Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1

Percentage Accurate: 98.3% → 98.7%

Time: 6.6s

Alternatives: 5

Speedup: 1.1×

• 35.4% 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: 98.3% 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: 98.7% accurate, 1.1× 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 98.8%

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

   1. sub-neg N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. *-commutative N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. *-lowering-*.f64 N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. *-lowering-*.f64 N/A

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

   8. metadata-eval 99.2

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

4. Applied egg-rr 99.2%

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

Alternative 2: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-47}:\\ \;\;\;\;y \cdot \left(z \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 1e-47) (* y (* z -4.0)) (* x x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 1e-47) {
		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) <= 1d-47) 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) <= 1e-47) {
		tmp = y * (z * -4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 1e-47:
		tmp = y * (z * -4.0)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 1e-47)
		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) <= 1e-47)
		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], 1e-47], N[(y * N[(z * -4.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 9.9999999999999997e-48

   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

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

      1. +-rgt-identity N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-lft-identity N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 87.2

         \[\leadsto \mathsf{fma}\left(-4, \color{blue}{\mathsf{fma}\left(y, z, 0\right)}, 0\right) \]

   5. Simplified 87.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4, \mathsf{fma}\left(y, z, 0\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. +-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 87.2

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

   7. Applied egg-rr 87.2%

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

   if 9.9999999999999997e-48 < (*.f64 x x)

   1. Initial program 97.9%

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

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{{x}^{2} + 0} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{x \cdot x} + 0 \]

      3. accelerator-lowering-fma.f64 86.4

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]

   5. Simplified 86.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 86.4

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

   7. Applied egg-rr 86.4%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-47}:\\ \;\;\;\;y \cdot \left(z \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.2% accurate, 3.2× 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 98.8%

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

3. Taylor expanded in x around inf

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

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{{x}^{2} + 0} \]

   2. unpow2 N/A

      \[\leadsto \color{blue}{x \cdot x} + 0 \]

   3. accelerator-lowering-fma.f64 57.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]

5. Simplified 57.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. *-lowering-*.f64 57.8

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

7. Applied egg-rr 57.8%

   \[\leadsto \color{blue}{x \cdot x} \]
8. Add Preprocessing

Alternative 4: 3.6% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 x)

C

double code(double x, double y, double z) {
	return 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
end function

Java

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

Python

def code(x, y, z):
	return x

Julia

function code(x, y, z)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in x around inf

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

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{{x}^{2} + 0} \]

   2. unpow2 N/A

      \[\leadsto \color{blue}{x \cdot x} + 0 \]

   3. accelerator-lowering-fma.f64 57.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]

5. Simplified 57.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. *-lowering-*.f64 57.8

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

7. Applied egg-rr 57.8%

   \[\leadsto \color{blue}{x \cdot x} \]
8. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto \color{blue}{{x}^{2}} \]

   2. pow-to-exp N/A

      \[\leadsto \color{blue}{e^{\log x \cdot 2}} \]

   3. *-commutative N/A

      \[\leadsto e^{\color{blue}{2 \cdot \log x}} \]

   4. count-2 N/A

      \[\leadsto e^{\color{blue}{\log x + \log x}} \]

   5. flip-+ N/A

      \[\leadsto e^{\color{blue}{\frac{\log x \cdot \log x - \log x \cdot \log x}{\log x - \log x}}} \]

   6. +-inverses N/A

      \[\leadsto e^{\frac{\color{blue}{0}}{\log x - \log x}} \]

   7. +-inverses N/A

      \[\leadsto e^{\frac{\color{blue}{\log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot \log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) - \log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot \log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)}}{\log x - \log x}} \]

   8. +-inverses N/A

      \[\leadsto e^{\frac{\log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot \log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) - \log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot \log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)}{\color{blue}{0}}} \]

   9. +-inverses N/A

      \[\leadsto e^{\frac{\log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot \log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) - \log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot \log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)}{\color{blue}{\log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) - \log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)}}} \]

   10. flip-+ N/A

       \[\leadsto e^{\color{blue}{\log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) + \log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)}} \]

   11. log-prod N/A

       \[\leadsto e^{\color{blue}{\log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot {x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)}} \]

   12. sqr-pow N/A

       \[\leadsto e^{\log \color{blue}{\left({x}^{\left(\frac{2}{2}\right)}\right)}} \]

   13. metadata-eval N/A

       \[\leadsto e^{\log \left({x}^{\color{blue}{1}}\right)} \]

   14. unpow1 N/A

       \[\leadsto e^{\log \color{blue}{x}} \]

   15. rem-exp-log 3.7

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

9. Applied egg-rr 3.7%

   \[\leadsto \color{blue}{x} \]
10. Add Preprocessing

Alternative 5: 3.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 1.0)

C

double code(double x, double y, double z) {
	return 1.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0
end function

Java

public static double code(double x, double y, double z) {
	return 1.0;
}

Python

def code(x, y, z):
	return 1.0

Julia

function code(x, y, z)
	return 1.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0;
end

Wolfram

code[x_, y_, z_] := 1.0

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in x around inf

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

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{{x}^{2} + 0} \]

   2. unpow2 N/A

      \[\leadsto \color{blue}{x \cdot x} + 0 \]

   3. accelerator-lowering-fma.f64 57.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]

5. Simplified 57.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. *-lowering-*.f64 57.8

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

7. Applied egg-rr 57.8%

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

9. Applied egg-rr 3.6%

   \[\leadsto \color{blue}{1} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024196 
(FPCore (x y z)
  :name "Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1"
  :precision binary64
  (- (* x x) (* (* y 4.0) z)))