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

Percentage Accurate: 100.0% → 100.0%

Time: 4.3s

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 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%

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

   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: 71.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+14} \lor \neg \left(x \leq 2.8 \cdot 10^{-60}\right):\\ \;\;\;\;y \cdot \left(x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot -0.125\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.1e+14) (not (<= x 2.8e-60))) (* y (* x 0.5)) (* z -0.125)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.1e+14) || !(x <= 2.8e-60)) {
		tmp = y * (x * 0.5);
	} else {
		tmp = z * -0.125;
	}
	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 <= (-1.1d+14)) .or. (.not. (x <= 2.8d-60))) then
        tmp = y * (x * 0.5d0)
    else
        tmp = z * (-0.125d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.1e+14) || !(x <= 2.8e-60)) {
		tmp = y * (x * 0.5);
	} else {
		tmp = z * -0.125;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.1e+14) or not (x <= 2.8e-60):
		tmp = y * (x * 0.5)
	else:
		tmp = z * -0.125
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.1e+14) || !(x <= 2.8e-60))
		tmp = Float64(y * Float64(x * 0.5));
	else
		tmp = Float64(z * -0.125);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.1e+14) || ~((x <= 2.8e-60)))
		tmp = y * (x * 0.5);
	else
		tmp = z * -0.125;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.1e+14], N[Not[LessEqual[x, 2.8e-60]], $MachinePrecision]], N[(y * N[(x * 0.5), $MachinePrecision]), $MachinePrecision], N[(z * -0.125), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1e14 or 2.8000000000000002e-60 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 91.6%

      \[\leadsto \color{blue}{y \cdot \left(-0.125 \cdot \frac{z}{y} + 0.5 \cdot x\right)} \]

   4. Taylor expanded in y around inf 79.5%

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

      1. associate-*r* 79.5%

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

   6. Simplified 79.5%

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

   if -1.1e14 < x < 2.8000000000000002e-60

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 76.3%

      \[\leadsto \color{blue}{-0.125 \cdot z} \]
   4. Step-by-step derivation

      1. *-commutative 76.3%

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

   5. Simplified 76.3%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+14} \lor \neg \left(x \leq 2.8 \cdot 10^{-60}\right):\\ \;\;\;\;y \cdot \left(x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot -0.125\\ \end{array} \]
5. Add Preprocessing

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. Add Preprocessing

3. Final simplification 100.0%

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

Alternative 4: 50.2% 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 100.0%

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

3. Taylor expanded in x around 0 50.9%

   \[\leadsto \color{blue}{-0.125 \cdot z} \]
4. Step-by-step derivation

   1. *-commutative 50.9%

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

5. Simplified 50.9%

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

Reproduce

herbie shell --seed 2024130 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, D"
  :precision binary64
  (- (/ (* x y) 2.0) (/ z 8.0)))