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

Percentage Accurate: 99.9% → 100.0%

Time: 3.0s

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: 99.9% 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. fmm-def 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. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(y, \frac{x}{2}, \frac{z}{-8}\right) \]
6. Add Preprocessing

Alternative 2: 76.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot x \leq -1.65 \cdot 10^{-98} \lor \neg \left(y \cdot x \leq 2.6 \cdot 10^{-90} \lor \neg \left(y \cdot x \leq 5 \cdot 10^{-7}\right) \land y \cdot x \leq 7 \cdot 10^{+53}\right):\\ \;\;\;\;\left(y \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot -0.125\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* y x) -1.65e-98)
         (not
          (or (<= (* y x) 2.6e-90)
              (and (not (<= (* y x) 5e-7)) (<= (* y x) 7e+53)))))
   (* (* y x) 0.5)
   (* z -0.125)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((y * x) <= -1.65e-98) || !(((y * x) <= 2.6e-90) || (!((y * x) <= 5e-7) && ((y * x) <= 7e+53)))) {
		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 (((y * x) <= (-1.65d-98)) .or. (.not. ((y * x) <= 2.6d-90) .or. (.not. ((y * x) <= 5d-7)) .and. ((y * x) <= 7d+53))) 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 (((y * x) <= -1.65e-98) || !(((y * x) <= 2.6e-90) || (!((y * x) <= 5e-7) && ((y * x) <= 7e+53)))) {
		tmp = (y * x) * 0.5;
	} else {
		tmp = z * -0.125;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((y * x) <= -1.65e-98) or not (((y * x) <= 2.6e-90) or (not ((y * x) <= 5e-7) and ((y * x) <= 7e+53))):
		tmp = (y * x) * 0.5
	else:
		tmp = z * -0.125
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(y * x) <= -1.65e-98) || !((Float64(y * x) <= 2.6e-90) || (!(Float64(y * x) <= 5e-7) && (Float64(y * x) <= 7e+53))))
		tmp = Float64(Float64(y * 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 (((y * x) <= -1.65e-98) || ~((((y * x) <= 2.6e-90) || (~(((y * x) <= 5e-7)) && ((y * x) <= 7e+53)))))
		tmp = (y * x) * 0.5;
	else
		tmp = z * -0.125;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(y * x), $MachinePrecision], -1.65e-98], N[Not[Or[LessEqual[N[(y * x), $MachinePrecision], 2.6e-90], And[N[Not[LessEqual[N[(y * x), $MachinePrecision], 5e-7]], $MachinePrecision], LessEqual[N[(y * x), $MachinePrecision], 7e+53]]]], $MachinePrecision]], N[(N[(y * x), $MachinePrecision] * 0.5), $MachinePrecision], N[(z * -0.125), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.6500000000000001e-98 or 2.6e-90 < (*.f64 x y) < 4.99999999999999977e-7 or 7.00000000000000038e53 < (*.f64 x y)

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 84.8%

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

   4. Taylor expanded in z around 0 86.0%

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

   if -1.6500000000000001e-98 < (*.f64 x y) < 2.6e-90 or 4.99999999999999977e-7 < (*.f64 x y) < 7.00000000000000038e53

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 90.6%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -1.65 \cdot 10^{-98} \lor \neg \left(y \cdot x \leq 2.6 \cdot 10^{-90} \lor \neg \left(y \cdot x \leq 5 \cdot 10^{-7}\right) \land y \cdot x \leq 7 \cdot 10^{+53}\right):\\ \;\;\;\;\left(y \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot -0.125\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% 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: 50.7% 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 49.3%

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

4. Final simplification 49.3%

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

Reproduce

herbie shell --seed 2024115 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, D"
  :precision binary64
  (- (/ (* x y) 2.0) (/ z 8.0)))