Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1

Percentage Accurate: 98.3% → 99.5%

Time: 3.1s

Alternatives: 6

Speedup: 1.0×

• 34.8% 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: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 5 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(x_m, x_m, \left(z \cdot -4\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;{x_m}^{2}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z)
 :precision binary64
 (if (<= x_m 5e+167) (fma x_m x_m (* (* z -4.0) y)) (pow x_m 2.0)))

C

x_m = fabs(x);
double code(double x_m, double y, double z) {
	double tmp;
	if (x_m <= 5e+167) {
		tmp = fma(x_m, x_m, ((z * -4.0) * y));
	} else {
		tmp = pow(x_m, 2.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y, z)
	tmp = 0.0
	if (x_m <= 5e+167)
		tmp = fma(x_m, x_m, Float64(Float64(z * -4.0) * y));
	else
		tmp = x_m ^ 2.0;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_] := If[LessEqual[x$95$m, 5e+167], N[(x$95$m * x$95$m + N[(N[(z * -4.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[Power[x$95$m, 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.9999999999999997e167

   1. Initial program 98.2%

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

      1. fma-neg 98.7%

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

      2. associate-*l* 98.7%

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

      3. *-commutative 98.7%

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

      4. distribute-rgt-neg-in 98.7%

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

      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%

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

   if 4.9999999999999997e167 < x

   1. Initial program 93.1%

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

   3. 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot -4\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z) :precision binary64 (fma (* z -4.0) y (pow x_m 2.0)))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_] := N[(N[(z * -4.0), $MachinePrecision] * y + N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.6%

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

   1. sub-neg 97.6%

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

   2. +-commutative 97.6%

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

   3. distribute-lft-neg-in 97.6%

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

   4. distribute-rgt-neg-in 97.6%

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

   5. metadata-eval 97.6%

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

   6. associate-*r* 97.6%

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

   7. *-commutative 97.6%

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

   8. *-commutative 97.6%

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

   9. fma-def 99.6%

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

   10. pow2 99.6%

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

4. Applied egg-rr 99.6%

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

5. Final simplification 99.6%

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

Alternative 3: 99.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 2.8 \cdot 10^{+138}:\\ \;\;\;\;x_m \cdot x_m - z \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;{x_m}^{2}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z)
 :precision binary64
 (if (<= x_m 2.8e+138) (- (* x_m x_m) (* z (* y 4.0))) (pow x_m 2.0)))

C

x_m = fabs(x);
double code(double x_m, double y, double z) {
	double tmp;
	if (x_m <= 2.8e+138) {
		tmp = (x_m * x_m) - (z * (y * 4.0));
	} else {
		tmp = pow(x_m, 2.0);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y, z)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 2.8d+138) then
        tmp = (x_m * x_m) - (z * (y * 4.0d0))
    else
        tmp = x_m ** 2.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y, double z) {
	double tmp;
	if (x_m <= 2.8e+138) {
		tmp = (x_m * x_m) - (z * (y * 4.0));
	} else {
		tmp = Math.pow(x_m, 2.0);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z):
	tmp = 0
	if x_m <= 2.8e+138:
		tmp = (x_m * x_m) - (z * (y * 4.0))
	else:
		tmp = math.pow(x_m, 2.0)
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z)
	tmp = 0.0
	if (x_m <= 2.8e+138)
		tmp = Float64(Float64(x_m * x_m) - Float64(z * Float64(y * 4.0)));
	else
		tmp = x_m ^ 2.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z)
	tmp = 0.0;
	if (x_m <= 2.8e+138)
		tmp = (x_m * x_m) - (z * (y * 4.0));
	else
		tmp = x_m ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_] := If[LessEqual[x$95$m, 2.8e+138], N[(N[(x$95$m * x$95$m), $MachinePrecision] - N[(z * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[x$95$m, 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.8000000000000001e138

   1. Initial program 98.6%

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

   if 2.8000000000000001e138 < x

   1. Initial program 91.2%

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

   3. Taylor expanded in x around inf 97.1%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{+138}:\\ \;\;\;\;x \cdot x - z \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z) :precision binary64 (- (* x_m x_m) (* z (* y 4.0))))

C

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

Fortran

x_m = abs(x)
real(8) function code(x_m, y, z)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x_m * x_m) - (z * (y * 4.0d0))
end function

Java

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

Python

x_m = math.fabs(x)
def code(x_m, y, z):
	return (x_m * x_m) - (z * (y * 4.0))

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_] := N[(N[(x$95$m * x$95$m), $MachinePrecision] - N[(z * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.6%

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

3. Final simplification 97.6%

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

Alternative 5: 53.6% accurate, 1.8× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z) :precision binary64 (* -4.0 (* z y)))

C

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

Fortran

x_m = abs(x)
real(8) function code(x_m, y, z)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (-4.0d0) * (z * y)
end function

Java

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

Python

x_m = math.fabs(x)
def code(x_m, y, z):
	return -4.0 * (z * y)

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_] := N[(-4.0 * N[(z * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in x around 0 49.2%

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

4. Final simplification 49.2%

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

Alternative 6: 53.6% accurate, 1.8× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z) :precision binary64 (* (* z -4.0) y))

C

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

Fortran

x_m = abs(x)
real(8) function code(x_m, y, z)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (z * (-4.0d0)) * y
end function

Java

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

Python

x_m = math.fabs(x)
def code(x_m, y, z):
	return (z * -4.0) * y

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_] := N[(N[(z * -4.0), $MachinePrecision] * y), $MachinePrecision]

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in x around 0 49.2%

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

   1. add-log-exp 22.9%

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

   2. *-un-lft-identity 22.9%

      \[\leadsto \log \color{blue}{\left(1 \cdot e^{-4 \cdot \left(y \cdot z\right)}\right)} \]

   3. log-prod 22.9%

      \[\leadsto \color{blue}{\log 1 + \log \left(e^{-4 \cdot \left(y \cdot z\right)}\right)} \]

   4. metadata-eval 22.9%

      \[\leadsto \color{blue}{0} + \log \left(e^{-4 \cdot \left(y \cdot z\right)}\right) \]

   5. add-log-exp 49.2%

      \[\leadsto 0 + \color{blue}{-4 \cdot \left(y \cdot z\right)} \]

   6. *-commutative 49.2%

      \[\leadsto 0 + \color{blue}{\left(y \cdot z\right) \cdot -4} \]

   7. associate-*l* 49.2%

      \[\leadsto 0 + \color{blue}{y \cdot \left(z \cdot -4\right)} \]

   8. *-commutative 49.2%

      \[\leadsto 0 + y \cdot \color{blue}{\left(-4 \cdot z\right)} \]

5. Applied egg-rr 49.2%

   \[\leadsto \color{blue}{0 + y \cdot \left(-4 \cdot z\right)} \]

6. Final simplification 49.2%

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

Reproduce

herbie shell --seed 2024011 
(FPCore (x y z)
  :name "Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1"
  :precision binary64
  (- (* x x) (* (* y 4.0) z)))