Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1

Percentage Accurate: 98.4% → 99.5%

Time: 2.6s

Alternatives: 5

Speedup: 0.7×

• 35.1% 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: 99.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

   1. sub-neg 97.7%

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

   2. +-commutative 97.7%

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

   3. associate-*l* 97.3%

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

   4. distribute-rgt-neg-in 97.3%

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

   5. fma-def 98.9%

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

   6. *-commutative 98.9%

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

   7. distribute-rgt-neg-in 98.9%

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

   8. metadata-eval 98.9%

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

3. Simplified 98.9%

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

4. Final simplification 98.9%

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

Alternative 2: 98.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.7%

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

   1. fma-neg 98.4%

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

   2. *-commutative 98.4%

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

   3. distribute-rgt-neg-in 98.4%

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

   4. distribute-rgt-neg-in 98.4%

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

   5. metadata-eval 98.4%

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

3. Simplified 98.4%

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

4. Final simplification 98.4%

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

Alternative 3: 83.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [y, z] = \mathsf{sort}([y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.3 \cdot 10^{-64} \lor \neg \left(x \cdot x \leq 140\right) \land x \cdot x \leq 2.2 \cdot 10^{+94}:\\ \;\;\;\;y \cdot \left(z \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

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) 1.3e-64)
         (and (not (<= (* x x) 140.0)) (<= (* x x) 2.2e+94)))
   (* y (* z -4.0))
   (* x x)))

C

assert(y < z);
double code(double x, double y, double z) {
	double tmp;
	if (((x * x) <= 1.3e-64) || (!((x * x) <= 140.0) && ((x * x) <= 2.2e+94))) {
		tmp = y * (z * -4.0);
	} 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) <= 1.3d-64) .or. (.not. ((x * x) <= 140.0d0)) .and. ((x * x) <= 2.2d+94)) then
        tmp = y * (z * (-4.0d0))
    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) <= 1.3e-64) || (!((x * x) <= 140.0) && ((x * x) <= 2.2e+94))) {
		tmp = y * (z * -4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

[y, z] = sort([y, z])
def code(x, y, z):
	tmp = 0
	if ((x * x) <= 1.3e-64) or (not ((x * x) <= 140.0) and ((x * x) <= 2.2e+94)):
		tmp = y * (z * -4.0)
	else:
		tmp = x * x
	return tmp

Julia

y, z = sort([y, z])
function code(x, y, z)
	tmp = 0.0
	if ((Float64(x * x) <= 1.3e-64) || (!(Float64(x * x) <= 140.0) && (Float64(x * x) <= 2.2e+94)))
		tmp = Float64(y * Float64(z * -4.0));
	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) <= 1.3e-64) || (~(((x * x) <= 140.0)) && ((x * x) <= 2.2e+94)))
		tmp = y * (z * -4.0);
	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], 1.3e-64], And[N[Not[LessEqual[N[(x * x), $MachinePrecision], 140.0]], $MachinePrecision], LessEqual[N[(x * x), $MachinePrecision], 2.2e+94]]], N[(y * N[(z * -4.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.3e-64 or 140 < (*.f64 x x) < 2.20000000000000012e94

   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 87.5%

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

      1. associate-*r* 87.5%

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

      2. *-commutative 87.5%

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

      3. associate-*l* 87.5%

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

   6. Simplified 87.5%

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

   if 1.3e-64 < (*.f64 x x) < 140 or 2.20000000000000012e94 < (*.f64 x x)

   1. Initial program 95.6%

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

      1. associate-*l* 94.9%

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

   3. Simplified 94.9%

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

   4. Taylor expanded in x around inf 89.4%

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

      1. unpow2 89.4%

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

   6. Simplified 89.4%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.3 \cdot 10^{-64} \lor \neg \left(x \cdot x \leq 140\right) \land x \cdot x \leq 2.2 \cdot 10^{+94}:\\ \;\;\;\;y \cdot \left(z \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 4: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [y, z] = \mathsf{sort}([y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.6 \cdot 10^{+294}:\\ \;\;\;\;x \cdot x - y \cdot \left(z \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: y and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 1.6e+294) (- (* x x) (* y (* z 4.0))) (* x x)))

C

assert(y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 1.6e+294) {
		tmp = (x * x) - (y * (z * 4.0));
	} 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) <= 1.6d+294) then
        tmp = (x * x) - (y * (z * 4.0d0))
    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) <= 1.6e+294) {
		tmp = (x * x) - (y * (z * 4.0));
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

[y, z] = sort([y, z])
def code(x, y, z):
	tmp = 0
	if (x * x) <= 1.6e+294:
		tmp = (x * x) - (y * (z * 4.0))
	else:
		tmp = x * x
	return tmp

Julia

y, z = sort([y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 1.6e+294)
		tmp = Float64(Float64(x * x) - Float64(y * Float64(z * 4.0)));
	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) <= 1.6e+294)
		tmp = (x * x) - (y * (z * 4.0));
	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], 1.6e+294], N[(N[(x * x), $MachinePrecision] - N[(y * N[(z * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.5999999999999999e294

   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%

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

   if 1.5999999999999999e294 < (*.f64 x x)

   1. Initial program 92.7%

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

      1. associate-*l* 91.6%

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

   3. Simplified 91.6%

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

   4. Taylor expanded in x around inf 97.6%

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

      1. unpow2 97.6%

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

   6. Simplified 97.6%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.6 \cdot 10^{+294}:\\ \;\;\;\;x \cdot x - y \cdot \left(z \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 5: 53.5% 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 97.7%

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

   1. associate-*l* 97.3%

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

3. Simplified 97.3%

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

4. Taylor expanded in x around inf 56.1%

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

   1. unpow2 56.1%

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

6. Simplified 56.1%

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

7. Final simplification 56.1%

   \[\leadsto x \cdot x \]

Reproduce

herbie shell --seed 2023230 
(FPCore (x y z)
  :name "Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1"
  :precision binary64
  (- (* x x) (* (* y 4.0) z)))