Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1

Percentage Accurate: 98.6% → 99.5%

Time: 2.8s

Alternatives: 5

Speedup: 1.0×

• 34.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 98.6% 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

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot -4\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= x 1e+167) (fma x x (* (* z -4.0) y)) (pow x 2.0)))

C

x = abs(x);
double code(double x, double y, double z) {
	double tmp;
	if (x <= 1e+167) {
		tmp = fma(x, x, ((z * -4.0) * y));
	} else {
		tmp = pow(x, 2.0);
	}
	return tmp;
}

Julia

x = abs(x)
function code(x, y, z)
	tmp = 0.0
	if (x <= 1e+167)
		tmp = fma(x, x, Float64(Float64(z * -4.0) * y));
	else
		tmp = x ^ 2.0;
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[x, 1e+167], N[(x * x + N[(N[(z * -4.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[Power[x, 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1e167

   1. Initial program 97.4%

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

      1. fma-neg 98.3%

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

      2. associate-*l* 98.7%

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

      3. distribute-rgt-neg-in 98.7%

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

      4. *-commutative 98.7%

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

      5. distribute-rgt-neg-in 98.7%

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

      6. metadata-eval 98.7%

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

   3. Simplified 98.7%

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

   if 1e167 < x

   1. Initial program 92.3%

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

   2. Taylor expanded in x around inf 100.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot -4\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{2}\\ \end{array} \]

Alternative 2: 99.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \mathsf{fma}\left(z \cdot -4, y, {x}^{2}\right) \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z) :precision binary64 (fma (* z -4.0) y (pow x 2.0)))

C

x = abs(x);
double code(double x, double y, double z) {
	return fma((z * -4.0), y, pow(x, 2.0));
}

Julia

x = abs(x)
function code(x, y, z)
	return fma(Float64(z * -4.0), y, (x ^ 2.0))
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_] := N[(N[(z * -4.0), $MachinePrecision] * y + N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.9%

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

   1. sub-neg 96.9%

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

   2. +-commutative 96.9%

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

   3. distribute-lft-neg-in 96.9%

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

   4. distribute-rgt-neg-in 96.9%

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

   5. metadata-eval 96.9%

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

   6. associate-*r* 97.2%

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

   7. *-commutative 97.2%

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

   8. *-commutative 97.2%

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

   9. fma-def 99.2%

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

   10. pow2 99.2%

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

3. Applied egg-rr 99.2%

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

4. Final simplification 99.2%

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

Alternative 3: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+154}:\\ \;\;\;\;x \cdot x - z \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= x 1e+154) (- (* x x) (* z (* y 4.0))) (pow x 2.0)))

C

x = abs(x);
double code(double x, double y, double z) {
	double tmp;
	if (x <= 1e+154) {
		tmp = (x * x) - (z * (y * 4.0));
	} else {
		tmp = pow(x, 2.0);
	}
	return tmp;
}

Fortran

NOTE: x should be positive 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 <= 1d+154) then
        tmp = (x * x) - (z * (y * 4.0d0))
    else
        tmp = x ** 2.0d0
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 1e+154) {
		tmp = (x * x) - (z * (y * 4.0));
	} else {
		tmp = Math.pow(x, 2.0);
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y, z):
	tmp = 0
	if x <= 1e+154:
		tmp = (x * x) - (z * (y * 4.0))
	else:
		tmp = math.pow(x, 2.0)
	return tmp

Julia

x = abs(x)
function code(x, y, z)
	tmp = 0.0
	if (x <= 1e+154)
		tmp = Float64(Float64(x * x) - Float64(z * Float64(y * 4.0)));
	else
		tmp = x ^ 2.0;
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 1e+154)
		tmp = (x * x) - (z * (y * 4.0));
	else
		tmp = x ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[x, 1e+154], N[(N[(x * x), $MachinePrecision] - N[(z * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[x, 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000004e154

   1. Initial program 97.4%

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

   if 1.00000000000000004e154 < x

   1. Initial program 92.9%

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

   2. Taylor expanded in x around inf 100.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+154}:\\ \;\;\;\;x \cdot x - z \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{2}\\ \end{array} \]

Alternative 4: 98.6% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z) :precision binary64 (- (* x x) (* z (* y 4.0))))

C

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

Fortran

NOTE: x should be positive 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) - (z * (y * 4.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_] := N[(N[(x * x), $MachinePrecision] - N[(z * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.9%

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

2. Final simplification 96.9%

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

Alternative 5: 53.5% accurate, 1.8× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z) :precision binary64 (* -4.0 (* z y)))

C

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

Fortran

NOTE: x should be positive 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 = (-4.0d0) * (z * y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_] := N[(-4.0 * N[(z * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.9%

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

2. Taylor expanded in x around 0 51.6%

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

3. Final simplification 51.6%

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

Reproduce

herbie shell --seed 2023308 
(FPCore (x y z)
  :name "Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1"
  :precision binary64
  (- (* x x) (* (* y 4.0) z)))