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

Percentage Accurate: 100.0% → 100.0%

Time: 2.2s

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

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{y}{2}, x, -0.125 \cdot z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (/ y 2.0) x (* -0.125 z)))

C

double code(double x, double y, double z) {
	return fma((y / 2.0), x, (-0.125 * z));
}

Julia

function code(x, y, z)
	return fma(Float64(y / 2.0), x, Float64(-0.125 * z))
end

Wolfram

code[x_, y_, z_] := N[(N[(y / 2.0), $MachinePrecision] * x + N[(-0.125 * z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-*l/ 100.0%

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

   3. fma-neg 100.0%

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

   4. distribute-frac-neg 100.0%

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

   5. neg-mul-1 100.0%

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

   6. associate-/l* 99.9%

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

   7. associate-/r/ 100.0%

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

   8. metadata-eval 100.0%

      \[\leadsto \mathsf{fma}\left(\frac{y}{2}, x, \color{blue}{-0.125} \cdot z\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{2}, x, -0.125 \cdot z\right)} \]

4. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(\frac{y}{2}, x, -0.125 \cdot z\right) \]

Alternative 2: 79.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot x \leq -4.7 \cdot 10^{+33} \lor \neg \left(y \cdot x \leq 1.6 \cdot 10^{-107}\right) \land \left(y \cdot x \leq 1.65 \cdot 10^{-101} \lor \neg \left(y \cdot x \leq 7.2 \cdot 10^{+23}\right)\right):\\ \;\;\;\;\left(y \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* y x) -4.7e+33)
         (and (not (<= (* y x) 1.6e-107))
              (or (<= (* y x) 1.65e-101) (not (<= (* y x) 7.2e+23)))))
   (* (* y x) 0.5)
   (* -0.125 z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((y * x) <= -4.7e+33) || (!((y * x) <= 1.6e-107) && (((y * x) <= 1.65e-101) || !((y * x) <= 7.2e+23)))) {
		tmp = (y * x) * 0.5;
	} 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 (((y * x) <= (-4.7d+33)) .or. (.not. ((y * x) <= 1.6d-107)) .and. ((y * x) <= 1.65d-101) .or. (.not. ((y * x) <= 7.2d+23))) then
        tmp = (y * x) * 0.5d0
    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 (((y * x) <= -4.7e+33) || (!((y * x) <= 1.6e-107) && (((y * x) <= 1.65e-101) || !((y * x) <= 7.2e+23)))) {
		tmp = (y * x) * 0.5;
	} else {
		tmp = -0.125 * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((y * x) <= -4.7e+33) or (not ((y * x) <= 1.6e-107) and (((y * x) <= 1.65e-101) or not ((y * x) <= 7.2e+23))):
		tmp = (y * x) * 0.5
	else:
		tmp = -0.125 * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(y * x) <= -4.7e+33) || (!(Float64(y * x) <= 1.6e-107) && ((Float64(y * x) <= 1.65e-101) || !(Float64(y * x) <= 7.2e+23))))
		tmp = Float64(Float64(y * x) * 0.5);
	else
		tmp = Float64(-0.125 * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((y * x) <= -4.7e+33) || (~(((y * x) <= 1.6e-107)) && (((y * x) <= 1.65e-101) || ~(((y * x) <= 7.2e+23)))))
		tmp = (y * x) * 0.5;
	else
		tmp = -0.125 * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(y * x), $MachinePrecision], -4.7e+33], And[N[Not[LessEqual[N[(y * x), $MachinePrecision], 1.6e-107]], $MachinePrecision], Or[LessEqual[N[(y * x), $MachinePrecision], 1.65e-101], N[Not[LessEqual[N[(y * x), $MachinePrecision], 7.2e+23]], $MachinePrecision]]]], N[(N[(y * x), $MachinePrecision] * 0.5), $MachinePrecision], N[(-0.125 * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -4.6999999999999998e33 or 1.60000000000000006e-107 < (*.f64 x y) < 1.64999999999999992e-101 or 7.1999999999999997e23 < (*.f64 x y)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 83.2%

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

   if -4.6999999999999998e33 < (*.f64 x y) < 1.60000000000000006e-107 or 1.64999999999999992e-101 < (*.f64 x y) < 7.1999999999999997e23

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 82.0%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -4.7 \cdot 10^{+33} \lor \neg \left(y \cdot x \leq 1.6 \cdot 10^{-107}\right) \land \left(y \cdot x \leq 1.65 \cdot 10^{-101} \lor \neg \left(y \cdot x \leq 7.2 \cdot 10^{+23}\right)\right):\\ \;\;\;\;\left(y \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot z\\ \end{array} \]

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return ((y * x) / 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 = ((y * x) / 2.0d0) - (z / 8.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 4: 50.7% accurate, 3.0× 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. Taylor expanded in x around 0 53.4%

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

3. Final simplification 53.4%

   \[\leadsto -0.125 \cdot z \]

Reproduce

herbie shell --seed 2023290 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, D"
  :precision binary64
  (- (/ (* x y) 2.0) (/ z 8.0)))