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

Percentage Accurate: 90.3% → 95.1%

Time: 8.3s

Alternatives: 9

Speedup: 0.8×

• 48.4% 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: 90.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+281}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\frac{-0.25}{y}}{z}}{z}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+281)
   (- (* x x) (* (* y 4.0) (- (* z z) t)))
   (pow (/ (/ (/ -0.25 y) z) z) -1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+281) {
		tmp = (x * x) - ((y * 4.0) * ((z * z) - t));
	} else {
		tmp = pow((((-0.25 / y) / z) / z), -1.0);
	}
	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) <= 2d+281) then
        tmp = (x * x) - ((y * 4.0d0) * ((z * z) - t))
    else
        tmp = ((((-0.25d0) / y) / z) / z) ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+281) {
		tmp = (x * x) - ((y * 4.0) * ((z * z) - t));
	} else {
		tmp = Math.pow((((-0.25 / y) / z) / z), -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 2e+281:
		tmp = (x * x) - ((y * 4.0) * ((z * z) - t))
	else:
		tmp = math.pow((((-0.25 / y) / z) / z), -1.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+281)
		tmp = Float64(Float64(x * x) - Float64(Float64(y * 4.0) * Float64(Float64(z * z) - t)));
	else
		tmp = Float64(Float64(Float64(-0.25 / y) / z) / z) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 2e+281)
		tmp = (x * x) - ((y * 4.0) * ((z * z) - t));
	else
		tmp = (((-0.25 / y) / z) / z) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+281], N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[(-0.25 / y), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000001e281

   1. Initial program 98.8%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   if 2.0000000000000001e281 < (*.f64 z z)

   1. Initial program 67.1%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 67.1

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

   4. Applied rewrites 67.1%

      \[\leadsto \color{blue}{\frac{1}{{\left(\mathsf{fma}\left(\left(z \cdot z - t\right) \cdot y, -4, x \cdot x\right)\right)}^{-1}}} \]

   5. Taylor expanded in z around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{4}}{y \cdot {z}^{2}}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{4}}{y \cdot {z}^{2}}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\frac{-1}{4}}{\color{blue}{{z}^{2} \cdot y}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\frac{-1}{4}}{\color{blue}{{z}^{2} \cdot y}}} \]

      4. unpow2 N/A

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

      5. lower-*.f64 75.7

         \[\leadsto \frac{1}{\frac{-0.25}{\color{blue}{\left(z \cdot z\right)} \cdot y}} \]

   7. Applied rewrites 75.7%

      \[\leadsto \frac{1}{\color{blue}{\frac{-0.25}{\left(z \cdot z\right) \cdot y}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 91.5%

      \[\leadsto \frac{1}{\frac{-0.25}{\left(z \cdot y\right) \cdot \color{blue}{z}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 91.5%

       \[\leadsto \frac{1}{\frac{\frac{\frac{-0.25}{y}}{z}}{\color{blue}{z}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+281}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\frac{-0.25}{y}}{z}}{z}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+281}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{-0.25}{\left(z \cdot y\right) \cdot z}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+281)
   (- (* x x) (* (* y 4.0) (- (* z z) t)))
   (pow (/ -0.25 (* (* z y) z)) -1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+281) {
		tmp = (x * x) - ((y * 4.0) * ((z * z) - t));
	} else {
		tmp = pow((-0.25 / ((z * y) * z)), -1.0);
	}
	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) <= 2d+281) then
        tmp = (x * x) - ((y * 4.0d0) * ((z * z) - t))
    else
        tmp = ((-0.25d0) / ((z * y) * z)) ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+281) {
		tmp = (x * x) - ((y * 4.0) * ((z * z) - t));
	} else {
		tmp = Math.pow((-0.25 / ((z * y) * z)), -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 2e+281:
		tmp = (x * x) - ((y * 4.0) * ((z * z) - t))
	else:
		tmp = math.pow((-0.25 / ((z * y) * z)), -1.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+281)
		tmp = Float64(Float64(x * x) - Float64(Float64(y * 4.0) * Float64(Float64(z * z) - t)));
	else
		tmp = Float64(-0.25 / Float64(Float64(z * y) * z)) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 2e+281)
		tmp = (x * x) - ((y * 4.0) * ((z * z) - t));
	else
		tmp = (-0.25 / ((z * y) * z)) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+281], N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(-0.25 / N[(N[(z * y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000001e281

   1. Initial program 98.8%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   if 2.0000000000000001e281 < (*.f64 z z)

   1. Initial program 67.1%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 67.1

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

   4. Applied rewrites 67.1%

      \[\leadsto \color{blue}{\frac{1}{{\left(\mathsf{fma}\left(\left(z \cdot z - t\right) \cdot y, -4, x \cdot x\right)\right)}^{-1}}} \]

   5. Taylor expanded in z around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{4}}{y \cdot {z}^{2}}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{4}}{y \cdot {z}^{2}}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\frac{-1}{4}}{\color{blue}{{z}^{2} \cdot y}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\frac{-1}{4}}{\color{blue}{{z}^{2} \cdot y}}} \]

      4. unpow2 N/A

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

      5. lower-*.f64 75.7

         \[\leadsto \frac{1}{\frac{-0.25}{\color{blue}{\left(z \cdot z\right)} \cdot y}} \]

   7. Applied rewrites 75.7%

      \[\leadsto \frac{1}{\color{blue}{\frac{-0.25}{\left(z \cdot z\right) \cdot y}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 91.5%

      \[\leadsto \frac{1}{\frac{-0.25}{\left(z \cdot y\right) \cdot \color{blue}{z}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+281}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{-0.25}{\left(z \cdot y\right) \cdot z}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot z - t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+25}:\\ \;\;\;\;\left(t \cdot 4\right) \cdot y\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+100}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z \cdot z\right) \cdot y\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z z) t)))
   (if (<= t_1 -5e+25)
     (* (* t 4.0) y)
     (if (<= t_1 2e+100) (* 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+25) {
		tmp = (t * 4.0) * y;
	} else if (t_1 <= 2e+100) {
		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) :: t_1
    real(8) :: tmp
    t_1 = (z * z) - t
    if (t_1 <= (-5d+25)) then
        tmp = (t * 4.0d0) * y
    else if (t_1 <= 2d+100) 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 t_1 = (z * z) - t;
	double tmp;
	if (t_1 <= -5e+25) {
		tmp = (t * 4.0) * y;
	} else if (t_1 <= 2e+100) {
		tmp = x * x;
	} else {
		tmp = ((z * z) * y) * -4.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) - t
	tmp = 0
	if t_1 <= -5e+25:
		tmp = (t * 4.0) * y
	elif t_1 <= 2e+100:
		tmp = x * x
	else:
		tmp = ((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+25)
		tmp = Float64(Float64(t * 4.0) * y);
	elseif (t_1 <= 2e+100)
		tmp = Float64(x * x);
	else
		tmp = Float64(Float64(Float64(z * z) * y) * -4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * z) - t;
	tmp = 0.0;
	if (t_1 <= -5e+25)
		tmp = (t * 4.0) * y;
	elseif (t_1 <= 2e+100)
		tmp = x * x;
	else
		tmp = ((z * z) * y) * -4.0;
	end
	tmp_2 = 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+25], N[(N[(t * 4.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, 2e+100], N[(x * x), $MachinePrecision], N[(N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision] * -4.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 z z) t) < -5.00000000000000024e25

   1. Initial program 100.0%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 80.0

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

   5. Applied rewrites 80.0%

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

   7. Applied rewrites 80.0%

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

   if -5.00000000000000024e25 < (-.f64 (*.f64 z z) t) < 2.00000000000000003e100

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. neg-mul-1 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-neg-in N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 99.9

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right)}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right) \cdot \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right)}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right) \cdot \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right)}}} \]

      5. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right) \cdot \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right)}{x \cdot x + \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right)}}}} \]

      6. flip-- N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot x - \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right)}}} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot x - \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right)}}} \]

      8. lower-/.f64 99.8

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{x \cdot x - \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right)}}} \]

   6. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(t \cdot y, 4, x \cdot x\right) - \left(z \cdot 4\right) \cdot \left(z \cdot y\right)}}} \]

   7. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 70.0

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

   9. Applied rewrites 70.0%

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

   if 2.00000000000000003e100 < (-.f64 (*.f64 z z) t)

   1. Initial program 81.3%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 60.7

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

   5. Applied rewrites 60.7%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z - t \leq -5 \cdot 10^{+25}:\\ \;\;\;\;\left(t \cdot 4\right) \cdot y\\ \mathbf{elif}\;z \cdot z - t \leq 2 \cdot 10^{+100}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z \cdot z\right) \cdot y\right) \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+281}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, z \cdot -4, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+281)
   (- (* x x) (* (* y 4.0) (- (* z z) t)))
   (fma (* z y) (* z -4.0) (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+281) {
		tmp = (x * x) - ((y * 4.0) * ((z * z) - t));
	} else {
		tmp = fma((z * y), (z * -4.0), (x * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+281)
		tmp = Float64(Float64(x * x) - Float64(Float64(y * 4.0) * Float64(Float64(z * z) - t)));
	else
		tmp = fma(Float64(z * y), Float64(z * -4.0), Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+281], N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * y), $MachinePrecision] * N[(z * -4.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000001e281

   1. Initial program 98.8%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   if 2.0000000000000001e281 < (*.f64 z z)

   1. Initial program 67.1%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 68.6

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

   5. Applied rewrites 68.6%

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

   7. Applied rewrites 91.3%

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

Alternative 5: 90.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-44}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot y, 4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, z \cdot -4, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e-44)
   (fma (* t y) 4.0 (* x x))
   (fma (* z y) (* z -4.0) (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e-44) {
		tmp = fma((t * y), 4.0, (x * x));
	} else {
		tmp = fma((z * y), (z * -4.0), (x * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e-44)
		tmp = fma(Float64(t * y), 4.0, Float64(x * x));
	else
		tmp = fma(Float64(z * y), Float64(z * -4.0), Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e-44], N[(N[(t * y), $MachinePrecision] * 4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(z * y), $MachinePrecision] * N[(z * -4.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.99999999999999953e-45

   1. Initial program 100.0%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{{x}^{2} + \left(\mathsf{neg}\left(-4\right)\right) \cdot \left(t \cdot y\right)} \]

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot y, 4, {x}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot y}, 4, {x}^{2}\right) \]

      7. unpow2 N/A

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

      8. lower-*.f64 97.4

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

   5. Applied rewrites 97.4%

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

   if 9.99999999999999953e-45 < (*.f64 z z)

   1. Initial program 83.2%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 76.0

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

   5. Applied rewrites 76.0%

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

   7. Applied rewrites 86.0%

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

Alternative 6: 83.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5.4 \cdot 10^{+107}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot y, 4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z \cdot z - t\right) \cdot y\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5.4e+107)
   (fma (* t y) 4.0 (* x x))
   (* (* (- (* z z) t) y) -4.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5.4e+107) {
		tmp = fma((t * y), 4.0, (x * x));
	} else {
		tmp = (((z * z) - t) * y) * -4.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5.4e+107)
		tmp = fma(Float64(t * y), 4.0, Float64(x * x));
	else
		tmp = Float64(Float64(Float64(Float64(z * z) - t) * y) * -4.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5.4e+107], N[(N[(t * y), $MachinePrecision] * 4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.4000000000000003e107

   1. Initial program 99.9%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{{x}^{2} + \left(\mathsf{neg}\left(-4\right)\right) \cdot \left(t \cdot y\right)} \]

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot y, 4, {x}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot y}, 4, {x}^{2}\right) \]

      7. unpow2 N/A

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

      8. lower-*.f64 92.3

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

   5. Applied rewrites 92.3%

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

   if 5.4000000000000003e107 < (*.f64 z z)

   1. Initial program 79.1%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 74.3

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

   5. Applied rewrites 74.3%

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

Alternative 7: 82.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1.3 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot y, 4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z \cdot z\right) \cdot y\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1.3e+132) (fma (* t y) 4.0 (* x x)) (* (* (* z z) y) -4.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1.3e+132) {
		tmp = fma((t * y), 4.0, (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) <= 1.3e+132)
		tmp = fma(Float64(t * y), 4.0, Float64(x * x));
	else
		tmp = Float64(Float64(Float64(z * z) * y) * -4.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1.3e+132], N[(N[(t * y), $MachinePrecision] * 4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.3e132

   1. Initial program 99.3%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{{x}^{2} + \left(\mathsf{neg}\left(-4\right)\right) \cdot \left(t \cdot y\right)} \]

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot y, 4, {x}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot y}, 4, {x}^{2}\right) \]

      7. unpow2 N/A

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

      8. lower-*.f64 89.2

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

   5. Applied rewrites 89.2%

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

   if 1.3e132 < (*.f64 z z)

   1. Initial program 77.1%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 70.0

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

   5. Applied rewrites 70.0%

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

Alternative 8: 59.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 8 \cdot 10^{-13}:\\ \;\;\;\;\left(t \cdot 4\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 8e-13) (* (* t 4.0) y) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 8e-13) {
		tmp = (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) <= 8d-13) then
        tmp = (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) <= 8e-13) {
		tmp = (t * 4.0) * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 8e-13:
		tmp = (t * 4.0) * y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 8e-13)
		tmp = Float64(Float64(t * 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) <= 8e-13)
		tmp = (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], 8e-13], N[(N[(t * 4.0), $MachinePrecision] * y), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 8.0000000000000002e-13

   1. Initial program 93.4%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 47.2

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

   5. Applied rewrites 47.2%

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

   7. Applied rewrites 47.2%

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

   if 8.0000000000000002e-13 < (*.f64 x x)

   1. Initial program 86.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. neg-mul-1 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-neg-in N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 89.7

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 89.7

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

   4. Applied rewrites 89.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right)}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right) \cdot \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right)}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right) \cdot \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right)}}} \]

      5. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right) \cdot \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right)}{x \cdot x + \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right)}}}} \]

      6. flip-- N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot x - \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right)}}} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot x - \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right)}}} \]

      8. lower-/.f64 89.7

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{x \cdot x - \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right)}}} \]

   6. Applied rewrites 86.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(t \cdot y, 4, x \cdot x\right) - \left(z \cdot 4\right) \cdot \left(z \cdot y\right)}}} \]

   7. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 70.3

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

   9. Applied rewrites 70.3%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 8 \cdot 10^{-13}:\\ \;\;\;\;\left(t \cdot 4\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 41.4% accurate, 4.5× 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 90.2%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. neg-mul-1 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. neg-mul-1 N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. distribute-lft-neg-in N/A

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

   14. lower-*.f64 N/A

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

   15. metadata-eval N/A

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

   16. lift-*.f64 N/A

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

   17. associate-*r* N/A

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

   18. lower-*.f64 N/A

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

   19. lower-*.f64 94.5

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

   20. lift-*.f64 N/A

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

   21. *-commutative N/A

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

   22. lower-*.f64 94.5

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

4. Applied rewrites 94.5%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right)}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right) \cdot \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right)}}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right)}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right) \cdot \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right)}}} \]

   5. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right) \cdot \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right)}{x \cdot x + \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right)}}}} \]

   6. flip-- N/A

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot x - \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right)}}} \]

   7. lift--.f64 N/A

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot x - \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right)}}} \]

   8. lower-/.f64 94.3

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x \cdot x - \mathsf{fma}\left(-4 \cdot y, t, \left(\left(4 \cdot y\right) \cdot z\right) \cdot z\right)}}} \]

6. Applied rewrites 89.6%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(t \cdot y, 4, x \cdot x\right) - \left(z \cdot 4\right) \cdot \left(z \cdot y\right)}}} \]

7. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. lower-*.f64 40.2

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

9. Applied rewrites 40.2%

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

10. Final simplification 40.2%

    \[\leadsto x \cdot x \]
11. Add Preprocessing

Developer Target 1: 90.3% 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 2024313 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* x x) (* 4 (* y (- (* z z) t)))))

  (- (* x x) (* (* y 4.0) (- (* z z) t))))