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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[(y * 0.5), $MachinePrecision] * x + 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. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. div-inv N/A

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

   9. lower-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. lift-/.f64 N/A

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

   12. clear-num N/A

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

   13. associate-/r/ N/A

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

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

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

   19. lower-*.f64 N/A

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

   20. metadata-eval N/A

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

   21. metadata-eval 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 78.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+75} \lor \neg \left(x \cdot y \leq 2 \cdot 10^{-15}\right):\\ \;\;\;\;0.5 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* x y) -1e+75) (not (<= (* x y) 2e-15)))
   (* 0.5 (* y x))
   (* -0.125 z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x * y) <= -1e+75) || !((x * y) <= 2e-15)) {
		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) <= (-1d+75)) .or. (.not. ((x * y) <= 2d-15))) 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) <= -1e+75) || !((x * y) <= 2e-15)) {
		tmp = 0.5 * (y * x);
	} else {
		tmp = -0.125 * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((x * y) <= -1e+75) or not ((x * y) <= 2e-15):
		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) <= -1e+75) || !(Float64(x * y) <= 2e-15))
		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) <= -1e+75) || ~(((x * y) <= 2e-15)))
		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], -1e+75], N[Not[LessEqual[N[(x * y), $MachinePrecision], 2e-15]], $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) < -9.99999999999999927e74 or 2.0000000000000002e-15 < (*.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 15.3

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

   5. Applied rewrites 15.3%

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

   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 85.8

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

   8. Applied rewrites 85.8%

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

   if -9.99999999999999927e74 < (*.f64 x y) < 2.0000000000000002e-15

   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 76.2

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

   5. Applied rewrites 76.2%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+75} \lor \neg \left(x \cdot y \leq 2 \cdot 10^{-15}\right):\\ \;\;\;\;0.5 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.9% 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 45.2

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

5. Applied rewrites 45.2%

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

Reproduce

herbie shell --seed 2024313 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, D"
  :precision binary64
  (- (/ (* x y) 2.0) (/ z 8.0)))