Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, A

Percentage Accurate: 99.9% → 100.0%

Time: 5.6s

Alternatives: 4

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return 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 - ((y * 4.0d0) * z)
end function

Java

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

Python

def code(x, y, z):
	return x - ((y * 4.0) * z)

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return 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 - ((y * 4.0d0) * z)
end function

Java

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

Python

def code(x, y, z):
	return x - ((y * 4.0) * z)

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

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

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (fma (* -4.0 z) y x))

C

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

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return fma(Float64(-4.0 * z), y, x)
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(N[(-4.0 * z), $MachinePrecision] * y + x), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. sub-neg N/A

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

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\left(y \cdot 4\right) \cdot z\right)\right)}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\left(4 \cdot y\right) \cdot z\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(4 \cdot \left(y \cdot z\right)\right)\right)\right) \]

   5. distribute-lft-neg-in N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(\left(\mathsf{neg}\left(4\right)\right) \cdot \color{blue}{\left(y \cdot z\right)}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(\left(\mathsf{neg}\left(4\right)\right) \cdot \left(z \cdot \color{blue}{y}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\left(\mathsf{neg}\left(4\right)\right), \color{blue}{\left(z \cdot y\right)}\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(-4, \left(\color{blue}{z} \cdot y\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(-4, \left(y \cdot \color{blue}{z}\right)\right)\right) \]

   10. *-lowering-*.f64 100.0%

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \color{blue}{z}\right)\right)\right) \]

3. Simplified 100.0%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. fma-define N/A

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

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

      \[\leadsto \mathsf{fma.f64}\left(\left(-4 \cdot z\right), \color{blue}{y}, x\right) \]

   6. *-lowering-*.f64 99.6%

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(-4, z\right), y, x\right) \]

6. Applied egg-rr 99.6%

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

Alternative 2: 70.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := -4 \cdot \left(z \cdot y\right)\\ \mathbf{if}\;z \leq -1.56 \cdot 10^{-21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-44}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* -4.0 (* z y))))
   (if (<= z -1.56e-21) t_0 (if (<= z 4.6e-44) x t_0))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = -4.0 * (z * y);
	double tmp;
	if (z <= -1.56e-21) {
		tmp = t_0;
	} else if (z <= 4.6e-44) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
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 = (-4.0d0) * (z * y)
    if (z <= (-1.56d-21)) then
        tmp = t_0
    else if (z <= 4.6d-44) then
        tmp = x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double t_0 = -4.0 * (z * y);
	double tmp;
	if (z <= -1.56e-21) {
		tmp = t_0;
	} else if (z <= 4.6e-44) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	t_0 = -4.0 * (z * y)
	tmp = 0
	if z <= -1.56e-21:
		tmp = t_0
	elif z <= 4.6e-44:
		tmp = x
	else:
		tmp = t_0
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(-4.0 * Float64(z * y))
	tmp = 0.0
	if (z <= -1.56e-21)
		tmp = t_0;
	elseif (z <= 4.6e-44)
		tmp = x;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	t_0 = -4.0 * (z * y);
	tmp = 0.0;
	if (z <= -1.56e-21)
		tmp = t_0;
	elseif (z <= 4.6e-44)
		tmp = x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(-4.0 * N[(z * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.56e-21], t$95$0, If[LessEqual[z, 4.6e-44], x, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.55999999999999999e-21 or 4.59999999999999996e-44 < z

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\left(y \cdot 4\right) \cdot z\right)\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\left(4 \cdot y\right) \cdot z\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(4 \cdot \left(y \cdot z\right)\right)\right)\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\left(\mathsf{neg}\left(4\right)\right) \cdot \color{blue}{\left(y \cdot z\right)}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\left(\mathsf{neg}\left(4\right)\right) \cdot \left(z \cdot \color{blue}{y}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\left(\mathsf{neg}\left(4\right)\right), \color{blue}{\left(z \cdot y\right)}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(-4, \left(\color{blue}{z} \cdot y\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(-4, \left(y \cdot \color{blue}{z}\right)\right)\right) \]

      10. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \color{blue}{z}\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(-4, \color{blue}{\left(y \cdot z\right)}\right) \]

      2. *-lowering-*.f64 68.7%

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \color{blue}{z}\right)\right) \]

   7. Simplified 68.7%

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

   if -1.55999999999999999e-21 < z < 4.59999999999999996e-44

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\left(y \cdot 4\right) \cdot z\right)\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\left(4 \cdot y\right) \cdot z\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(4 \cdot \left(y \cdot z\right)\right)\right)\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\left(\mathsf{neg}\left(4\right)\right) \cdot \color{blue}{\left(y \cdot z\right)}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\left(\mathsf{neg}\left(4\right)\right) \cdot \left(z \cdot \color{blue}{y}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\left(\mathsf{neg}\left(4\right)\right), \color{blue}{\left(z \cdot y\right)}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(-4, \left(\color{blue}{z} \cdot y\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(-4, \left(y \cdot \color{blue}{z}\right)\right)\right) \]

      10. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \color{blue}{z}\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 76.4%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.56 \cdot 10^{-21}:\\ \;\;\;\;-4 \cdot \left(z \cdot y\right)\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-44}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(z \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (+ x (* -4.0 (* z y))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return x + (-4.0 * (z * y));
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + ((-4.0d0) * (z * y))
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	return x + (-4.0 * (z * y));
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return x + (-4.0 * (z * y))

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(x + Float64(-4.0 * Float64(z * y)))
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = x + (-4.0 * (z * y));
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(x + N[(-4.0 * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. sub-neg N/A

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

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\left(y \cdot 4\right) \cdot z\right)\right)}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\left(4 \cdot y\right) \cdot z\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(4 \cdot \left(y \cdot z\right)\right)\right)\right) \]

   5. distribute-lft-neg-in N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(\left(\mathsf{neg}\left(4\right)\right) \cdot \color{blue}{\left(y \cdot z\right)}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(\left(\mathsf{neg}\left(4\right)\right) \cdot \left(z \cdot \color{blue}{y}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\left(\mathsf{neg}\left(4\right)\right), \color{blue}{\left(z \cdot y\right)}\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(-4, \left(\color{blue}{z} \cdot y\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(-4, \left(y \cdot \color{blue}{z}\right)\right)\right) \]

   10. *-lowering-*.f64 100.0%

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \color{blue}{z}\right)\right)\right) \]

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 4: 50.7% accurate, 7.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ x \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 x)

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return x;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
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

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

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return x

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return x
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = x;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := x

Derivation

1. Initial program 100.0%

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

   1. sub-neg N/A

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

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\left(y \cdot 4\right) \cdot z\right)\right)}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\left(4 \cdot y\right) \cdot z\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(4 \cdot \left(y \cdot z\right)\right)\right)\right) \]

   5. distribute-lft-neg-in N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(\left(\mathsf{neg}\left(4\right)\right) \cdot \color{blue}{\left(y \cdot z\right)}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(\left(\mathsf{neg}\left(4\right)\right) \cdot \left(z \cdot \color{blue}{y}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\left(\mathsf{neg}\left(4\right)\right), \color{blue}{\left(z \cdot y\right)}\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(-4, \left(\color{blue}{z} \cdot y\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(-4, \left(y \cdot \color{blue}{z}\right)\right)\right) \]

   10. *-lowering-*.f64 100.0%

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(y, \color{blue}{z}\right)\right)\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in x around inf

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

7. Simplified 49.8%

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

Reproduce

herbie shell --seed 2024160 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, A"
  :precision binary64
  (- x (* (* y 4.0) z)))