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

Percentage Accurate: 99.9% → 100.0%

Time: 2.1s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. sub-neg 100.0%

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

   2. distribute-rgt-neg-out 100.0%

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

   3. +-commutative 100.0%

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

   4. associate-*l* 100.0%

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

   5. distribute-rgt-neg-in 100.0%

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

   6. *-commutative 100.0%

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

   7. fma-define 100.0%

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

   8. distribute-rgt-neg-in 100.0%

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

   9. metadata-eval 100.0%

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

3. Simplified 100.0%

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

Alternative 2: 68.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -4 \cdot \left(y \cdot z\right)\\ \mathbf{if}\;y \leq -1.5 \cdot 10^{+69}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -1 \cdot 10^{+30}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -3.8 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{-172}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* -4.0 (* y z))))
   (if (<= y -1.5e+69)
     t_0
     (if (<= y -1e+30)
       x
       (if (<= y -3.8e+14) t_0 (if (<= y 5.1e-172) x t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = -4.0 * (y * z);
	double tmp;
	if (y <= -1.5e+69) {
		tmp = t_0;
	} else if (y <= -1e+30) {
		tmp = x;
	} else if (y <= -3.8e+14) {
		tmp = t_0;
	} else if (y <= 5.1e-172) {
		tmp = x;
	} else {
		tmp = t_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 = (-4.0d0) * (y * z)
    if (y <= (-1.5d+69)) then
        tmp = t_0
    else if (y <= (-1d+30)) then
        tmp = x
    else if (y <= (-3.8d+14)) then
        tmp = t_0
    else if (y <= 5.1d-172) then
        tmp = x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -4.0 * (y * z);
	double tmp;
	if (y <= -1.5e+69) {
		tmp = t_0;
	} else if (y <= -1e+30) {
		tmp = x;
	} else if (y <= -3.8e+14) {
		tmp = t_0;
	} else if (y <= 5.1e-172) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -4.0 * (y * z)
	tmp = 0
	if y <= -1.5e+69:
		tmp = t_0
	elif y <= -1e+30:
		tmp = x
	elif y <= -3.8e+14:
		tmp = t_0
	elif y <= 5.1e-172:
		tmp = x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-4.0 * Float64(y * z))
	tmp = 0.0
	if (y <= -1.5e+69)
		tmp = t_0;
	elseif (y <= -1e+30)
		tmp = x;
	elseif (y <= -3.8e+14)
		tmp = t_0;
	elseif (y <= 5.1e-172)
		tmp = x;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -4.0 * (y * z);
	tmp = 0.0;
	if (y <= -1.5e+69)
		tmp = t_0;
	elseif (y <= -1e+30)
		tmp = x;
	elseif (y <= -3.8e+14)
		tmp = t_0;
	elseif (y <= 5.1e-172)
		tmp = x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(-4.0 * N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.5e+69], t$95$0, If[LessEqual[y, -1e+30], x, If[LessEqual[y, -3.8e+14], t$95$0, If[LessEqual[y, 5.1e-172], x, t$95$0]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.49999999999999992e69 or -1e30 < y < -3.8e14 or 5.0999999999999998e-172 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 64.5%

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

   if -1.49999999999999992e69 < y < -1e30 or -3.8e14 < y < 5.0999999999999998e-172

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 78.3%

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

Alternative 3: 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]

Derivation

1. Initial program 100.0%

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

Alternative 4: 51.8% accurate, 7.0× 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 100.0%

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

3. Taylor expanded in x around inf 54.0%

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

Reproduce

herbie shell --seed 2024034 -o generate:simplify
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, A"
  :precision binary64
  (- x (* (* y 4.0) z)))