (y * -4.0));
	} else {
		tmp = (x * x) - (z * (z * (y * 4.0)));
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 5e+28:
		tmp = (x * x) - (t * (y * -4.0))
	else:
		tmp = (x * x) - (z * (z * (y * 4.0)))
	return tmp

Julia

z = abs(z)
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e+28)
		tmp = Float64(Float64(x * x) - Float64(t * Float64(y * -4.0)));
	else
		tmp = Float64(Float64(x * x) - Float64(z * Float64(z * Float64(y * 4.0))));
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 5e+28)
		tmp = (x * x) - (t * (y * -4.0));
	else
		tmp = (x * x) - (z * (z * (y * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+28], N[(N[(x * x), $MachinePrecision] - N[(t * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(z * N[(z * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.99999999999999957e28

   1. Initial program 97.2%

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

   2. Taylor expanded in z around 0 90.3%

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

      1. *-commutative 90.3%

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

      2. *-commutative 90.3%

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

      3. associate-*l* 90.3%

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

   4. Simplified 90.3%

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

   if 4.99999999999999957e28 < (*.f64 z z)

   1. Initial program 81.9%

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

   2. Taylor expanded in z around inf 78.5%

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

      1. unpow2 78.5%

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

      2. associate-*r* 78.5%

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

      3. *-commutative 78.5%

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

      4. associate-*r* 87.6%

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

      5. *-commutative 87.6%

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

   4. Simplified 87.6%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+28}:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \]

Alternative 8: 95.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 9 \cdot 10^{+119}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= z 9e+119)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (- (* x x) (* z (* z (* y 4.0))))))

C

z = abs(z);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 9e+119) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (x * x) - (z * (z * (y * 4.0)));
	}
	return tmp;
}

Fortran

NOTE: z 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+119) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = (x * x) - (z * (z * (y * 4.0d0)))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 9e+119) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (x * x) - (z * (z * (y * 4.0)));
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z, t):
	tmp = 0
	if z <= 9e+119:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = (x * x) - (z * (z * (y * 4.0)))
	return tmp

Julia

z = abs(z)
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 9e+119)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(Float64(x * x) - Float64(z * Float64(z * Float64(y * 4.0))));
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 9e+119)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = (x * x) - (z * (z * (y * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[z, 9e+119], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(z * N[(z * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 9.00000000000000039e119

   1. Initial program 91.6%

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

   if 9.00000000000000039e119 < z

   1. Initial program 83.4%

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

   2. Taylor expanded in z around inf 83.4%

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

      1. unpow2 83.4%

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

      2. associate-*r* 83.4%

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

      3. *-commutative 83.4%

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

      4. associate-*r* 97.0%

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

      5. *-commutative 97.0%

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

   4. Simplified 97.0%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9 \cdot 10^{+119}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \]

Alternative 9: 81.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.4 \cdot 10^{+149}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 1.4e+149) (* (- (* z z) t) (* y -4.0)) (* x x)))

C

z = abs(z);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 1.4e+149) {
		tmp = ((z * z) - t) * (y * -4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

NOTE: z 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 * x) <= 1.4d+149) then
        tmp = ((z * z) - t) * (y * (-4.0d0))
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 1.4e+149) {
		tmp = ((z * z) - t) * (y * -4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 1.4e+149:
		tmp = ((z * z) - t) * (y * -4.0)
	else:
		tmp = x * x
	return tmp

Julia

z = abs(z)
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 1.4e+149)
		tmp = Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 1.4e+149)
		tmp = ((z * z) - t) * (y * -4.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 1.4e+149], 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) < 1.4e149

   1. Initial program 94.5%

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

   2. Taylor expanded in x around 0 82.8%

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

      1. *-commutative 82.8%

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

      2. *-commutative 82.8%

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

      3. unpow2 82.8%

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

      4. *-commutative 82.8%

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

      5. associate-*l* 82.8%

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

   4. Simplified 82.8%

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

   if 1.4e149 < (*.f64 x x)

   1. Initial program 84.3%

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

   2. Taylor expanded in x around inf 84.6%

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

      1. unpow2 84.6%

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

   4. Simplified 84.6%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.4 \cdot 10^{+149}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 10: 58.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 7.2 \cdot 10^{-23}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 7.2e-23) (* t (* y 4.0)) (* x x)))

C

z = abs(z);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 7.2e-23) {
		tmp = t * (y * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

NOTE: z 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 * x) <= 7.2d-23) then
        tmp = t * (y * 4.0d0)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 7.2e-23) {
		tmp = t * (y * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 7.2e-23:
		tmp = t * (y * 4.0)
	else:
		tmp = x * x
	return tmp

Julia

z = abs(z)
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 7.2e-23)
		tmp = Float64(t * Float64(y * 4.0));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 7.2e-23)
		tmp = t * (y * 4.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 7.2e-23], N[(t * N[(y * 4.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 7.1999999999999996e-23

   1. Initial program 94.9%

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

   2. Taylor expanded in t around inf 51.6%

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

      1. associate-*r* 51.6%

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

   4. Simplified 51.6%

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

   if 7.1999999999999996e-23 < (*.f64 x x)

   1. Initial program 86.2%

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

   2. Taylor expanded in x around inf 75.0%

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

      1. unpow2 75.0%

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

   4. Simplified 75.0%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 7.2 \cdot 10^{-23}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 11: 40.7% accurate, 4.3× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ x \cdot x \end{array} \]

FPCore

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

C

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

Fortran

NOTE: z 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 = x * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.5%

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

2. Taylor expanded in x around inf 43.3%

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

   1. unpow2 43.3%

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

4. Simplified 43.3%

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

5. Final simplification 43.3%

   \[\leadsto x \cdot x \]

Developer target: 90.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023274 
(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))))