Diagrams.Solve.Polynomial:quartForm from diagrams-solve-0.1, D

Percentage Accurate: 100.0% → 100.0%

Time: 2.9s

Alternatives: 4

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{2} - \frac{z}{8} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (/ (* x y) 2.0) (/ z 8.0)))

C

double code(double x, double y, double z) {
	return ((x * y) / 2.0) - (z / 8.0);
}

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) / 2.0d0) - (z / 8.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return ((x * y) / 2.0) - (z / 8.0);
}

Python

def code(x, y, z):
	return ((x * y) / 2.0) - (z / 8.0)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x * y) / 2.0) - Float64(z / 8.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x * y) / 2.0) - (z / 8.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x * y), $MachinePrecision] / 2.0), $MachinePrecision] - N[(z / 8.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{2} - \frac{z}{8} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (/ (* x y) 2.0) (/ z 8.0)))

C

double code(double x, double y, double z) {
	return ((x * y) / 2.0) - (z / 8.0);
}

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) / 2.0d0) - (z / 8.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return ((x * y) / 2.0) - (z / 8.0);
}

Python

def code(x, y, z):
	return ((x * y) / 2.0) - (z / 8.0)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x * y) / 2.0) - Float64(z / 8.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x * y) / 2.0) - (z / 8.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x * y), $MachinePrecision] / 2.0), $MachinePrecision] - N[(z / 8.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, \frac{x}{2}, \frac{z}{-8}\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma y (/ x 2.0) (/ z -8.0)))

C

double code(double x, double y, double z) {
	return fma(y, (x / 2.0), (z / -8.0));
}

Julia

function code(x, y, z)
	return fma(y, Float64(x / 2.0), Float64(z / -8.0))
end

Wolfram

code[x_, y_, z_] := N[(y * N[(x / 2.0), $MachinePrecision] + N[(z / -8.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\frac{x \cdot y}{2} - \frac{z}{8} \]
2. Step-by-step derivation

   1. *-commutative 99.4%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{2} - \frac{z}{8} \]

   2. associate-/l* 100.0%

      \[\leadsto \color{blue}{y \cdot \frac{x}{2}} - \frac{z}{8} \]

   3. fma-neg 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{x}{2}, -\frac{z}{8}\right)} \]

   4. distribute-neg-frac2 100.0%

      \[\leadsto \mathsf{fma}\left(y, \frac{x}{2}, \color{blue}{\frac{z}{-8}}\right) \]

   5. metadata-eval 100.0%

      \[\leadsto \mathsf{fma}\left(y, \frac{x}{2}, \frac{z}{\color{blue}{-8}}\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{x}{2}, \frac{z}{-8}\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 69.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.16 \cdot 10^{+33} \lor \neg \left(z \leq -1.2 \cdot 10^{-104}\right) \land \left(z \leq -1.4 \cdot 10^{-117} \lor \neg \left(z \leq -4.5 \cdot 10^{-197}\right) \land \left(z \leq -2.5 \cdot 10^{-209} \lor \neg \left(z \leq -4.3 \cdot 10^{-232}\right) \land \left(z \leq -1.52 \cdot 10^{-235} \lor \neg \left(z \leq 1.08 \cdot 10^{+38}\right)\right)\right)\right):\\ \;\;\;\;z \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.16e+33)
         (and (not (<= z -1.2e-104))
              (or (<= z -1.4e-117)
                  (and (not (<= z -4.5e-197))
                       (or (<= z -2.5e-209)
                           (and (not (<= z -4.3e-232))
                                (or (<= z -1.52e-235)
                                    (not (<= z 1.08e+38)))))))))
   (* z -0.125)
   (* x (* y 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.16e+33) || (!(z <= -1.2e-104) && ((z <= -1.4e-117) || (!(z <= -4.5e-197) && ((z <= -2.5e-209) || (!(z <= -4.3e-232) && ((z <= -1.52e-235) || !(z <= 1.08e+38)))))))) {
		tmp = z * -0.125;
	} else {
		tmp = x * (y * 0.5);
	}
	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 <= (-1.16d+33)) .or. (.not. (z <= (-1.2d-104))) .and. (z <= (-1.4d-117)) .or. (.not. (z <= (-4.5d-197))) .and. (z <= (-2.5d-209)) .or. (.not. (z <= (-4.3d-232))) .and. (z <= (-1.52d-235)) .or. (.not. (z <= 1.08d+38))) then
        tmp = z * (-0.125d0)
    else
        tmp = x * (y * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.16e+33) || (!(z <= -1.2e-104) && ((z <= -1.4e-117) || (!(z <= -4.5e-197) && ((z <= -2.5e-209) || (!(z <= -4.3e-232) && ((z <= -1.52e-235) || !(z <= 1.08e+38)))))))) {
		tmp = z * -0.125;
	} else {
		tmp = x * (y * 0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.16e+33) or (not (z <= -1.2e-104) and ((z <= -1.4e-117) or (not (z <= -4.5e-197) and ((z <= -2.5e-209) or (not (z <= -4.3e-232) and ((z <= -1.52e-235) or not (z <= 1.08e+38))))))):
		tmp = z * -0.125
	else:
		tmp = x * (y * 0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.16e+33) || (!(z <= -1.2e-104) && ((z <= -1.4e-117) || (!(z <= -4.5e-197) && ((z <= -2.5e-209) || (!(z <= -4.3e-232) && ((z <= -1.52e-235) || !(z <= 1.08e+38))))))))
		tmp = Float64(z * -0.125);
	else
		tmp = Float64(x * Float64(y * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.16e+33) || (~((z <= -1.2e-104)) && ((z <= -1.4e-117) || (~((z <= -4.5e-197)) && ((z <= -2.5e-209) || (~((z <= -4.3e-232)) && ((z <= -1.52e-235) || ~((z <= 1.08e+38)))))))))
		tmp = z * -0.125;
	else
		tmp = x * (y * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.16e+33], And[N[Not[LessEqual[z, -1.2e-104]], $MachinePrecision], Or[LessEqual[z, -1.4e-117], And[N[Not[LessEqual[z, -4.5e-197]], $MachinePrecision], Or[LessEqual[z, -2.5e-209], And[N[Not[LessEqual[z, -4.3e-232]], $MachinePrecision], Or[LessEqual[z, -1.52e-235], N[Not[LessEqual[z, 1.08e+38]], $MachinePrecision]]]]]]]], N[(z * -0.125), $MachinePrecision], N[(x * N[(y * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.16000000000000001e33 or -1.2e-104 < z < -1.4e-117 or -4.5000000000000001e-197 < z < -2.5000000000000002e-209 or -4.2999999999999997e-232 < z < -1.52e-235 or 1.07999999999999995e38 < z

   1. Initial program 100.0%

      \[\frac{x \cdot y}{2} - \frac{z}{8} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

         \[\leadsto \color{blue}{x \cdot \frac{y}{2}} - \frac{z}{8} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{2} - \frac{z}{8}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 75.9%

      \[\leadsto \color{blue}{-0.125 \cdot z} \]

   if -1.16000000000000001e33 < z < -1.2e-104 or -1.4e-117 < z < -4.5000000000000001e-197 or -2.5000000000000002e-209 < z < -4.2999999999999997e-232 or -1.52e-235 < z < 1.07999999999999995e38

   1. Initial program 98.7%

      \[\frac{x \cdot y}{2} - \frac{z}{8} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

         \[\leadsto \color{blue}{x \cdot \frac{y}{2}} - \frac{z}{8} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{2} - \frac{z}{8}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 74.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot y\right)} \]
   6. Step-by-step derivation

      1. *-commutative 74.7%

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot 0.5} \]

      2. associate-*r* 75.9%

         \[\leadsto \color{blue}{x \cdot \left(y \cdot 0.5\right)} \]

      3. *-commutative 75.9%

         \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot y\right)} \]

   7. Simplified 75.9%

      \[\leadsto \color{blue}{x \cdot \left(0.5 \cdot y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.16 \cdot 10^{+33} \lor \neg \left(z \leq -1.2 \cdot 10^{-104}\right) \land \left(z \leq -1.4 \cdot 10^{-117} \lor \neg \left(z \leq -4.5 \cdot 10^{-197}\right) \land \left(z \leq -2.5 \cdot 10^{-209} \lor \neg \left(z \leq -4.3 \cdot 10^{-232}\right) \land \left(z \leq -1.52 \cdot 10^{-235} \lor \neg \left(z \leq 1.08 \cdot 10^{+38}\right)\right)\right)\right):\\ \;\;\;\;z \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{y}{2} - \frac{z}{8} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (* x (/ y 2.0)) (/ z 8.0)))

C

double code(double x, double y, double z) {
	return (x * (y / 2.0)) - (z / 8.0);
}

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 / 2.0d0)) - (z / 8.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (x * (y / 2.0)) - (z / 8.0);
}

Python

def code(x, y, z):
	return (x * (y / 2.0)) - (z / 8.0)

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(y / 2.0)) - Float64(z / 8.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (y / 2.0)) - (z / 8.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(y / 2.0), $MachinePrecision]), $MachinePrecision] - N[(z / 8.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\frac{x \cdot y}{2} - \frac{z}{8} \]
2. Step-by-step derivation

   1. associate-/l* 100.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{2}} - \frac{z}{8} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot \frac{y}{2} - \frac{z}{8}} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 4: 51.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ z \cdot -0.125 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* z -0.125))

C

double code(double x, double y, double z) {
	return z * -0.125;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = z * (-0.125d0)
end function

Java

public static double code(double x, double y, double z) {
	return z * -0.125;
}

Python

def code(x, y, z):
	return z * -0.125

Julia

function code(x, y, z)
	return Float64(z * -0.125)
end

MATLAB

function tmp = code(x, y, z)
	tmp = z * -0.125;
end

Wolfram

code[x_, y_, z_] := N[(z * -0.125), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\frac{x \cdot y}{2} - \frac{z}{8} \]
2. Step-by-step derivation

   1. associate-/l* 100.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{2}} - \frac{z}{8} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot \frac{y}{2} - \frac{z}{8}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 51.9%

   \[\leadsto \color{blue}{-0.125 \cdot z} \]

6. Final simplification 51.9%

   \[\leadsto z \cdot -0.125 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, D"
  :precision binary64
  (- (/ (* x y) 2.0) (/ z 8.0)))