Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1

Percentage Accurate: 98.4% → 99.3%

Time: 5.3s

Alternatives: 5

Speedup: 0.4×

• 34.7% 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: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot x - z \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{+303}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\frac{x \cdot x}{y} - z \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* x x) (* z (* y 4.0)))))
   (if (<= t_0 4e+303) t_0 (* y (- (/ (* x x) y) (* z 4.0))))))

C

double code(double x, double y, double z) {
	double t_0 = (x * x) - (z * (y * 4.0));
	double tmp;
	if (t_0 <= 4e+303) {
		tmp = t_0;
	} else {
		tmp = y * (((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) :: t_0
    real(8) :: tmp
    t_0 = (x * x) - (z * (y * 4.0d0))
    if (t_0 <= 4d+303) then
        tmp = t_0
    else
        tmp = y * (((x * x) / y) - (z * 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x * x) - (z * (y * 4.0));
	double tmp;
	if (t_0 <= 4e+303) {
		tmp = t_0;
	} else {
		tmp = y * (((x * x) / y) - (z * 4.0));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x * x) - (z * (y * 4.0))
	tmp = 0
	if t_0 <= 4e+303:
		tmp = t_0
	else:
		tmp = y * (((x * x) / y) - (z * 4.0))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * x) - Float64(z * Float64(y * 4.0)))
	tmp = 0.0
	if (t_0 <= 4e+303)
		tmp = t_0;
	else
		tmp = Float64(y * Float64(Float64(Float64(x * x) / y) - Float64(z * 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x * x) - (z * (y * 4.0));
	tmp = 0.0;
	if (t_0 <= 4e+303)
		tmp = t_0;
	else
		tmp = y * (((x * x) / y) - (z * 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] - N[(z * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e+303], t$95$0, N[(y * N[(N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision] - N[(z * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) z)) < 4e303

   1. Initial program 100.0%

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

   if 4e303 < (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) z))

   1. Initial program 93.8%

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

      1. pow2 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x - z \cdot \left(y \cdot 4\right) \leq 4 \cdot 10^{+303}:\\ \;\;\;\;x \cdot x - z \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\frac{x \cdot x}{y} - z \cdot 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% 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. fmm-def 98.4%

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

   2. associate-*l* 98.1%

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

   3. *-commutative 98.1%

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

   4. distribute-rgt-neg-in 98.1%

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

   5. distribute-rgt-neg-in 98.1%

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

   6. metadata-eval 98.1%

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

3. Simplified 98.1%

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

Alternative 3: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot x - z \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* x x) (* z (* y 4.0)))))
   (if (<= t_0 INFINITY) t_0 (* 4.0 (* y z)))))

C

double code(double x, double y, double z) {
	double t_0 = (x * x) - (z * (y * 4.0));
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = 4.0 * (y * z);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = (x * x) - (z * (y * 4.0));
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else {
		tmp = 4.0 * (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x * x) - (z * (y * 4.0))
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0
	else:
		tmp = 4.0 * (y * z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * x) - Float64(z * Float64(y * 4.0)))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = Float64(4.0 * Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x * x) - (z * (y * 4.0));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = 4.0 * (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] - N[(z * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], t$95$0, N[(4.0 * N[(y * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) z)) < +inf.0

   1. Initial program 100.0%

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

   if +inf.0 < (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) z))

   1. Initial program 0.0%

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

   3. Taylor expanded in y around 0 0.0%

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

      1. rem-square-sqrt 0.0%

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

      2. fabs-sqr 0.0%

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

      3. rem-square-sqrt 0.0%

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

      4. fabs-neg 0.0%

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

      5. distribute-lft-neg-in 0.0%

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

      6. metadata-eval 0.0%

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

      7. *-commutative 0.0%

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

      8. *-commutative 0.0%

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

      9. associate-*r* 0.0%

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

      10. rem-square-sqrt 0.0%

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

      11. fabs-sqr 0.0%

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

      12. rem-square-sqrt 100.0%

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

      13. associate-*r* 100.0%

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

      14. *-commutative 100.0%

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

      15. associate-*r* 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 100.0%

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

4. Final simplification 100.0%

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

Alternative 4: 83.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5.4 \cdot 10^{+16}:\\ \;\;\;\;-4 \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - y \cdot \left(z \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 5.4e+16) (* -4.0 (* y z)) (- (* x x) (* y (* z -4.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 5.4e+16) {
		tmp = -4.0 * (y * z);
	} 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) <= 5.4d+16) then
        tmp = (-4.0d0) * (y * z)
    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) <= 5.4e+16) {
		tmp = -4.0 * (y * z);
	} else {
		tmp = (x * x) - (y * (z * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 5.4e+16:
		tmp = -4.0 * (y * z)
	else:
		tmp = (x * x) - (y * (z * -4.0))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 5.4e+16)
		tmp = Float64(-4.0 * Float64(y * z));
	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) <= 5.4e+16)
		tmp = -4.0 * (y * z);
	else
		tmp = (x * x) - (y * (z * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 5.4e+16], N[(-4.0 * N[(y * z), $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) < 5.4e16

   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 88.0%

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

   if 5.4e16 < (*.f64 x x)

   1. Initial program 96.1%

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

   3. Taylor expanded in y around 0 96.1%

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

      1. rem-square-sqrt 47.1%

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

      2. fabs-sqr 47.1%

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

      3. rem-square-sqrt 78.9%

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

      4. fabs-neg 78.9%

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

      5. distribute-lft-neg-in 78.9%

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

      6. metadata-eval 78.9%

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

      7. *-commutative 78.9%

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

      8. *-commutative 78.9%

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

      9. associate-*r* 78.9%

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

      10. rem-square-sqrt 47.1%

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

      11. fabs-sqr 47.1%

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

      12. rem-square-sqrt 77.9%

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

      13. associate-*r* 77.9%

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

      14. *-commutative 77.9%

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

      15. associate-*r* 77.9%

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

   5. Simplified 77.9%

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

Alternative 5: 53.0% 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 98.4%

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

3. Taylor expanded in x around 0 61.3%

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

Reproduce

herbie shell --seed 2024170 
(FPCore (x y z)
  :name "Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1"
  :precision binary64
  (- (* x x) (* (* y 4.0) z)))