Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1

Percentage Accurate: 98.4% → 98.3%

Time: 3.2s

Alternatives: 4

Speedup: 0.6×

• 34.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \left(y \cdot 4\right) \cdot z\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-166}:\\ \;\;\;\;x \cdot x - t\_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{x \cdot x}{z} - y \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* y 4.0) z)))
   (if (<= t_0 5e-166) (- (* x x) t_0) (* z (- (/ (* x x) z) (* y 4.0))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = (y * 4.0) * z;
	double tmp;
	if (t_0 <= 5e-166) {
		tmp = (x * x) - t_0;
	} else {
		tmp = z * (((x * x) / z) - (y * 4.0));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y * 4.0d0) * z
    if (t_0 <= 5d-166) then
        tmp = (x * x) - t_0
    else
        tmp = z * (((x * x) / z) - (y * 4.0d0))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double t_0 = (y * 4.0) * z;
	double tmp;
	if (t_0 <= 5e-166) {
		tmp = (x * x) - t_0;
	} else {
		tmp = z * (((x * x) / z) - (y * 4.0));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	t_0 = (y * 4.0) * z
	tmp = 0
	if t_0 <= 5e-166:
		tmp = (x * x) - t_0
	else:
		tmp = z * (((x * x) / z) - (y * 4.0))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(Float64(y * 4.0) * z)
	tmp = 0.0
	if (t_0 <= 5e-166)
		tmp = Float64(Float64(x * x) - t_0);
	else
		tmp = Float64(z * Float64(Float64(Float64(x * x) / z) - Float64(y * 4.0)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	t_0 = (y * 4.0) * z;
	tmp = 0.0;
	if (t_0 <= 5e-166)
		tmp = (x * x) - t_0;
	else
		tmp = z * (((x * x) / z) - (y * 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * 4.0), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-166], N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision], N[(z * N[(N[(N[(x * x), $MachinePrecision] / z), $MachinePrecision] - N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) z) < 5e-166

   1. Initial program 100.0%

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

   if 5e-166 < (*.f64 (*.f64 y #s(literal 4 binary64)) z)

   1. Initial program 94.5%

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

   3. Taylor expanded in z around inf 100.0%

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

      1. unpow2 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot 4\right) \cdot z \leq 5 \cdot 10^{-166}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{x \cdot x}{z} - y \cdot 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2.1 \cdot 10^{+277}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 2.1e+277) (- (* x x) (* (* y 4.0) z)) (* x x)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 2.1e+277) {
		tmp = (x * x) - ((y * 4.0) * z);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x * x) <= 2.1d+277) then
        tmp = (x * x) - ((y * 4.0d0) * z)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 2.1e+277) {
		tmp = (x * x) - ((y * 4.0) * z);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if (x * x) <= 2.1e+277:
		tmp = (x * x) - ((y * 4.0) * z)
	else:
		tmp = x * x
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 2.1e+277)
		tmp = Float64(Float64(x * x) - Float64(Float64(y * 4.0) * z));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 2.1e+277)
		tmp = (x * x) - ((y * 4.0) * z);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 2.1e+277], N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.09999999999999999e277

   1. Initial program 100.0%

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

   if 2.09999999999999999e277 < (*.f64 x x)

   1. Initial program 94.0%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. unpow2 96.8%

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

   5. Applied egg-rr 100.0%

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

Alternative 3: 85.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 7.5 \cdot 10^{-36}:\\ \;\;\;\;-4 \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 7.5e-36) (* -4.0 (* y z)) (* x x)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 7.5e-36) {
		tmp = -4.0 * (y * z);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x * x) <= 7.5d-36) then
        tmp = (-4.0d0) * (y * z)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 7.5e-36) {
		tmp = -4.0 * (y * z);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if (x * x) <= 7.5e-36:
		tmp = -4.0 * (y * z)
	else:
		tmp = x * x
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 7.5e-36)
		tmp = Float64(-4.0 * Float64(y * z));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 7.5e-36)
		tmp = -4.0 * (y * z);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 7.5e-36], N[(-4.0 * N[(y * z), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 7.49999999999999972e-36

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 86.2%

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

   if 7.49999999999999972e-36 < (*.f64 x x)

   1. Initial program 96.4%

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

   3. Taylor expanded in x around inf 88.9%

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

      1. unpow2 87.9%

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

   5. Applied egg-rr 88.9%

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

Alternative 4: 53.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ x \cdot x \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (* x x))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return x * x;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x * x
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	return x * x;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return x * x

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(x * x)
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = x * x;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(x * x), $MachinePrecision]

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in x around inf 58.1%

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

   1. unpow2 93.4%

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

5. Applied egg-rr 58.1%

   \[\leadsto \color{blue}{x \cdot x} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024100 
(FPCore (x y z)
  :name "Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1"
  :precision binary64
  (- (* x x) (* (* y 4.0) z)))