Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1

Percentage Accurate: 98.1% → 98.6%

Time: 5.8s

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: 98.1% 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: 98.6% 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 99.2%

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

   1. fmm-def 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. Add Preprocessing

Alternative 2: 83.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 6.8 \cdot 10^{-46}:\\ \;\;\;\;z \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - y \cdot \left(z \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 6.8e-46) (* z (* y -4.0)) (- (* x x) (* y (* z -4.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 6.8e-46) {
		tmp = z * (y * -4.0);
	} else {
		tmp = (x * x) - (y * (z * -4.0));
	}
	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) <= 6.8d-46) then
        tmp = z * (y * (-4.0d0))
    else
        tmp = (x * x) - (y * (z * (-4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 6.8e-46) {
		tmp = z * (y * -4.0);
	} else {
		tmp = (x * x) - (y * (z * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 6.8e-46:
		tmp = z * (y * -4.0)
	else:
		tmp = (x * x) - (y * (z * -4.0))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 6.8e-46)
		tmp = Float64(z * Float64(y * -4.0));
	else
		tmp = Float64(Float64(x * x) - Float64(y * Float64(z * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 6.8e-46)
		tmp = z * (y * -4.0);
	else
		tmp = (x * x) - (y * (z * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 6.8e-46], N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(y * N[(z * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 6.79999999999999992e-46

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 100.0%

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

   4. Taylor expanded in y around inf 90.7%

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

      1. associate-*r* 90.7%

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

      2. *-commutative 90.7%

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

   6. Simplified 90.7%

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

   if 6.79999999999999992e-46 < (*.f64 x x)

   1. Initial program 98.5%

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

   3. Taylor expanded in y around 0 98.5%

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

      1. rem-square-sqrt 55.5%

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

      2. fabs-sqr 55.5%

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

      3. rem-square-sqrt 88.4%

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

      4. fabs-neg 88.4%

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

      5. distribute-lft-neg-in 88.4%

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

      6. metadata-eval 88.4%

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

      7. *-commutative 88.4%

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

      8. *-commutative 88.4%

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

      9. associate-*r* 88.4%

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

      10. rem-square-sqrt 44.7%

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

      11. fabs-sqr 44.7%

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

      12. rem-square-sqrt 80.4%

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

      13. associate-*r* 80.4%

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

      14. *-commutative 80.4%

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

      15. associate-*r* 80.4%

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

   5. Simplified 80.4%

      \[\leadsto x \cdot x - \color{blue}{y \cdot \left(z \cdot -4\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 6.8 \cdot 10^{-46}:\\ \;\;\;\;z \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - y \cdot \left(z \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% 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 99.2%

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

3. Final simplification 99.2%

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

Alternative 4: 52.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return 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 = z * (y * (-4.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in y around inf 94.1%

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

4. Taylor expanded in y around inf 53.5%

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

   1. associate-*r* 53.5%

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

   2. *-commutative 53.5%

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

6. Simplified 53.5%

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

7. Final simplification 53.5%

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

Alternative 5: 52.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return -4.0 * (y * 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 = (-4.0d0) * (y * z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in x around 0 53.5%

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

Reproduce

herbie shell --seed 2024180 
(FPCore (x y z)
  :name "Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1"
  :precision binary64
  (- (* x x) (* (* y 4.0) z)))