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

Percentage Accurate: 90.9% → 96.8%

Time: 18.5s

Alternatives: 10

Speedup: 0.7×

• 49.2% 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: 96.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+263}:\\ \;\;\;\;\mathsf{fma}\left(\left(z \cdot z - t\right) \cdot -4, y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+263)
   (fma (* (- (* z z) t) -4.0) y (* x x))
   (- (* x x) (* z (* z (* y 4.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+263) {
		tmp = fma((((z * z) - t) * -4.0), y, (x * x));
	} else {
		tmp = (x * x) - (z * (z * (y * 4.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+263)
		tmp = fma(Float64(Float64(Float64(z * z) - t) * -4.0), y, Float64(x * x));
	else
		tmp = Float64(Float64(x * x) - Float64(z * Float64(z * Float64(y * 4.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+263], N[(N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * -4.0), $MachinePrecision] * y + N[(x * x), $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) < 1.00000000000000002e263

   1. Initial program 97.7%

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

      1. sub-neg 97.7%

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

      2. +-commutative 97.7%

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

      3. *-commutative 97.7%

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

      4. distribute-rgt-neg-in 97.7%

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

      5. *-commutative 97.7%

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

      6. distribute-lft-neg-in 97.7%

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

      7. metadata-eval 97.7%

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

      8. associate-*r* 97.7%

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

      9. fma-def 99.4%

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

   3. Applied egg-rr 99.4%

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

   if 1.00000000000000002e263 < (*.f64 z z)

   1. Initial program 67.3%

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

   2. Taylor expanded in z around inf 67.3%

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

      1. *-commutative 67.3%

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

      2. unpow2 67.3%

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

      3. *-commutative 67.3%

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

      4. associate-*r* 67.3%

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

      5. associate-*l* 93.2%

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

      6. *-commutative 93.2%

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

   4. Simplified 93.2%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+263}:\\ \;\;\;\;\mathsf{fma}\left(\left(z \cdot z - t\right) \cdot -4, y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \]

Alternative 2: 96.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot z - t\\ \mathbf{if}\;t_1 \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\mathsf{fma}\left(x, x, t_1 \cdot \left(-4 \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z z) t)))
   (if (<= t_1 5e+305)
     (fma x x (* t_1 (* -4.0 y)))
     (- (* x x) (* z (* z (* y 4.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) - t;
	double tmp;
	if (t_1 <= 5e+305) {
		tmp = fma(x, x, (t_1 * (-4.0 * y)));
	} else {
		tmp = (x * x) - (z * (z * (y * 4.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) - t)
	tmp = 0.0
	if (t_1 <= 5e+305)
		tmp = fma(x, x, Float64(t_1 * Float64(-4.0 * y)));
	else
		tmp = Float64(Float64(x * x) - Float64(z * Float64(z * Float64(y * 4.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+305], N[(x * x + N[(t$95$1 * N[(-4.0 * y), $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 (*.f64 z z) t) < 5.00000000000000009e305

   1. Initial program 98.3%

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

      1. fma-neg 98.8%

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

      2. *-commutative 98.8%

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

      3. distribute-rgt-neg-in 98.8%

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

      4. distribute-rgt-neg-in 98.8%

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

      5. metadata-eval 98.8%

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

   3. Simplified 98.8%

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

   if 5.00000000000000009e305 < (-.f64 (*.f64 z z) t)

   1. Initial program 60.8%

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

   2. Taylor expanded in z around inf 60.8%

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

      1. *-commutative 60.8%

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

      2. unpow2 60.8%

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

      3. *-commutative 60.8%

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

      4. associate-*r* 60.8%

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

      5. associate-*l* 90.6%

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

      6. *-commutative 90.6%

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

   4. Simplified 90.6%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z - t \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(-4 \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \]

Alternative 3: 84.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot x - t \cdot \left(-4 \cdot y\right)\\ \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+230}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(-4 \cdot y\right)\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+265}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(z \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x x) (* t (* -4.0 y)))))
   (if (<= (* z z) 5e-8)
     t_1
     (if (<= (* z z) 2e+230)
       (* (- (* z z) t) (* -4.0 y))
       (if (<= (* z z) 5e+265) t_1 (* z (* y (* z -4.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) - (t * (-4.0 * y));
	double tmp;
	if ((z * z) <= 5e-8) {
		tmp = t_1;
	} else if ((z * z) <= 2e+230) {
		tmp = ((z * z) - t) * (-4.0 * y);
	} else if ((z * z) <= 5e+265) {
		tmp = t_1;
	} else {
		tmp = z * (y * (z * -4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * x) - (t * ((-4.0d0) * y))
    if ((z * z) <= 5d-8) then
        tmp = t_1
    else if ((z * z) <= 2d+230) then
        tmp = ((z * z) - t) * ((-4.0d0) * y)
    else if ((z * z) <= 5d+265) then
        tmp = t_1
    else
        tmp = z * (y * (z * (-4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * x) - (t * (-4.0 * y));
	double tmp;
	if ((z * z) <= 5e-8) {
		tmp = t_1;
	} else if ((z * z) <= 2e+230) {
		tmp = ((z * z) - t) * (-4.0 * y);
	} else if ((z * z) <= 5e+265) {
		tmp = t_1;
	} else {
		tmp = z * (y * (z * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * x) - (t * (-4.0 * y))
	tmp = 0
	if (z * z) <= 5e-8:
		tmp = t_1
	elif (z * z) <= 2e+230:
		tmp = ((z * z) - t) * (-4.0 * y)
	elif (z * z) <= 5e+265:
		tmp = t_1
	else:
		tmp = z * (y * (z * -4.0))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) - Float64(t * Float64(-4.0 * y)))
	tmp = 0.0
	if (Float64(z * z) <= 5e-8)
		tmp = t_1;
	elseif (Float64(z * z) <= 2e+230)
		tmp = Float64(Float64(Float64(z * z) - t) * Float64(-4.0 * y));
	elseif (Float64(z * z) <= 5e+265)
		tmp = t_1;
	else
		tmp = Float64(z * Float64(y * Float64(z * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * x) - (t * (-4.0 * y));
	tmp = 0.0;
	if ((z * z) <= 5e-8)
		tmp = t_1;
	elseif ((z * z) <= 2e+230)
		tmp = ((z * z) - t) * (-4.0 * y);
	elseif ((z * z) <= 5e+265)
		tmp = t_1;
	else
		tmp = z * (y * (z * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] - N[(t * N[(-4.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * z), $MachinePrecision], 5e-8], t$95$1, If[LessEqual[N[(z * z), $MachinePrecision], 2e+230], N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(-4.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 5e+265], t$95$1, N[(z * N[(y * N[(z * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 4.9999999999999998e-8 or 2.0000000000000002e230 < (*.f64 z z) < 5.0000000000000002e265

   1. Initial program 97.6%

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

   2. Taylor expanded in z around 0 94.3%

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

      1. associate-*r* 94.3%

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

   4. Simplified 94.3%

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

   if 4.9999999999999998e-8 < (*.f64 z z) < 2.0000000000000002e230

   1. Initial program 98.0%

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

   2. Taylor expanded in x around 0 76.6%

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

      1. *-commutative 76.6%

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

      2. *-commutative 76.6%

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

      3. unpow2 76.6%

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

      4. *-commutative 76.6%

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

      5. associate-*l* 76.6%

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

   4. Simplified 76.6%

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

   if 5.0000000000000002e265 < (*.f64 z z)

   1. Initial program 66.9%

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

   2. Taylor expanded in x around 0 71.1%

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

      1. *-commutative 71.1%

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

      2. *-commutative 71.1%

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

      3. unpow2 71.1%

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

      4. *-commutative 71.1%

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

      5. associate-*l* 71.1%

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

   4. Simplified 71.1%

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

      1. add-cbrt-cube 64.6%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right) \cdot \left(\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\right) \cdot \left(\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)}} \]

      2. pow3 64.6%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)}^{3}}} \]

      3. *-commutative 64.6%

         \[\leadsto \sqrt[3]{{\left(\left(z \cdot z - t\right) \cdot \color{blue}{\left(-4 \cdot y\right)}\right)}^{3}} \]

   6. Applied egg-rr 64.6%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\left(z \cdot z - t\right) \cdot \left(-4 \cdot y\right)\right)}^{3}}} \]

   7. Taylor expanded in z around inf 64.6%

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

      1. unpow2 64.6%

         \[\leadsto \sqrt[3]{{\left(\color{blue}{\left(z \cdot z\right)} \cdot \left(-4 \cdot y\right)\right)}^{3}} \]

   9. Simplified 64.6%

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

       1. rem-cbrt-cube 71.1%

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

       2. add-sqr-sqrt 45.9%

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

       3. pow2 45.9%

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

       4. sqrt-unprod 37.4%

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

       5. *-commutative 37.4%

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

       6. *-commutative 37.4%

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

       7. swap-sqr 37.4%

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

       8. metadata-eval 37.4%

          \[\leadsto {z}^{2} \cdot \sqrt{\left(y \cdot y\right) \cdot \color{blue}{16}} \]

       9. metadata-eval 37.4%

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

       10. swap-sqr 37.4%

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

       11. sqrt-unprod 5.9%

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

       12. pow2 5.9%

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

       13. *-commutative 5.9%

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

       14. sqrt-prod 5.9%

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

       15. metadata-eval 5.9%

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

       16. unpow-prod-down 0.7%

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

       17. unpow2 0.7%

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

       18. *-commutative 0.7%

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

       19. associate-*r* 0.7%

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

   11. Applied egg-rr 86.5%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x \cdot x - t \cdot \left(-4 \cdot y\right)\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+230}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(-4 \cdot y\right)\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+265}:\\ \;\;\;\;x \cdot x - t \cdot \left(-4 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(z \cdot -4\right)\right)\\ \end{array} \]

Alternative 4: 96.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z - t \leq 5 \cdot 10^{+305}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- (* z z) t) 5e+305)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (- (* x x) (* z (* z (* y 4.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) - t) <= 5e+305) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (x * x) - (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) - t) <= 5d+305) 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

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) - t) <= 5e+305) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (x * x) - (z * (z * (y * 4.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * z) - t) <= 5e+305:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = (x * x) - (z * (z * (y * 4.0)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(z * z) - t) <= 5e+305)
		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

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * z) - t) <= 5e+305)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = (x * x) - (z * (z * (y * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision], 5e+305], 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 (-.f64 (*.f64 z z) t) < 5.00000000000000009e305

   1. Initial program 98.3%

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

   if 5.00000000000000009e305 < (-.f64 (*.f64 z z) t)

   1. Initial program 60.8%

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

   2. Taylor expanded in z around inf 60.8%

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

      1. *-commutative 60.8%

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

      2. unpow2 60.8%

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

      3. *-commutative 60.8%

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

      4. associate-*r* 60.8%

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

      5. associate-*l* 90.6%

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

      6. *-commutative 90.6%

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

   4. Simplified 90.6%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z - t \leq 5 \cdot 10^{+305}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \]

Alternative 5: 89.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-106}:\\ \;\;\;\;x \cdot x - t \cdot \left(-4 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e-106)
   (- (* x x) (* t (* -4.0 y)))
   (- (* x x) (* z (* z (* y 4.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e-106) {
		tmp = (x * x) - (t * (-4.0 * y));
	} else {
		tmp = (x * x) - (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) <= 5d-106) then
        tmp = (x * x) - (t * ((-4.0d0) * y))
    else
        tmp = (x * x) - (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) <= 5e-106) {
		tmp = (x * x) - (t * (-4.0 * y));
	} else {
		tmp = (x * x) - (z * (z * (y * 4.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 5e-106:
		tmp = (x * x) - (t * (-4.0 * y))
	else:
		tmp = (x * x) - (z * (z * (y * 4.0)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e-106)
		tmp = Float64(Float64(x * x) - Float64(t * Float64(-4.0 * y)));
	else
		tmp = Float64(Float64(x * x) - Float64(z * Float64(z * Float64(y * 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 5e-106)
		tmp = (x * x) - (t * (-4.0 * y));
	else
		tmp = (x * x) - (z * (z * (y * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e-106], N[(N[(x * x), $MachinePrecision] - N[(t * N[(-4.0 * y), $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) < 4.99999999999999983e-106

   1. Initial program 99.0%

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

   2. Taylor expanded in z around 0 98.5%

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

      1. associate-*r* 98.5%

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

   4. Simplified 98.5%

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

   if 4.99999999999999983e-106 < (*.f64 z z)

   1. Initial program 82.7%

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

   2. Taylor expanded in z around inf 73.0%

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

      1. *-commutative 73.0%

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

      2. unpow2 73.0%

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

      3. *-commutative 73.0%

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

      4. associate-*r* 73.0%

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

      5. associate-*l* 85.1%

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

      6. *-commutative 85.1%

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

   4. Simplified 85.1%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-106}:\\ \;\;\;\;x \cdot x - t \cdot \left(-4 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \]

Alternative 6: 57.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 8 \cdot 10^{-109}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(-4 \cdot y\right)\\ \mathbf{elif}\;x \cdot x \leq 1.75 \cdot 10^{+67}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 8e-109)
   (* (* z z) (* -4.0 y))
   (if (<= (* x x) 1.75e+67) (* y (* t 4.0)) (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 8e-109) {
		tmp = (z * z) * (-4.0 * y);
	} else if ((x * x) <= 1.75e+67) {
		tmp = y * (t * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x * x) <= 8d-109) then
        tmp = (z * z) * ((-4.0d0) * y)
    else if ((x * x) <= 1.75d+67) then
        tmp = y * (t * 4.0d0)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 8e-109) {
		tmp = (z * z) * (-4.0 * y);
	} else if ((x * x) <= 1.75e+67) {
		tmp = y * (t * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 8e-109:
		tmp = (z * z) * (-4.0 * y)
	elif (x * x) <= 1.75e+67:
		tmp = y * (t * 4.0)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 8e-109)
		tmp = Float64(Float64(z * z) * Float64(-4.0 * y));
	elseif (Float64(x * x) <= 1.75e+67)
		tmp = Float64(y * Float64(t * 4.0));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 8e-109)
		tmp = (z * z) * (-4.0 * y);
	elseif ((x * x) <= 1.75e+67)
		tmp = y * (t * 4.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 8e-109], N[(N[(z * z), $MachinePrecision] * N[(-4.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 1.75e+67], N[(y * N[(t * 4.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 7.9999999999999999e-109

   1. Initial program 95.1%

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

   2. Taylor expanded in z around inf 53.0%

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

      1. *-commutative 53.0%

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

      2. unpow2 53.0%

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

      3. *-commutative 53.0%

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

      4. associate-*l* 53.0%

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

   4. Simplified 53.0%

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

   if 7.9999999999999999e-109 < (*.f64 x x) < 1.75e67

   1. Initial program 88.8%

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

   2. Taylor expanded in t around inf 44.6%

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

      1. *-commutative 44.6%

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

      2. associate-*r* 44.6%

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

      3. *-commutative 44.6%

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

   4. Simplified 44.6%

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

   if 1.75e67 < (*.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 77.0%

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

      1. unpow2 77.0%

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

   4. Simplified 77.0%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 8 \cdot 10^{-109}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(-4 \cdot y\right)\\ \mathbf{elif}\;x \cdot x \leq 1.75 \cdot 10^{+67}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 7: 60.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 6.1 \cdot 10^{-21}:\\ \;\;\;\;z \cdot \left(y \cdot \left(z \cdot -4\right)\right)\\ \mathbf{elif}\;x \cdot x \leq 7.8 \cdot 10^{+66}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 6.1e-21)
   (* z (* y (* z -4.0)))
   (if (<= (* x x) 7.8e+66) (* y (* t 4.0)) (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 6.1e-21) {
		tmp = z * (y * (z * -4.0));
	} else if ((x * x) <= 7.8e+66) {
		tmp = y * (t * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x * x) <= 6.1d-21) then
        tmp = z * (y * (z * (-4.0d0)))
    else if ((x * x) <= 7.8d+66) then
        tmp = y * (t * 4.0d0)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 6.1e-21) {
		tmp = z * (y * (z * -4.0));
	} else if ((x * x) <= 7.8e+66) {
		tmp = y * (t * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 6.1e-21:
		tmp = z * (y * (z * -4.0))
	elif (x * x) <= 7.8e+66:
		tmp = y * (t * 4.0)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 6.1e-21)
		tmp = Float64(z * Float64(y * Float64(z * -4.0)));
	elseif (Float64(x * x) <= 7.8e+66)
		tmp = Float64(y * Float64(t * 4.0));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 6.1e-21)
		tmp = z * (y * (z * -4.0));
	elseif ((x * x) <= 7.8e+66)
		tmp = y * (t * 4.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 6.1e-21], N[(z * N[(y * N[(z * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 7.8e+66], N[(y * N[(t * 4.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 6.10000000000000013e-21

   1. Initial program 93.0%

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

   2. Taylor expanded in x around 0 83.2%

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

      1. *-commutative 83.2%

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

      2. *-commutative 83.2%

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

      3. unpow2 83.2%

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

      4. *-commutative 83.2%

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

      5. associate-*l* 83.2%

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

   4. Simplified 83.2%

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

      1. add-cbrt-cube 55.0%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right) \cdot \left(\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\right) \cdot \left(\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)}} \]

      2. pow3 55.0%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)}^{3}}} \]

      3. *-commutative 55.0%

         \[\leadsto \sqrt[3]{{\left(\left(z \cdot z - t\right) \cdot \color{blue}{\left(-4 \cdot y\right)}\right)}^{3}} \]

   6. Applied egg-rr 55.0%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\left(z \cdot z - t\right) \cdot \left(-4 \cdot y\right)\right)}^{3}}} \]

   7. Taylor expanded in z around inf 37.6%

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

      1. unpow2 37.6%

         \[\leadsto \sqrt[3]{{\left(\color{blue}{\left(z \cdot z\right)} \cdot \left(-4 \cdot y\right)\right)}^{3}} \]

   9. Simplified 37.6%

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

       1. rem-cbrt-cube 49.3%

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

       2. add-sqr-sqrt 32.1%

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

       3. pow2 32.1%

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

       4. sqrt-unprod 26.1%

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

       5. *-commutative 26.1%

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

       6. *-commutative 26.1%

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

       7. swap-sqr 26.1%

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

       8. metadata-eval 26.1%

          \[\leadsto {z}^{2} \cdot \sqrt{\left(y \cdot y\right) \cdot \color{blue}{16}} \]

       9. metadata-eval 26.1%

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

       10. swap-sqr 26.1%

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

       11. sqrt-unprod 2.7%

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

       12. pow2 2.7%

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

       13. *-commutative 2.7%

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

       14. sqrt-prod 2.7%

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

       15. metadata-eval 2.7%

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

       16. unpow-prod-down 2.7%

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

       17. unpow2 2.7%

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

       18. *-commutative 2.7%

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

       19. associate-*r* 2.7%

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

   11. Applied egg-rr 56.2%

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

   if 6.10000000000000013e-21 < (*.f64 x x) < 7.8000000000000007e66

   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 62.1%

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

      1. *-commutative 62.1%

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

      2. associate-*r* 62.1%

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

      3. *-commutative 62.1%

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

   4. Simplified 62.1%

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

   if 7.8000000000000007e66 < (*.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 77.0%

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

      1. unpow2 77.0%

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

   4. Simplified 77.0%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 6.1 \cdot 10^{-21}:\\ \;\;\;\;z \cdot \left(y \cdot \left(z \cdot -4\right)\right)\\ \mathbf{elif}\;x \cdot x \leq 7.8 \cdot 10^{+66}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 8: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5.5 \cdot 10^{+71}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(-4 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 5.5e+71) (* (- (* z z) t) (* -4.0 y)) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 5.5e+71) {
		tmp = ((z * z) - t) * (-4.0 * y);
	} 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) <= 5.5d+71) then
        tmp = ((z * z) - t) * ((-4.0d0) * y)
    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) <= 5.5e+71) {
		tmp = ((z * z) - t) * (-4.0 * y);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 5.5e+71:
		tmp = ((z * z) - t) * (-4.0 * y)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 5.5e+71)
		tmp = Float64(Float64(Float64(z * z) - t) * Float64(-4.0 * y));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 5.5e+71)
		tmp = ((z * z) - t) * (-4.0 * y);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 5.5e+71], N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(-4.0 * y), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5.5e71

   1. Initial program 93.7%

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

   2. Taylor expanded in x around 0 82.7%

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

      1. *-commutative 82.7%

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

      2. *-commutative 82.7%

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

      3. unpow2 82.7%

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

      4. *-commutative 82.7%

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

      5. associate-*l* 82.7%

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

   4. Simplified 82.7%

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

   if 5.5e71 < (*.f64 x x)

   1. Initial program 82.0%

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

   2. Taylor expanded in x around inf 77.7%

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

      1. unpow2 77.7%

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

   4. Simplified 77.7%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5.5 \cdot 10^{+71}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(-4 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 9: 58.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5.8 \cdot 10^{+66}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 5.8e+66) (* y (* t 4.0)) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 5.8e+66) {
		tmp = y * (t * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x * x) <= 5.8d+66) then
        tmp = y * (t * 4.0d0)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 5.8e+66) {
		tmp = y * (t * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 5.8e+66:
		tmp = y * (t * 4.0)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 5.8e+66)
		tmp = Float64(y * Float64(t * 4.0));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 5.8e+66)
		tmp = y * (t * 4.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 5.8e+66], N[(y * N[(t * 4.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5.79999999999999972e66

   1. Initial program 93.7%

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

   2. Taylor expanded in t around inf 43.0%

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

      1. *-commutative 43.0%

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

      2. associate-*r* 43.0%

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

      3. *-commutative 43.0%

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

   4. Simplified 43.0%

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

   if 5.79999999999999972e66 < (*.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 77.0%

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

      1. unpow2 77.0%

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

   4. Simplified 77.0%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5.8 \cdot 10^{+66}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 10: 40.5% 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 88.9%

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

2. Taylor expanded in x around inf 41.7%

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

   1. unpow2 41.7%

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

4. Simplified 41.7%

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

5. Final simplification 41.7%

   \[\leadsto x \cdot x \]

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 2023279 
(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))))