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

Percentage Accurate: 100.0% → 100.0%

Time: 3.6s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. associate-/l* 100.0%

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

   2. fma-neg 100.0%

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

   3. remove-double-neg 100.0%

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

   4. distribute-frac-neg 100.0%

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

   5. neg-mul-1 100.0%

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

   6. *-commutative 100.0%

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

   7. associate-/l* 100.0%

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

   8. distribute-rgt-neg-in 100.0%

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

   9. metadata-eval 100.0%

      \[\leadsto \mathsf{fma}\left(x, y \cdot \left(-\color{blue}{-0.5}\right), -\frac{z}{8}\right) \]

   10. metadata-eval 100.0%

       \[\leadsto \mathsf{fma}\left(x, y \cdot \color{blue}{0.5}, -\frac{z}{8}\right) \]

   11. distribute-frac-neg 100.0%

       \[\leadsto \mathsf{fma}\left(x, y \cdot 0.5, \color{blue}{\frac{-z}{8}}\right) \]

   12. neg-mul-1 100.0%

       \[\leadsto \mathsf{fma}\left(x, y \cdot 0.5, \frac{\color{blue}{-1 \cdot z}}{8}\right) \]

   13. *-commutative 100.0%

       \[\leadsto \mathsf{fma}\left(x, y \cdot 0.5, \frac{\color{blue}{z \cdot -1}}{8}\right) \]

   14. associate-/l* 100.0%

       \[\leadsto \mathsf{fma}\left(x, y \cdot 0.5, \color{blue}{z \cdot \frac{-1}{8}}\right) \]

   15. metadata-eval 100.0%

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

3. Simplified 100.0%

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

Alternative 2: 78.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{-40} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{+55}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot -0.125\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* x y) -5e-40) (not (<= (* x y) 5e+55)))
   (* 0.5 (* x y))
   (* z -0.125)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x * y) <= -5e-40) || !((x * y) <= 5e+55)) {
		tmp = 0.5 * (x * y);
	} 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 * y) <= (-5d-40)) .or. (.not. ((x * y) <= 5d+55))) then
        tmp = 0.5d0 * (x * y)
    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 * y) <= -5e-40) || !((x * y) <= 5e+55)) {
		tmp = 0.5 * (x * y);
	} else {
		tmp = z * -0.125;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((x * y) <= -5e-40) or not ((x * y) <= 5e+55):
		tmp = 0.5 * (x * y)
	else:
		tmp = z * -0.125
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(x * y) <= -5e-40) || !(Float64(x * y) <= 5e+55))
		tmp = Float64(0.5 * Float64(x * y));
	else
		tmp = Float64(z * -0.125);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((x * y) <= -5e-40) || ~(((x * y) <= 5e+55)))
		tmp = 0.5 * (x * y);
	else
		tmp = z * -0.125;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -5e-40], N[Not[LessEqual[N[(x * y), $MachinePrecision], 5e+55]], $MachinePrecision]], N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision], N[(z * -0.125), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -4.99999999999999965e-40 or 5.00000000000000046e55 < (*.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 inf 80.3%

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

   if -4.99999999999999965e-40 < (*.f64 x y) < 5.00000000000000046e55

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 79.7%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{-40} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{+55}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot -0.125\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

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]

Derivation

1. Initial program 100.0%

   \[\frac{x \cdot y}{2} - \frac{z}{8} \]
2. Add Preprocessing
3. 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 50.7%

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

4. Final simplification 50.7%

   \[\leadsto z \cdot -0.125 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, D"
  :precision binary64
  (- (/ (* x y) 2.0) (/ z 8.0)))