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

Percentage Accurate: 99.9% → 100.0%

Time: 4.0s

Alternatives: 3

Speedup: 1.2×

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, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

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

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. distribute-rgt-neg-in N/A

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

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

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

   10. metadata-eval 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 79.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y \cdot 4\right)\\ t_1 := -4 \cdot \left(z \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+28}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+37}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* y 4.0))) (t_1 (* -4.0 (* z y))))
   (if (<= t_0 -5e+28) t_1 (if (<= t_0 1e+37) x t_1))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y * 4.0);
	double t_1 = -4.0 * (z * y);
	double tmp;
	if (t_0 <= -5e+28) {
		tmp = t_1;
	} else if (t_0 <= 1e+37) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = z * (y * 4.0d0)
    t_1 = (-4.0d0) * (z * y)
    if (t_0 <= (-5d+28)) then
        tmp = t_1
    else if (t_0 <= 1d+37) then
        tmp = x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (y * 4.0);
	double t_1 = -4.0 * (z * y);
	double tmp;
	if (t_0 <= -5e+28) {
		tmp = t_1;
	} else if (t_0 <= 1e+37) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (y * 4.0)
	t_1 = -4.0 * (z * y)
	tmp = 0
	if t_0 <= -5e+28:
		tmp = t_1
	elif t_0 <= 1e+37:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(y * 4.0))
	t_1 = Float64(-4.0 * Float64(z * y))
	tmp = 0.0
	if (t_0 <= -5e+28)
		tmp = t_1;
	elseif (t_0 <= 1e+37)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (y * 4.0);
	t_1 = -4.0 * (z * y);
	tmp = 0.0;
	if (t_0 <= -5e+28)
		tmp = t_1;
	elseif (t_0 <= 1e+37)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[(z * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+28], t$95$1, If[LessEqual[t$95$0, 1e+37], x, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) z) < -4.99999999999999957e28 or 9.99999999999999954e36 < (*.f64 (*.f64 y #s(literal 4 binary64)) z)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. *-lowering-*.f64 83.6

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

   5. Simplified 83.6%

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

   if -4.99999999999999957e28 < (*.f64 (*.f64 y #s(literal 4 binary64)) z) < 9.99999999999999954e36

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 79.6%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(y \cdot 4\right) \leq -5 \cdot 10^{+28}:\\ \;\;\;\;-4 \cdot \left(z \cdot y\right)\\ \mathbf{elif}\;z \cdot \left(y \cdot 4\right) \leq 10^{+37}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(z \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.5% accurate, 14.0× 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

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

5. Simplified 53.5%

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

Reproduce

herbie shell --seed 2024204 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, A"
  :precision binary64
  (- x (* (* y 4.0) z)))