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

Percentage Accurate: 100.0% → 99.9%

Time: 2.9s

Alternatives: 4

Speedup: N/A×

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 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. distribute-rgt-neg-out 99.6%

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

   3. +-commutative 99.6%

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

   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. Final simplification 100.0%

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

Alternative 2: 69.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.22 \cdot 10^{-70} \lor \neg \left(z \leq 102000000 \lor \neg \left(z \leq 4.7 \cdot 10^{+29}\right) \land \left(z \leq 4.7 \cdot 10^{+67} \lor \neg \left(z \leq 2.25 \cdot 10^{+247}\right) \land z \leq 1.04 \cdot 10^{+263}\right)\right):\\ \;\;\;\;-4 \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.22e-70)
         (not
          (or (<= z 102000000.0)
              (and (not (<= z 4.7e+29))
                   (or (<= z 4.7e+67)
                       (and (not (<= z 2.25e+247)) (<= z 1.04e+263)))))))
   (* -4.0 (* y z))
   x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.22e-70) || !((z <= 102000000.0) || (!(z <= 4.7e+29) && ((z <= 4.7e+67) || (!(z <= 2.25e+247) && (z <= 1.04e+263)))))) {
		tmp = -4.0 * (y * z);
	} 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 <= (-1.22d-70)) .or. (.not. (z <= 102000000.0d0) .or. (.not. (z <= 4.7d+29)) .and. (z <= 4.7d+67) .or. (.not. (z <= 2.25d+247)) .and. (z <= 1.04d+263))) then
        tmp = (-4.0d0) * (y * z)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.22e-70) || !((z <= 102000000.0) || (!(z <= 4.7e+29) && ((z <= 4.7e+67) || (!(z <= 2.25e+247) && (z <= 1.04e+263)))))) {
		tmp = -4.0 * (y * z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.22e-70) or not ((z <= 102000000.0) or (not (z <= 4.7e+29) and ((z <= 4.7e+67) or (not (z <= 2.25e+247) and (z <= 1.04e+263))))):
		tmp = -4.0 * (y * z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.22e-70) || !((z <= 102000000.0) || (!(z <= 4.7e+29) && ((z <= 4.7e+67) || (!(z <= 2.25e+247) && (z <= 1.04e+263))))))
		tmp = Float64(-4.0 * Float64(y * z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.22e-70) || ~(((z <= 102000000.0) || (~((z <= 4.7e+29)) && ((z <= 4.7e+67) || (~((z <= 2.25e+247)) && (z <= 1.04e+263)))))))
		tmp = -4.0 * (y * z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.22e-70], N[Not[Or[LessEqual[z, 102000000.0], And[N[Not[LessEqual[z, 4.7e+29]], $MachinePrecision], Or[LessEqual[z, 4.7e+67], And[N[Not[LessEqual[z, 2.25e+247]], $MachinePrecision], LessEqual[z, 1.04e+263]]]]]], $MachinePrecision]], N[(-4.0 * N[(y * z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -1.22e-70 or 1.02e8 < z < 4.7000000000000002e29 or 4.70000000000000017e67 < z < 2.25000000000000001e247 or 1.04e263 < z

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 77.2%

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

   if -1.22e-70 < z < 1.02e8 or 4.7000000000000002e29 < z < 4.70000000000000017e67 or 2.25000000000000001e247 < z < 1.04e263

   1. Initial program 99.2%

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

      1. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 82.2%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.22 \cdot 10^{-70} \lor \neg \left(z \leq 102000000 \lor \neg \left(z \leq 4.7 \cdot 10^{+29}\right) \land \left(z \leq 4.7 \cdot 10^{+67} \lor \neg \left(z \leq 2.25 \cdot 10^{+247}\right) \land z \leq 1.04 \cdot 10^{+263}\right)\right):\\ \;\;\;\;-4 \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-*l* 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 4: 49.9% 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 99.6%

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

   1. associate-*l* 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around inf 52.2%

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

6. Final simplification 52.2%

   \[\leadsto x \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024050 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, A"
  :precision binary64
  (- x (* (* y 4.0) z)))