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

Percentage Accurate: 99.9% → 99.9%

Time: 1.8s

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(z * N[(y * -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. +-commutative 100.0%

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

   3. *-commutative 100.0%

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

   4. distribute-rgt-neg-in 100.0%

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

   5. fma-def 100.0%

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

   6. distribute-rgt-neg-in 100.0%

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

   7. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 67.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-146} \lor \neg \left(z \leq 2.25 \cdot 10^{-108} \lor \neg \left(z \leq 8.8 \cdot 10^{-36}\right) \land z \leq 3.4 \cdot 10^{-16}\right):\\ \;\;\;\;-4 \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.1e-146)
         (not (or (<= z 2.25e-108) (and (not (<= z 8.8e-36)) (<= z 3.4e-16)))))
   (* -4.0 (* z y))
   x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.1e-146) || !((z <= 2.25e-108) || (!(z <= 8.8e-36) && (z <= 3.4e-16)))) {
		tmp = -4.0 * (z * y);
	} 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 ((z <= (-2.1d-146)) .or. (.not. (z <= 2.25d-108) .or. (.not. (z <= 8.8d-36)) .and. (z <= 3.4d-16))) then
        tmp = (-4.0d0) * (z * y)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.1e-146) || !((z <= 2.25e-108) || (!(z <= 8.8e-36) && (z <= 3.4e-16)))) {
		tmp = -4.0 * (z * y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.1e-146) or not ((z <= 2.25e-108) or (not (z <= 8.8e-36) and (z <= 3.4e-16))):
		tmp = -4.0 * (z * y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.1e-146) || !((z <= 2.25e-108) || (!(z <= 8.8e-36) && (z <= 3.4e-16))))
		tmp = Float64(-4.0 * Float64(z * y));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.1e-146) || ~(((z <= 2.25e-108) || (~((z <= 8.8e-36)) && (z <= 3.4e-16)))))
		tmp = -4.0 * (z * y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.1e-146], N[Not[Or[LessEqual[z, 2.25e-108], And[N[Not[LessEqual[z, 8.8e-36]], $MachinePrecision], LessEqual[z, 3.4e-16]]]], $MachinePrecision]], N[(-4.0 * N[(z * y), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -2.0999999999999999e-146 or 2.24999999999999985e-108 < z < 8.7999999999999997e-36 or 3.4e-16 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 74.2%

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

   if -2.0999999999999999e-146 < z < 2.24999999999999985e-108 or 8.7999999999999997e-36 < z < 3.4e-16

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 74.4%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-146} \lor \neg \left(z \leq 2.25 \cdot 10^{-108} \lor \neg \left(z \leq 8.8 \cdot 10^{-36}\right) \land z \leq 3.4 \cdot 10^{-16}\right):\\ \;\;\;\;-4 \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 99.9% 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: 50.3% 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 41.6%

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

3. Final simplification 41.6%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023229 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, A"
  :precision binary64
  (- x (* (* y 4.0) z)))