Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1

Percentage Accurate: 98.4% → 99.5%

Time: 2.9s

Alternatives: 6

Speedup: 1.0×

• 34.0% 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: 99.5% 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 98.4%

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

   1. sub-neg 98.4%

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

   2. +-commutative 98.4%

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

   3. associate-*l* 98.4%

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

   4. distribute-rgt-neg-in 98.4%

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

   5. fma-def 99.6%

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

   6. *-commutative 99.6%

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

   7. distribute-rgt-neg-in 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

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 98.4%

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

   1. fma-neg 98.8%

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

   2. *-commutative 98.8%

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

   3. distribute-rgt-neg-in 98.8%

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

   4. distribute-rgt-neg-in 98.8%

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

   5. metadata-eval 98.8%

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

3. Simplified 98.8%

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

4. Final simplification 98.8%

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

Alternative 3: 83.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{-38} \lor \neg \left(x \cdot x \leq 4.1 \cdot 10^{+63}\right) \land x \cdot x \leq 2.6 \cdot 10^{+113}:\\ \;\;\;\;-4 \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* x x) 4e-38)
         (and (not (<= (* x x) 4.1e+63)) (<= (* x x) 2.6e+113)))
   (* -4.0 (* y z))
   (* x x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x * x) <= 4e-38) || (!((x * x) <= 4.1e+63) && ((x * x) <= 2.6e+113))) {
		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) <= 4d-38) .or. (.not. ((x * x) <= 4.1d+63)) .and. ((x * x) <= 2.6d+113)) 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) <= 4e-38) || (!((x * x) <= 4.1e+63) && ((x * x) <= 2.6e+113))) {
		tmp = -4.0 * (y * z);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((x * x) <= 4e-38) or (not ((x * x) <= 4.1e+63) and ((x * x) <= 2.6e+113)):
		tmp = -4.0 * (y * z)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(x * x) <= 4e-38) || (!(Float64(x * x) <= 4.1e+63) && (Float64(x * x) <= 2.6e+113)))
		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) <= 4e-38) || (~(((x * x) <= 4.1e+63)) && ((x * x) <= 2.6e+113)))
		tmp = -4.0 * (y * z);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(x * x), $MachinePrecision], 4e-38], And[N[Not[LessEqual[N[(x * x), $MachinePrecision], 4.1e+63]], $MachinePrecision], LessEqual[N[(x * x), $MachinePrecision], 2.6e+113]]], N[(-4.0 * N[(y * z), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 3.9999999999999998e-38 or 4.09999999999999993e63 < (*.f64 x x) < 2.5999999999999999e113

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 85.4%

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

   if 3.9999999999999998e-38 < (*.f64 x x) < 4.09999999999999993e63 or 2.5999999999999999e113 < (*.f64 x x)

   1. Initial program 96.8%

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

   2. Taylor expanded in x around inf 84.5%

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

      1. unpow2 84.5%

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

   4. Simplified 84.5%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{-38} \lor \neg \left(x \cdot x \leq 4.1 \cdot 10^{+63}\right) \land x \cdot x \leq 2.6 \cdot 10^{+113}:\\ \;\;\;\;-4 \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 4: 83.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2.9 \cdot 10^{-42}:\\ \;\;\;\;y \cdot \left(z \cdot -4\right)\\ \mathbf{elif}\;x \cdot x \leq 1.05 \cdot 10^{+60}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \cdot x \leq 7 \cdot 10^{+112}:\\ \;\;\;\;-4 \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 2.9e-42)
   (* y (* z -4.0))
   (if (<= (* x x) 1.05e+60)
     (* x x)
     (if (<= (* x x) 7e+112) (* -4.0 (* y z)) (* x x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 2.9e-42) {
		tmp = y * (z * -4.0);
	} else if ((x * x) <= 1.05e+60) {
		tmp = x * x;
	} else if ((x * x) <= 7e+112) {
		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) <= 2.9d-42) then
        tmp = y * (z * (-4.0d0))
    else if ((x * x) <= 1.05d+60) then
        tmp = x * x
    else if ((x * x) <= 7d+112) 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) <= 2.9e-42) {
		tmp = y * (z * -4.0);
	} else if ((x * x) <= 1.05e+60) {
		tmp = x * x;
	} else if ((x * x) <= 7e+112) {
		tmp = -4.0 * (y * z);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 2.9e-42:
		tmp = y * (z * -4.0)
	elif (x * x) <= 1.05e+60:
		tmp = x * x
	elif (x * x) <= 7e+112:
		tmp = -4.0 * (y * z)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 2.9e-42)
		tmp = Float64(y * Float64(z * -4.0));
	elseif (Float64(x * x) <= 1.05e+60)
		tmp = Float64(x * x);
	elseif (Float64(x * x) <= 7e+112)
		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) <= 2.9e-42)
		tmp = y * (z * -4.0);
	elseif ((x * x) <= 1.05e+60)
		tmp = x * x;
	elseif ((x * x) <= 7e+112)
		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], 2.9e-42], N[(y * N[(z * -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 1.05e+60], N[(x * x), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 7e+112], N[(-4.0 * N[(y * z), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 2.9000000000000003e-42

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 86.1%

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

      1. expm1-log1p-u 59.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-4 \cdot \left(y \cdot z\right)\right)\right)} \]

      2. expm1-udef 28.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-4 \cdot \left(y \cdot z\right)\right)} - 1} \]

      3. log1p-udef 28.4%

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

      4. add-exp-log 55.2%

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

   4. Applied egg-rr 55.2%

      \[\leadsto \color{blue}{\left(1 + -4 \cdot \left(y \cdot z\right)\right) - 1} \]
   5. Step-by-step derivation

      1. +-commutative 55.2%

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

      2. associate--l+ 86.1%

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

      3. *-commutative 86.1%

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

      4. metadata-eval 86.1%

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

      5. associate-*l* 86.2%

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

   6. Simplified 86.2%

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

   if 2.9000000000000003e-42 < (*.f64 x x) < 1.0500000000000001e60 or 6.99999999999999994e112 < (*.f64 x x)

   1. Initial program 96.8%

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

   2. Taylor expanded in x around inf 84.5%

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

      1. unpow2 84.5%

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

   4. Simplified 84.5%

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

   if 1.0500000000000001e60 < (*.f64 x x) < 6.99999999999999994e112

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 75.6%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2.9 \cdot 10^{-42}:\\ \;\;\;\;y \cdot \left(z \cdot -4\right)\\ \mathbf{elif}\;x \cdot x \leq 1.05 \cdot 10^{+60}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \cdot x \leq 7 \cdot 10^{+112}:\\ \;\;\;\;-4 \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 5: 98.4% 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 98.4%

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

2. Final simplification 98.4%

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

Alternative 6: 53.1% 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 98.4%

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

2. Taylor expanded in x around inf 52.0%

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

   1. unpow2 52.0%

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

4. Simplified 52.0%

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

5. Final simplification 52.0%

   \[\leadsto x \cdot x \]

Reproduce

herbie shell --seed 2023200 
(FPCore (x y z)
  :name "Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1"
  :precision binary64
  (- (* x x) (* (* y 4.0) z)))