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

Percentage Accurate: 90.8% → 97.3%

Time: 16.0s

Alternatives: 10

Speedup: 0.8×

• 49.5% 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: 90.8% 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: 97.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot 4 \leq -1 \cdot 10^{+57} \lor \neg \left(y \cdot 4 \leq 10^{-131}\right):\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot -4\right) \cdot \left(z \cdot z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + \left(y \cdot \left(4 \cdot t\right) - \left(\left(y \cdot 4\right) \cdot z\right) \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* y 4.0) -1e+57) (not (<= (* y 4.0) 1e-131)))
   (fma x x (* (* y -4.0) (- (* z z) t)))
   (+ (* x x) (- (* y (* 4.0 t)) (* (* (* y 4.0) z) z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((y * 4.0) <= -1e+57) || !((y * 4.0) <= 1e-131)) {
		tmp = fma(x, x, ((y * -4.0) * ((z * z) - t)));
	} else {
		tmp = (x * x) + ((y * (4.0 * t)) - (((y * 4.0) * z) * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(y * 4.0) <= -1e+57) || !(Float64(y * 4.0) <= 1e-131))
		tmp = fma(x, x, Float64(Float64(y * -4.0) * Float64(Float64(z * z) - t)));
	else
		tmp = Float64(Float64(x * x) + Float64(Float64(y * Float64(4.0 * t)) - Float64(Float64(Float64(y * 4.0) * z) * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(y * 4.0), $MachinePrecision], -1e+57], N[Not[LessEqual[N[(y * 4.0), $MachinePrecision], 1e-131]], $MachinePrecision]], N[(x * x + N[(N[(y * -4.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] + N[(N[(y * N[(4.0 * t), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(y * 4.0), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y 4) < -1.00000000000000005e57 or 9.9999999999999999e-132 < (*.f64 y 4)

   1. Initial program 94.9%

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

      1. fma-neg 99.3%

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

      2. *-commutative 99.3%

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

      3. distribute-rgt-neg-in 99.3%

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

      4. distribute-rgt-neg-in 99.3%

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

      5. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   if -1.00000000000000005e57 < (*.f64 y 4) < 9.9999999999999999e-132

   1. Initial program 86.1%

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

      1. sub-neg 86.1%

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

      2. distribute-rgt-in 86.1%

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

      3. fma-def 86.1%

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

   3. Applied egg-rr 86.1%

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

      1. fma-udef 86.1%

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

      2. distribute-lft-neg-out 86.1%

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

      3. unsub-neg 86.1%

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

      4. associate-*l* 99.9%

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

      5. associate-*r* 99.9%

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

      6. *-commutative 99.9%

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

      7. *-commutative 99.9%

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

      8. associate-*r* 99.9%

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

      9. *-commutative 99.9%

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

      10. associate-*l* 99.9%

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

   5. Applied egg-rr 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot 4 \leq -1 \cdot 10^{+57} \lor \neg \left(y \cdot 4 \leq 10^{-131}\right):\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot -4\right) \cdot \left(z \cdot z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + \left(y \cdot \left(4 \cdot t\right) - \left(\left(y \cdot 4\right) \cdot z\right) \cdot z\right)\\ \end{array} \]

Alternative 2: 91.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot z - t\\ \mathbf{if}\;\left(y \cdot 4\right) \cdot t_1 \leq 2 \cdot 10^{+190}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -4\right) \cdot t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z z) t)))
   (if (<= (* (* y 4.0) t_1) 2e+190)
     (+ (* x x) (* (* y 4.0) (- t (* z z))))
     (* (* y -4.0) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) - t;
	double tmp;
	if (((y * 4.0) * t_1) <= 2e+190) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (y * -4.0) * t_1;
	}
	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 = (z * z) - t
    if (((y * 4.0d0) * t_1) <= 2d+190) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = (y * (-4.0d0)) * t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * z) - t;
	double tmp;
	if (((y * 4.0) * t_1) <= 2e+190) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (y * -4.0) * t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) - t
	tmp = 0
	if ((y * 4.0) * t_1) <= 2e+190:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = (y * -4.0) * t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) - t)
	tmp = 0.0
	if (Float64(Float64(y * 4.0) * t_1) <= 2e+190)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(Float64(y * -4.0) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * z) - t;
	tmp = 0.0;
	if (((y * 4.0) * t_1) <= 2e+190)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = (y * -4.0) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[N[(N[(y * 4.0), $MachinePrecision] * t$95$1), $MachinePrecision], 2e+190], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * -4.0), $MachinePrecision] * t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y 4) (-.f64 (*.f64 z z) t)) < 2.0000000000000001e190

   1. Initial program 96.6%

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

   if 2.0000000000000001e190 < (*.f64 (*.f64 y 4) (-.f64 (*.f64 z z) t))

   1. Initial program 70.7%

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

   2. Taylor expanded in x around 0 81.2%

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

      1. associate-*r* 81.2%

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

      2. unpow2 81.2%

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

      3. *-commutative 81.2%

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

      4. *-commutative 81.2%

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

   4. Simplified 81.2%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \leq 2 \cdot 10^{+190}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -4\right) \cdot \left(z \cdot z - t\right)\\ \end{array} \]

Alternative 3: 86.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 6.6 \cdot 10^{+78}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{elif}\;z \cdot z \leq 1.55 \cdot 10^{+300}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -4\right) \cdot \left(z \cdot z - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 6.6e+78)
   (- (* x x) (* y (* t -4.0)))
   (if (<= (* z z) 1.55e+300)
     (- (* x x) (* (* y 4.0) (* z z)))
     (* (* y -4.0) (- (* z z) t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 6.6e+78) {
		tmp = (x * x) - (y * (t * -4.0));
	} else if ((z * z) <= 1.55e+300) {
		tmp = (x * x) - ((y * 4.0) * (z * z));
	} else {
		tmp = (y * -4.0) * ((z * z) - t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z * z) <= 6.6d+78) then
        tmp = (x * x) - (y * (t * (-4.0d0)))
    else if ((z * z) <= 1.55d+300) then
        tmp = (x * x) - ((y * 4.0d0) * (z * z))
    else
        tmp = (y * (-4.0d0)) * ((z * z) - t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 6.6e+78) {
		tmp = (x * x) - (y * (t * -4.0));
	} else if ((z * z) <= 1.55e+300) {
		tmp = (x * x) - ((y * 4.0) * (z * z));
	} else {
		tmp = (y * -4.0) * ((z * z) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 6.6e+78:
		tmp = (x * x) - (y * (t * -4.0))
	elif (z * z) <= 1.55e+300:
		tmp = (x * x) - ((y * 4.0) * (z * z))
	else:
		tmp = (y * -4.0) * ((z * z) - t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 6.6e+78)
		tmp = Float64(Float64(x * x) - Float64(y * Float64(t * -4.0)));
	elseif (Float64(z * z) <= 1.55e+300)
		tmp = Float64(Float64(x * x) - Float64(Float64(y * 4.0) * Float64(z * z)));
	else
		tmp = Float64(Float64(y * -4.0) * Float64(Float64(z * z) - t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 6.6e+78)
		tmp = (x * x) - (y * (t * -4.0));
	elseif ((z * z) <= 1.55e+300)
		tmp = (x * x) - ((y * 4.0) * (z * z));
	else
		tmp = (y * -4.0) * ((z * z) - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 6.6e+78], N[(N[(x * x), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 1.55e+300], N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * -4.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 6.6e78

   1. Initial program 97.9%

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

   2. Taylor expanded in z around 0 93.5%

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

      1. associate-*r* 93.5%

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

   4. Simplified 93.5%

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

   if 6.6e78 < (*.f64 z z) < 1.5499999999999999e300

   1. Initial program 99.8%

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

   2. Taylor expanded in z around inf 92.4%

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

      1. unpow2 92.4%

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

   4. Simplified 92.4%

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

   if 1.5499999999999999e300 < (*.f64 z z)

   1. Initial program 70.1%

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

   2. Taylor expanded in x around 0 76.0%

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

      1. associate-*r* 76.0%

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

      2. unpow2 76.0%

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

      3. *-commutative 76.0%

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

      4. *-commutative 76.0%

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

   4. Simplified 76.0%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 6.6 \cdot 10^{+78}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{elif}\;z \cdot z \leq 1.55 \cdot 10^{+300}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -4\right) \cdot \left(z \cdot z - t\right)\\ \end{array} \]

Alternative 4: 58.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2.35 \cdot 10^{-63} \lor \neg \left(x \cdot x \leq 6.5 \cdot 10^{+31}\right) \land x \cdot x \leq 4.7 \cdot 10^{+60}:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* x x) 2.35e-63)
         (and (not (<= (* x x) 6.5e+31)) (<= (* x x) 4.7e+60)))
   (* 4.0 (* y t))
   (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) <= 2.35e-63) || (!((x * x) <= 6.5e+31) && ((x * x) <= 4.7e+60))) {
		tmp = 4.0 * (y * t);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x * x) <= 2.35d-63) .or. (.not. ((x * x) <= 6.5d+31)) .and. ((x * x) <= 4.7d+60)) then
        tmp = 4.0d0 * (y * t)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) <= 2.35e-63) || (!((x * x) <= 6.5e+31) && ((x * x) <= 4.7e+60))) {
		tmp = 4.0 * (y * t);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x * x) <= 2.35e-63) or (not ((x * x) <= 6.5e+31) and ((x * x) <= 4.7e+60)):
		tmp = 4.0 * (y * t)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x * x) <= 2.35e-63) || (!(Float64(x * x) <= 6.5e+31) && (Float64(x * x) <= 4.7e+60)))
		tmp = Float64(4.0 * Float64(y * t));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x * x) <= 2.35e-63) || (~(((x * x) <= 6.5e+31)) && ((x * x) <= 4.7e+60)))
		tmp = 4.0 * (y * t);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x * x), $MachinePrecision], 2.35e-63], And[N[Not[LessEqual[N[(x * x), $MachinePrecision], 6.5e+31]], $MachinePrecision], LessEqual[N[(x * x), $MachinePrecision], 4.7e+60]]], N[(4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.35e-63 or 6.5000000000000004e31 < (*.f64 x x) < 4.6999999999999998e60

   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 46.5%

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

      1. *-commutative 46.5%

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

   4. Simplified 46.5%

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

   if 2.35e-63 < (*.f64 x x) < 6.5000000000000004e31 or 4.6999999999999998e60 < (*.f64 x x)

   1. Initial program 90.0%

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

   2. Taylor expanded in x around inf 71.4%

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

      1. unpow2 71.4%

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

   4. Simplified 71.4%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2.35 \cdot 10^{-63} \lor \neg \left(x \cdot x \leq 6.5 \cdot 10^{+31}\right) \land x \cdot x \leq 4.7 \cdot 10^{+60}:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 5: 58.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.2 \cdot 10^{-63}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{elif}\;x \cdot x \leq 8.2 \cdot 10^{+31} \lor \neg \left(x \cdot x \leq 4.5 \cdot 10^{+60}\right):\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 1.2e-63)
   (* y (* 4.0 t))
   (if (or (<= (* x x) 8.2e+31) (not (<= (* x x) 4.5e+60)))
     (* x x)
     (* 4.0 (* y t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 1.2e-63) {
		tmp = y * (4.0 * t);
	} else if (((x * x) <= 8.2e+31) || !((x * x) <= 4.5e+60)) {
		tmp = x * x;
	} else {
		tmp = 4.0 * (y * t);
	}
	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) <= 1.2d-63) then
        tmp = y * (4.0d0 * t)
    else if (((x * x) <= 8.2d+31) .or. (.not. ((x * x) <= 4.5d+60))) then
        tmp = x * x
    else
        tmp = 4.0d0 * (y * t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 1.2e-63) {
		tmp = y * (4.0 * t);
	} else if (((x * x) <= 8.2e+31) || !((x * x) <= 4.5e+60)) {
		tmp = x * x;
	} else {
		tmp = 4.0 * (y * t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 1.2e-63:
		tmp = y * (4.0 * t)
	elif ((x * x) <= 8.2e+31) or not ((x * x) <= 4.5e+60):
		tmp = x * x
	else:
		tmp = 4.0 * (y * t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 1.2e-63)
		tmp = Float64(y * Float64(4.0 * t));
	elseif ((Float64(x * x) <= 8.2e+31) || !(Float64(x * x) <= 4.5e+60))
		tmp = Float64(x * x);
	else
		tmp = Float64(4.0 * Float64(y * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 1.2e-63)
		tmp = y * (4.0 * t);
	elseif (((x * x) <= 8.2e+31) || ~(((x * x) <= 4.5e+60)))
		tmp = x * x;
	else
		tmp = 4.0 * (y * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 1.2e-63], N[(y * N[(4.0 * t), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(x * x), $MachinePrecision], 8.2e+31], N[Not[LessEqual[N[(x * x), $MachinePrecision], 4.5e+60]], $MachinePrecision]], N[(x * x), $MachinePrecision], N[(4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 1.2e-63

   1. Initial program 92.9%

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

   2. Taylor expanded in t around inf 45.6%

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

      1. associate-*r* 45.6%

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

      2. *-commutative 45.6%

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

   4. Simplified 45.6%

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

   if 1.2e-63 < (*.f64 x x) < 8.2000000000000003e31 or 4.50000000000000013e60 < (*.f64 x x)

   1. Initial program 90.0%

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

   2. Taylor expanded in x around inf 71.4%

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

      1. unpow2 71.4%

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

   4. Simplified 71.4%

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

   if 8.2000000000000003e31 < (*.f64 x x) < 4.50000000000000013e60

   1. Initial program 79.2%

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

   2. Taylor expanded in t around inf 57.2%

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

      1. *-commutative 57.2%

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

   4. Simplified 57.2%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.2 \cdot 10^{-63}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{elif}\;x \cdot x \leq 8.2 \cdot 10^{+31} \lor \neg \left(x \cdot x \leq 4.5 \cdot 10^{+60}\right):\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 6: 93.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

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 * t)) - (((y * 4.0d0) * z) * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

   1. sub-neg 90.8%

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

   2. distribute-rgt-in 89.7%

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

   3. fma-def 89.7%

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

3. Applied egg-rr 89.7%

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

   1. fma-udef 89.7%

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

   2. distribute-lft-neg-out 89.7%

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

   3. unsub-neg 89.7%

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

   4. associate-*l* 96.0%

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

   5. associate-*r* 96.0%

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

   6. *-commutative 96.0%

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

   7. *-commutative 96.0%

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

   8. associate-*r* 96.0%

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

   9. *-commutative 96.0%

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

   10. associate-*l* 96.0%

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

5. Applied egg-rr 96.0%

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

6. Final simplification 96.0%

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

Alternative 7: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 3.8 \cdot 10^{+222}:\\ \;\;\;\;\left(y \cdot -4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 3.8e+222) (* (* y -4.0) (- (* z z) t)) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 3.8e+222) {
		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 * x) <= 3.8d+222) 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 * x) <= 3.8e+222) {
		tmp = (y * -4.0) * ((z * z) - t);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 3.8e+222:
		tmp = (y * -4.0) * ((z * z) - t)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 3.8e+222)
		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 * x) <= 3.8e+222)
		tmp = (y * -4.0) * ((z * z) - t);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 3.8e+222], 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 (*.f64 x x) < 3.80000000000000018e222

   1. Initial program 93.2%

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

   2. Taylor expanded in x around 0 76.6%

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

      1. associate-*r* 76.6%

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

      2. unpow2 76.6%

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

      3. *-commutative 76.6%

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

      4. *-commutative 76.6%

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

   4. Simplified 76.6%

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

   if 3.80000000000000018e222 < (*.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 86.4%

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

      1. unpow2 86.4%

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

   4. Simplified 86.4%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 3.8 \cdot 10^{+222}:\\ \;\;\;\;\left(y \cdot -4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 8: 59.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4.8 \cdot 10^{+78}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(y \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 4.8e+78) (* x x) (* (* z z) (* y -4.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 4.8e+78) {
		tmp = x * x;
	} else {
		tmp = (z * z) * (y * -4.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z * z) <= 4.8d+78) then
        tmp = x * x
    else
        tmp = (z * z) * (y * (-4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 4.8e+78) {
		tmp = x * x;
	} else {
		tmp = (z * z) * (y * -4.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 4.8e+78:
		tmp = x * x
	else:
		tmp = (z * z) * (y * -4.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 4.8e+78)
		tmp = Float64(x * x);
	else
		tmp = Float64(Float64(z * z) * Float64(y * -4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 4.8e+78)
		tmp = x * x;
	else
		tmp = (z * z) * (y * -4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 4.8e+78], N[(x * x), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.7999999999999997e78

   1. Initial program 97.9%

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

   2. Taylor expanded in x around inf 55.9%

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

      1. unpow2 55.9%

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

   4. Simplified 55.9%

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

   if 4.7999999999999997e78 < (*.f64 z z)

   1. Initial program 81.3%

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

   2. Taylor expanded in z around inf 71.4%

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

      1. *-commutative 71.4%

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

      2. unpow2 71.4%

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

      3. *-commutative 71.4%

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

      4. associate-*l* 71.4%

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

   4. Simplified 71.4%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4.8 \cdot 10^{+78}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(y \cdot -4\right)\\ \end{array} \]

Alternative 9: 74.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 3.2 \cdot 10^{+39}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -4\right) \cdot \left(z \cdot z - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 3.2e+39) (- (* x x) (* y (* t -4.0))) (* (* y -4.0) (- (* z z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3.2e+39) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = (y * -4.0) * ((z * z) - t);
	}
	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 <= 3.2d+39) then
        tmp = (x * x) - (y * (t * (-4.0d0)))
    else
        tmp = (y * (-4.0d0)) * ((z * z) - t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3.2e+39) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = (y * -4.0) * ((z * z) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 3.2e+39:
		tmp = (x * x) - (y * (t * -4.0))
	else:
		tmp = (y * -4.0) * ((z * z) - t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 3.2e+39)
		tmp = Float64(Float64(x * x) - Float64(y * Float64(t * -4.0)));
	else
		tmp = Float64(Float64(y * -4.0) * Float64(Float64(z * z) - t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 3.2e+39)
		tmp = (x * x) - (y * (t * -4.0));
	else
		tmp = (y * -4.0) * ((z * z) - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 3.2e+39], N[(N[(x * x), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * -4.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.19999999999999993e39

   1. Initial program 92.3%

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

   2. Taylor expanded in z around 0 75.1%

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

      1. associate-*r* 75.1%

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

   4. Simplified 75.1%

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

   if 3.19999999999999993e39 < z

   1. Initial program 85.5%

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

   2. Taylor expanded in x around 0 74.6%

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

      1. associate-*r* 74.6%

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

      2. unpow2 74.6%

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

      3. *-commutative 74.6%

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

      4. *-commutative 74.6%

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

   4. Simplified 74.6%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.2 \cdot 10^{+39}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -4\right) \cdot \left(z \cdot z - t\right)\\ \end{array} \]

Alternative 10: 41.4% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

2. Taylor expanded in x around inf 42.2%

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

   1. unpow2 42.2%

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

4. Simplified 42.2%

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

5. Final simplification 42.2%

   \[\leadsto x \cdot x \]

Developer target: 90.8% 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 2023277 
(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))))