Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1

Percentage Accurate: 98.5% → 99.0%

Time: 3.3s

Alternatives: 4

Speedup: 1.0×

• 34.0% 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: 98.5% 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: 99.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

   1. associate-*l* 98.0%

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

   2. *-commutative 98.0%

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

   3. *-commutative 98.0%

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

   4. fma-neg 99.2%

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

   5. *-commutative 99.2%

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

   6. distribute-rgt-neg-in 99.2%

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

   7. distribute-rgt-neg-in 99.2%

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

   8. metadata-eval 99.2%

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

3. Simplified 99.2%

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

5. Final simplification 99.2%

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

Alternative 2: 83.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.8 \cdot 10^{-113} \lor \neg \left(x \cdot x \leq 9 \cdot 10^{-50}\right) \land x \cdot x \leq 280000000000:\\ \;\;\;\;y \cdot \left(z \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* x x) 1.8e-113)
         (and (not (<= (* x x) 9e-50)) (<= (* x x) 280000000000.0)))
   (* y (* z -4.0))
   (* x x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x * x) <= 1.8e-113) || (!((x * x) <= 9e-50) && ((x * x) <= 280000000000.0))) {
		tmp = y * (z * -4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

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) <= 1.8d-113) .or. (.not. ((x * x) <= 9d-50)) .and. ((x * x) <= 280000000000.0d0)) then
        tmp = y * (z * (-4.0d0))
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (((x * x) <= 1.8e-113) || (!((x * x) <= 9e-50) && ((x * x) <= 280000000000.0))) {
		tmp = y * (z * -4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((x * x) <= 1.8e-113) or (not ((x * x) <= 9e-50) and ((x * x) <= 280000000000.0)):
		tmp = y * (z * -4.0)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(x * x) <= 1.8e-113) || (!(Float64(x * x) <= 9e-50) && (Float64(x * x) <= 280000000000.0)))
		tmp = Float64(y * Float64(z * -4.0));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((x * x) <= 1.8e-113) || (~(((x * x) <= 9e-50)) && ((x * x) <= 280000000000.0)))
		tmp = y * (z * -4.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(x * x), $MachinePrecision], 1.8e-113], And[N[Not[LessEqual[N[(x * x), $MachinePrecision], 9e-50]], $MachinePrecision], LessEqual[N[(x * x), $MachinePrecision], 280000000000.0]]], N[(y * N[(z * -4.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.79999999999999987e-113 or 8.99999999999999924e-50 < (*.f64 x x) < 2.8e11

   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 91.5%

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

      1. *-commutative 91.5%

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

      2. associate-*r* 91.5%

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

      3. *-commutative 91.5%

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

   5. Simplified 91.5%

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

   if 1.79999999999999987e-113 < (*.f64 x x) < 8.99999999999999924e-50 or 2.8e11 < (*.f64 x x)

   1. Initial program 96.3%

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

   3. Taylor expanded in x around inf 81.4%

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

      1. unpow2 81.5%

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

   5. Applied egg-rr 81.5%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.8 \cdot 10^{-113} \lor \neg \left(x \cdot x \leq 9 \cdot 10^{-50}\right) \land x \cdot x \leq 280000000000:\\ \;\;\;\;y \cdot \left(z \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

3. Final simplification 98.0%

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

Alternative 4: 52.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ x \cdot x \end{array} \]

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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 50.4%

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

   1. unpow2 50.4%

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

5. Applied egg-rr 50.4%

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

6. Final simplification 50.4%

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

Reproduce

herbie shell --seed 2024046 
(FPCore (x y z)
  :name "Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1"
  :precision binary64
  (- (* x x) (* (* y 4.0) z)))