Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1

Percentage Accurate: 98.4% → 98.8%

Time: 4.1s

Alternatives: 5

Speedup: 0.1×

• 34.7% 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: 98.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} [y, z] = \mathsf{sort}([y, z])\\ \\ \mathsf{fma}\left(x, x, y \cdot \left(z \cdot -4\right)\right) \end{array} \]

FPCore

NOTE: y and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (fma x x (* y (* z -4.0))))

C

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

Julia

y, z = sort([y, z])
function code(x, y, z)
	return fma(x, x, Float64(y * Float64(z * -4.0)))
end

Wolfram

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

Derivation

1. Initial program 98.1%

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

   1. fma-neg 98.8%

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

   2. associate-*l* 98.8%

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

   3. *-commutative 98.8%

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

   4. distribute-rgt-neg-in 98.8%

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

   5. distribute-rgt-neg-in 98.8%

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

   6. metadata-eval 98.8%

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

3. Simplified 98.8%

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

4. Final simplification 98.8%

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

Alternative 2: 83.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [y, z] = \mathsf{sort}([y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4.3 \cdot 10^{-203} \lor \neg \left(x \cdot x \leq 7 \cdot 10^{-191}\right) \land x \cdot x \leq 5 \cdot 10^{-132}:\\ \;\;\;\;-4 \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: y and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= (* x x) 4.3e-203)
         (and (not (<= (* x x) 7e-191)) (<= (* x x) 5e-132)))
   (* -4.0 (* y z))
   (* x x)))

C

assert(y < z);
double code(double x, double y, double z) {
	double tmp;
	if (((x * x) <= 4.3e-203) || (!((x * x) <= 7e-191) && ((x * x) <= 5e-132))) {
		tmp = -4.0 * (y * z);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

NOTE: 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) <= 4.3d-203) .or. (.not. ((x * x) <= 7d-191)) .and. ((x * x) <= 5d-132)) then
        tmp = (-4.0d0) * (y * z)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

assert y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (((x * x) <= 4.3e-203) || (!((x * x) <= 7e-191) && ((x * x) <= 5e-132))) {
		tmp = -4.0 * (y * z);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

[y, z] = sort([y, z])
def code(x, y, z):
	tmp = 0
	if ((x * x) <= 4.3e-203) or (not ((x * x) <= 7e-191) and ((x * x) <= 5e-132)):
		tmp = -4.0 * (y * z)
	else:
		tmp = x * x
	return tmp

Julia

y, z = sort([y, z])
function code(x, y, z)
	tmp = 0.0
	if ((Float64(x * x) <= 4.3e-203) || (!(Float64(x * x) <= 7e-191) && (Float64(x * x) <= 5e-132)))
		tmp = Float64(-4.0 * Float64(y * z));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

y, z = num2cell(sort([y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((x * x) <= 4.3e-203) || (~(((x * x) <= 7e-191)) && ((x * x) <= 5e-132)))
		tmp = -4.0 * (y * z);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[Or[LessEqual[N[(x * x), $MachinePrecision], 4.3e-203], And[N[Not[LessEqual[N[(x * x), $MachinePrecision], 7e-191]], $MachinePrecision], LessEqual[N[(x * x), $MachinePrecision], 5e-132]]], N[(-4.0 * N[(y * z), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.30000000000000027e-203 or 7.00000000000000013e-191 < (*.f64 x x) < 4.9999999999999999e-132

   1. Initial program 99.0%

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

      1. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 97.0%

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

   if 4.30000000000000027e-203 < (*.f64 x x) < 7.00000000000000013e-191 or 4.9999999999999999e-132 < (*.f64 x x)

   1. Initial program 97.5%

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

      1. associate-*l* 96.9%

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

   3. Simplified 96.9%

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

   4. Taylor expanded in x around inf 79.9%

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

      1. unpow2 79.9%

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

   6. Simplified 79.9%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4.3 \cdot 10^{-203} \lor \neg \left(x \cdot x \leq 7 \cdot 10^{-191}\right) \land x \cdot x \leq 5 \cdot 10^{-132}:\\ \;\;\;\;-4 \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 3: 83.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [y, z] = \mathsf{sort}([y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4.3 \cdot 10^{-203}:\\ \;\;\;\;y \cdot \left(z \cdot -4\right)\\ \mathbf{elif}\;x \cdot x \leq 7 \cdot 10^{-191}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \cdot x \leq 4.8 \cdot 10^{-132}:\\ \;\;\;\;-4 \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: y and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 4.3e-203)
   (* y (* z -4.0))
   (if (<= (* x x) 7e-191)
     (* x x)
     (if (<= (* x x) 4.8e-132) (* -4.0 (* y z)) (* x x)))))

C

assert(y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 4.3e-203) {
		tmp = y * (z * -4.0);
	} else if ((x * x) <= 7e-191) {
		tmp = x * x;
	} else if ((x * x) <= 4.8e-132) {
		tmp = -4.0 * (y * z);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

NOTE: 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) <= 4.3d-203) then
        tmp = y * (z * (-4.0d0))
    else if ((x * x) <= 7d-191) then
        tmp = x * x
    else if ((x * x) <= 4.8d-132) then
        tmp = (-4.0d0) * (y * z)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

assert y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 4.3e-203) {
		tmp = y * (z * -4.0);
	} else if ((x * x) <= 7e-191) {
		tmp = x * x;
	} else if ((x * x) <= 4.8e-132) {
		tmp = -4.0 * (y * z);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

[y, z] = sort([y, z])
def code(x, y, z):
	tmp = 0
	if (x * x) <= 4.3e-203:
		tmp = y * (z * -4.0)
	elif (x * x) <= 7e-191:
		tmp = x * x
	elif (x * x) <= 4.8e-132:
		tmp = -4.0 * (y * z)
	else:
		tmp = x * x
	return tmp

Julia

y, z = sort([y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 4.3e-203)
		tmp = Float64(y * Float64(z * -4.0));
	elseif (Float64(x * x) <= 7e-191)
		tmp = Float64(x * x);
	elseif (Float64(x * x) <= 4.8e-132)
		tmp = Float64(-4.0 * Float64(y * z));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

y, z = num2cell(sort([y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 4.3e-203)
		tmp = y * (z * -4.0);
	elseif ((x * x) <= 7e-191)
		tmp = x * x;
	elseif ((x * x) <= 4.8e-132)
		tmp = -4.0 * (y * z);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 4.3e-203], N[(y * N[(z * -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 7e-191], N[(x * x), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 4.8e-132], N[(-4.0 * N[(y * z), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 4.30000000000000027e-203

   1. Initial program 98.9%

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

      1. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 97.8%

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

      1. expm1-log1p-u 71.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-4 \cdot \left(y \cdot z\right)\right)\right)} \]

      2. expm1-udef 39.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-4 \cdot \left(y \cdot z\right)\right)} - 1} \]

      3. log1p-udef 39.6%

         \[\leadsto e^{\color{blue}{\log \left(1 + -4 \cdot \left(y \cdot z\right)\right)}} - 1 \]

      4. add-exp-log 65.5%

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

   6. Applied egg-rr 65.5%

      \[\leadsto \color{blue}{\left(1 + -4 \cdot \left(y \cdot z\right)\right) - 1} \]
   7. Step-by-step derivation

      1. +-commutative 65.5%

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

      2. associate--l+ 97.8%

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

      3. *-commutative 97.8%

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

      4. metadata-eval 97.8%

         \[\leadsto \left(y \cdot z\right) \cdot -4 + \color{blue}{0} \]

      5. associate-*l* 97.9%

         \[\leadsto \color{blue}{y \cdot \left(z \cdot -4\right)} + 0 \]

   8. Simplified 97.9%

      \[\leadsto \color{blue}{y \cdot \left(z \cdot -4\right) + 0} \]

   if 4.30000000000000027e-203 < (*.f64 x x) < 7.00000000000000013e-191 or 4.80000000000000031e-132 < (*.f64 x x)

   1. Initial program 97.5%

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

      1. associate-*l* 96.9%

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

   3. Simplified 96.9%

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

   4. Taylor expanded in x around inf 79.9%

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

      1. unpow2 79.9%

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

   6. Simplified 79.9%

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

   if 7.00000000000000013e-191 < (*.f64 x x) < 4.80000000000000031e-132

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 83.9%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4.3 \cdot 10^{-203}:\\ \;\;\;\;y \cdot \left(z \cdot -4\right)\\ \mathbf{elif}\;x \cdot x \leq 7 \cdot 10^{-191}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \cdot x \leq 4.8 \cdot 10^{-132}:\\ \;\;\;\;-4 \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 4: 98.4% accurate, 1.0× speedup

Math

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

FPCore

NOTE: y and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (- (* x x) (* y (* z 4.0))))

C

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

Fortran

NOTE: 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) - (y * (z * 4.0d0))
end function

Java

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

Python

[y, z] = sort([y, z])
def code(x, y, z):
	return (x * x) - (y * (z * 4.0))

Julia

y, z = sort([y, z])
function code(x, y, z)
	return Float64(Float64(x * x) - Float64(y * Float64(z * 4.0)))
end

MATLAB

y, z = num2cell(sort([y, z])){:}
function tmp = code(x, y, z)
	tmp = (x * x) - (y * (z * 4.0));
end

Wolfram

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

Derivation

1. Initial program 98.1%

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

   1. associate-*l* 98.1%

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

3. Simplified 98.1%

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

4. Final simplification 98.1%

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

Alternative 5: 53.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: 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 98.1%

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

   1. associate-*l* 98.1%

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

3. Simplified 98.1%

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

4. Taylor expanded in x around inf 56.5%

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

   1. unpow2 56.5%

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

6. Simplified 56.5%

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

7. Final simplification 56.5%

   \[\leadsto x \cdot x \]

Reproduce

herbie shell --seed 2023279 
(FPCore (x y z)
  :name "Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1"
  :precision binary64
  (- (* x x) (* (* y 4.0) z)))