Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1

Percentage Accurate: 98.3% → 99.4%

Time: 2.3s

Alternatives: 5

Speedup: 1.0×

• 33.8% 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: 98.3% 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: 99.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. sub-neg 99.6%

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

   2. +-commutative 99.6%

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

   3. associate-*l* 99.6%

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

   4. distribute-rgt-neg-in 99.6%

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

   5. fma-def 100.0%

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

   6. *-commutative 100.0%

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

   7. distribute-rgt-neg-in 100.0%

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

   8. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 98.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. fma-neg 99.6%

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

   2. *-commutative 99.6%

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

   3. distribute-rgt-neg-in 99.6%

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

   4. distribute-rgt-neg-in 99.6%

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

   5. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 3: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.25 \cdot 10^{-89}:\\ \;\;\;\;-4 \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 1.25e-89) (* -4.0 (* y z)) (* x x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 1.25e-89) {
		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) <= 1.25d-89) 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) <= 1.25e-89) {
		tmp = -4.0 * (y * z);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 1.25e-89:
		tmp = -4.0 * (y * z)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 1.25e-89)
		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) <= 1.25e-89)
		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], 1.25e-89], N[(-4.0 * N[(y * z), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.24999999999999992e-89

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 90.0%

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

   if 1.24999999999999992e-89 < (*.f64 x x)

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf 80.3%

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

      1. unpow2 80.3%

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

   4. Simplified 80.3%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.25 \cdot 10^{-89}:\\ \;\;\;\;-4 \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 4: 98.3% 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 99.6%

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

2. Final simplification 99.6%

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

Alternative 5: 53.8% 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 99.6%

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

2. Taylor expanded in x around inf 53.9%

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

   1. unpow2 53.9%

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

4. Simplified 53.9%

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

5. Final simplification 53.9%

   \[\leadsto x \cdot x \]

Reproduce

herbie shell --seed 2023257 
(FPCore (x y z)
  :name "Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1"
  :precision binary64
  (- (* x x) (* (* y 4.0) z)))