Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1

Percentage Accurate: 98.4% → 98.8%

Time: 3.9s

Alternatives: 5

Speedup: 1.0×

• 34.2% 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

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

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 98.4%

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

   1. fma-neg 99.2%

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

   2. associate-*l* 99.2%

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

   3. *-commutative 99.2%

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

   4. distribute-rgt-neg-in 99.2%

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

   5. distribute-rgt-neg-in 99.2%

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

   6. 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: 85.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.35 \cdot 10^{-14}:\\ \;\;\;\;-4 \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 1.35e-14) (* -4.0 (* y z)) (* x x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 1.35e-14) {
		tmp = -4.0 * (y * z);
	} 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) <= 1.35d-14) then
        tmp = (-4.0d0) * (y * z)
    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) <= 1.35e-14) {
		tmp = -4.0 * (y * z);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 1.35e-14:
		tmp = -4.0 * (y * z)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 1.35e-14)
		tmp = Float64(-4.0 * Float64(y * z));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 1.35e-14)
		tmp = -4.0 * (y * z);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 1.35e-14], N[(-4.0 * N[(y * z), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.3499999999999999e-14

   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 89.5%

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

   if 1.3499999999999999e-14 < (*.f64 x x)

   1. Initial program 97.1%

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

   3. Taylor expanded in x around inf 80.6%

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

      1. unpow2 80.6%

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

   5. Applied egg-rr 80.6%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.35 \cdot 10^{-14}:\\ \;\;\;\;-4 \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4.4 \cdot 10^{-14}:\\ \;\;\;\;y \cdot \left(z \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 4.4e-14) (* y (* z -4.0)) (* x x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 4.4e-14) {
		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) <= 4.4d-14) 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) <= 4.4e-14) {
		tmp = y * (z * -4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 4.4e-14:
		tmp = y * (z * -4.0)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 4.4e-14)
		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) <= 4.4e-14)
		tmp = y * (z * -4.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 4.4e-14], N[(y * N[(z * -4.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.4000000000000002e-14

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*r* 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. fma-def 100.0%

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

      10. pow2 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. unpow2 19.3%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in z around inf 89.5%

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

      1. associate-*r* 89.5%

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

      2. *-commutative 89.5%

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

      3. associate-*l* 89.5%

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

   9. Simplified 89.5%

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

   if 4.4000000000000002e-14 < (*.f64 x x)

   1. Initial program 97.1%

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

   3. Taylor expanded in x around inf 80.6%

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

      1. unpow2 80.6%

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

   5. Applied egg-rr 80.6%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4.4 \cdot 10^{-14}:\\ \;\;\;\;y \cdot \left(z \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% 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 98.4%

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

3. Final simplification 98.4%

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

Alternative 5: 53.6% 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 98.4%

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

3. Taylor expanded in x around inf 52.6%

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

   1. unpow2 52.6%

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

5. Applied egg-rr 52.6%

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

6. Final simplification 52.6%

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

Reproduce

herbie shell --seed 2024020 
(FPCore (x y z)
  :name "Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1"
  :precision binary64
  (- (* x x) (* (* y 4.0) z)))