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

Percentage Accurate: 100.0% → 100.0%

Time: 4.8s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(x \cdot y\right) - \frac{1}{8} \cdot z} \]
4. Step-by-step derivation

   1. fp-cancel-sub-sign-inv N/A

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

   2. associate-*r* N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 100.0

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

5. Applied rewrites 100.0%

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

Alternative 2: 79.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot y}{2}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-20} \lor \neg \left(t\_0 \leq 4 \cdot 10^{+17}\right):\\ \;\;\;\;0.5 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x y) 2.0)))
   (if (or (<= t_0 -1e-20) (not (<= t_0 4e+17)))
     (* 0.5 (* y x))
     (* -0.125 z))))

C

double code(double x, double y, double z) {
	double t_0 = (x * y) / 2.0;
	double tmp;
	if ((t_0 <= -1e-20) || !(t_0 <= 4e+17)) {
		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) :: t_0
    real(8) :: tmp
    t_0 = (x * y) / 2.0d0
    if ((t_0 <= (-1d-20)) .or. (.not. (t_0 <= 4d+17))) 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 t_0 = (x * y) / 2.0;
	double tmp;
	if ((t_0 <= -1e-20) || !(t_0 <= 4e+17)) {
		tmp = 0.5 * (y * x);
	} else {
		tmp = -0.125 * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x * y) / 2.0
	tmp = 0
	if (t_0 <= -1e-20) or not (t_0 <= 4e+17):
		tmp = 0.5 * (y * x)
	else:
		tmp = -0.125 * z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * y) / 2.0)
	tmp = 0.0
	if ((t_0 <= -1e-20) || !(t_0 <= 4e+17))
		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)
	t_0 = (x * y) / 2.0;
	tmp = 0.0;
	if ((t_0 <= -1e-20) || ~((t_0 <= 4e+17)))
		tmp = 0.5 * (y * x);
	else
		tmp = -0.125 * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * y), $MachinePrecision] / 2.0), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e-20], N[Not[LessEqual[t$95$0, 4e+17]], $MachinePrecision]], N[(0.5 * N[(y * x), $MachinePrecision]), $MachinePrecision], N[(-0.125 * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) #s(literal 2 binary64)) < -9.99999999999999945e-21 or 4e17 < (/.f64 (*.f64 x y) #s(literal 2 binary64))

   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 17.9

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

   5. Applied rewrites 17.9%

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

   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 84.7

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

   8. Applied rewrites 84.7%

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

   if -9.99999999999999945e-21 < (/.f64 (*.f64 x y) #s(literal 2 binary64)) < 4e17

   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 83.0

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

   5. Applied rewrites 83.0%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{2} \leq -1 \cdot 10^{-20} \lor \neg \left(\frac{x \cdot y}{2} \leq 4 \cdot 10^{+17}\right):\\ \;\;\;\;0.5 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.3% 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 49.4

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

5. Applied rewrites 49.4%

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

Reproduce

herbie shell --seed 2024337 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, D"
  :precision binary64
  (- (/ (* x y) 2.0) (/ z 8.0)))