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

Percentage Accurate: 100.0% → 100.0%

Time: 2.8s

Alternatives: 3

Speedup: 1.8×

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, 1.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y \cdot x, 0.5, -0.125 \cdot z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (* y x) 0.5 (* -0.125 z)))

C

double code(double x, double y, double z) {
	return fma((y * x), 0.5, (-0.125 * z));
}

Julia

function code(x, y, z)
	return fma(Float64(y * x), 0.5, Float64(-0.125 * z))
end

Wolfram

code[x_, y_, z_] := N[(N[(y * x), $MachinePrecision] * 0.5 + N[(-0.125 * z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

      \[\leadsto \color{blue}{\frac{x \cdot y}{2} + \left(\mathsf{neg}\left(\frac{z}{8}\right)\right)} \]

   3. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x \cdot y}{2}} + \left(\mathsf{neg}\left(\frac{z}{8}\right)\right) \]

   4. div-inv N/A

      \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\frac{z}{8}\right)\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot y, \frac{1}{2}, \mathsf{neg}\left(\frac{z}{8}\right)\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot y}, \frac{1}{2}, \mathsf{neg}\left(\frac{z}{8}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot x}, \frac{1}{2}, \mathsf{neg}\left(\frac{z}{8}\right)\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot x}, \frac{1}{2}, \mathsf{neg}\left(\frac{z}{8}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(y \cdot x, \color{blue}{\frac{1}{2}}, \mathsf{neg}\left(\frac{z}{8}\right)\right) \]

   10. lift-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(y \cdot x, \frac{1}{2}, \mathsf{neg}\left(\color{blue}{\frac{z}{8}}\right)\right) \]

   11. clear-num N/A

       \[\leadsto \mathsf{fma}\left(y \cdot x, \frac{1}{2}, \mathsf{neg}\left(\color{blue}{\frac{1}{\frac{8}{z}}}\right)\right) \]

   12. associate-/r/ N/A

       \[\leadsto \mathsf{fma}\left(y \cdot x, \frac{1}{2}, \mathsf{neg}\left(\color{blue}{\frac{1}{8} \cdot z}\right)\right) \]

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

       \[\leadsto \mathsf{fma}\left(y \cdot x, \frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot z}\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(y \cdot x, \frac{1}{2}, \left(\mathsf{neg}\left(\color{blue}{\frac{1}{8}}\right)\right) \cdot z\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(y \cdot x, \frac{1}{2}, \color{blue}{\frac{-1}{8}} \cdot z\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(y \cdot x, \frac{1}{2}, \color{blue}{\frac{1}{-8}} \cdot z\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(y \cdot x, \frac{1}{2}, \frac{1}{\color{blue}{\mathsf{neg}\left(8\right)}} \cdot z\right) \]

   18. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(y \cdot x, \frac{1}{2}, \color{blue}{\frac{1}{\mathsf{neg}\left(8\right)} \cdot z}\right) \]

   19. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(y \cdot x, \frac{1}{2}, \frac{1}{\color{blue}{-8}} \cdot z\right) \]

   20. metadata-eval 100.0

       \[\leadsto \mathsf{fma}\left(y \cdot x, 0.5, \color{blue}{-0.125} \cdot z\right) \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot x, 0.5, -0.125 \cdot z\right)} \]
5. Add Preprocessing

Alternative 2: 76.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+134} \lor \neg \left(x \cdot y \leq 1.85 \cdot 10^{+32}\right):\\ \;\;\;\;0.5 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* x y) -5e+134) (not (<= (* x y) 1.85e+32)))
   (* 0.5 (* y x))
   (* -0.125 z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x * y) <= -5e+134) || !((x * y) <= 1.85e+32)) {
		tmp = 0.5 * (y * x);
	} else {
		tmp = -0.125 * z;
	}
	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 * y) <= (-5d+134)) .or. (.not. ((x * y) <= 1.85d+32))) then
        tmp = 0.5d0 * (y * x)
    else
        tmp = (-0.125d0) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (((x * y) <= -5e+134) || !((x * y) <= 1.85e+32)) {
		tmp = 0.5 * (y * x);
	} else {
		tmp = -0.125 * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((x * y) <= -5e+134) or not ((x * y) <= 1.85e+32):
		tmp = 0.5 * (y * x)
	else:
		tmp = -0.125 * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(x * y) <= -5e+134) || !(Float64(x * y) <= 1.85e+32))
		tmp = Float64(0.5 * Float64(y * x));
	else
		tmp = Float64(-0.125 * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((x * y) <= -5e+134) || ~(((x * y) <= 1.85e+32)))
		tmp = 0.5 * (y * x);
	else
		tmp = -0.125 * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -5e+134], N[Not[LessEqual[N[(x * y), $MachinePrecision], 1.85e+32]], $MachinePrecision]], N[(0.5 * N[(y * x), $MachinePrecision]), $MachinePrecision], N[(-0.125 * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -4.99999999999999981e134 or 1.85e32 < (*.f64 x y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot z} \]
   4. Step-by-step derivation

      1. lower-*.f64 11.7

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

   5. Applied rewrites 11.7%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(x \cdot y\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(x \cdot y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(y \cdot x\right)} \]

      3. lower-*.f64 90.1

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

   8. Applied rewrites 90.1%

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

   if -4.99999999999999981e134 < (*.f64 x y) < 1.85e32

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot z} \]
   4. Step-by-step derivation

      1. lower-*.f64 78.5

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

   5. Applied rewrites 78.5%

      \[\leadsto \color{blue}{-0.125 \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+134} \lor \neg \left(x \cdot y \leq 1.85 \cdot 10^{+32}\right):\\ \;\;\;\;0.5 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.0% accurate, 5.2× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return -0.125 * 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 = (-0.125d0) * z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-1}{8} \cdot z} \]
4. Step-by-step derivation

   1. lower-*.f64 55.0

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

5. Applied rewrites 55.0%

   \[\leadsto \color{blue}{-0.125 \cdot z} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024309 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, D"
  :precision binary64
  (- (/ (* x y) 2.0) (/ z 8.0)))