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

Percentage Accurate: 100.0% → 99.9%

Time: 2.0s

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: 100.0% 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

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

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-def 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. Final simplification 100.0%

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

Alternative 2: 70.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-43}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-77} \lor \neg \left(x \leq -2.8 \cdot 10^{-119}\right) \land x \leq 2.3 \cdot 10^{+31}:\\ \;\;\;\;z \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -8.2e-43)
   x
   (if (or (<= x -6.5e-77) (and (not (<= x -2.8e-119)) (<= x 2.3e+31)))
     (* z (* y -4.0))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -8.2e-43) {
		tmp = x;
	} else if ((x <= -6.5e-77) || (!(x <= -2.8e-119) && (x <= 2.3e+31))) {
		tmp = z * (y * -4.0);
	} else {
		tmp = 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 <= (-8.2d-43)) then
        tmp = x
    else if ((x <= (-6.5d-77)) .or. (.not. (x <= (-2.8d-119))) .and. (x <= 2.3d+31)) then
        tmp = z * (y * (-4.0d0))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -8.2e-43) {
		tmp = x;
	} else if ((x <= -6.5e-77) || (!(x <= -2.8e-119) && (x <= 2.3e+31))) {
		tmp = z * (y * -4.0);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -8.2e-43:
		tmp = x
	elif (x <= -6.5e-77) or (not (x <= -2.8e-119) and (x <= 2.3e+31)):
		tmp = z * (y * -4.0)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -8.2e-43)
		tmp = x;
	elseif ((x <= -6.5e-77) || (!(x <= -2.8e-119) && (x <= 2.3e+31)))
		tmp = Float64(z * Float64(y * -4.0));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -8.2e-43)
		tmp = x;
	elseif ((x <= -6.5e-77) || (~((x <= -2.8e-119)) && (x <= 2.3e+31)))
		tmp = z * (y * -4.0);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -8.2e-43], x, If[Or[LessEqual[x, -6.5e-77], And[N[Not[LessEqual[x, -2.8e-119]], $MachinePrecision], LessEqual[x, 2.3e+31]]], N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -8.1999999999999996e-43 or -6.4999999999999999e-77 < x < -2.8e-119 or 2.3e31 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 75.2%

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

   if -8.1999999999999996e-43 < x < -6.4999999999999999e-77 or -2.8e-119 < x < 2.3e31

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 82.6%

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

      1. associate-*r* 82.6%

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

      2. *-commutative 82.6%

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

      3. *-commutative 82.6%

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

      4. *-commutative 82.6%

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

   4. Simplified 82.6%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-43}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-77} \lor \neg \left(x \leq -2.8 \cdot 10^{-119}\right) \land x \leq 2.3 \cdot 10^{+31}:\\ \;\;\;\;z \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 4: 51.3% 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. Taylor expanded in x around inf 48.2%

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

3. Final simplification 48.2%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023338 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, A"
  :precision binary64
  (- x (* (* y 4.0) z)))