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

Percentage Accurate: 99.9% → 99.9%

Time: 1.9s

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: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. sub-neg 99.6%

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

   2. +-commutative 99.6%

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

   3. *-commutative 99.6%

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

   4. distribute-rgt-neg-in 99.6%

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

   5. fma-def 99.6%

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

   6. distribute-rgt-neg-in 99.6%

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

   7. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 71.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [y, z] = \mathsf{sort}([y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.72 \cdot 10^{-15}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 0.0026:\\ \;\;\;\;-4 \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

NOTE: y and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -1.72e-15) x (if (<= x 0.0026) (* -4.0 (* z y)) x)))

C

assert(y < z);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.72e-15) {
		tmp = x;
	} else if (x <= 0.0026) {
		tmp = -4.0 * (z * y);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

NOTE: 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) :: tmp
    if (x <= (-1.72d-15)) then
        tmp = x
    else if (x <= 0.0026d0) then
        tmp = (-4.0d0) * (z * y)
    else
        tmp = x
    end if
    code = tmp
end function

Java

assert y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.72e-15) {
		tmp = x;
	} else if (x <= 0.0026) {
		tmp = -4.0 * (z * y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

[y, z] = sort([y, z])
def code(x, y, z):
	tmp = 0
	if x <= -1.72e-15:
		tmp = x
	elif x <= 0.0026:
		tmp = -4.0 * (z * y)
	else:
		tmp = x
	return tmp

Julia

y, z = sort([y, z])
function code(x, y, z)
	tmp = 0.0
	if (x <= -1.72e-15)
		tmp = x;
	elseif (x <= 0.0026)
		tmp = Float64(-4.0 * Float64(z * y));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

y, z = num2cell(sort([y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.72e-15)
		tmp = x;
	elseif (x <= 0.0026)
		tmp = -4.0 * (z * y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[x, -1.72e-15], x, If[LessEqual[x, 0.0026], N[(-4.0 * N[(z * y), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7199999999999999e-15 or 0.0025999999999999999 < x

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf 78.6%

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

   if -1.7199999999999999e-15 < x < 0.0025999999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 77.9%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.72 \cdot 10^{-15}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 0.0026:\\ \;\;\;\;-4 \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: 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 - (z * (y * 4.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Final simplification 99.6%

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

Alternative 4: 51.0% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in x around inf 53.1%

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

3. Final simplification 53.1%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023214 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, A"
  :precision binary64
  (- x (* (* y 4.0) z)))