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

Percentage Accurate: 100.0% → 100.0%

Time: 4.6s

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[(x / 2.0), $MachinePrecision] * y + 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. associate-*l/ 100.0%

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

   2. fma-neg 100.0%

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

   3. distribute-frac-neg 100.0%

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

   4. neg-mul-1 100.0%

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

   5. associate-/l* 99.9%

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

   6. associate-/r/ 100.0%

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

   7. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 77.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1.6 \cdot 10^{+142} \lor \neg \left(x \cdot y \leq -4.15 \cdot 10^{+40} \lor \neg \left(x \cdot y \leq -11000000\right) \land x \cdot y \leq 1.6 \cdot 10^{-42}\right):\\ \;\;\;\;\left(x \cdot y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* x y) -1.6e+142)
         (not
          (or (<= (* x y) -4.15e+40)
              (and (not (<= (* x y) -11000000.0)) (<= (* x y) 1.6e-42)))))
   (* (* x y) 0.5)
   (* -0.125 z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x * y) <= -1.6e+142) || !(((x * y) <= -4.15e+40) || (!((x * y) <= -11000000.0) && ((x * y) <= 1.6e-42)))) {
		tmp = (x * y) * 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 (((x * y) <= (-1.6d+142)) .or. (.not. ((x * y) <= (-4.15d+40)) .or. (.not. ((x * y) <= (-11000000.0d0))) .and. ((x * y) <= 1.6d-42))) then
        tmp = (x * y) * 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 (((x * y) <= -1.6e+142) || !(((x * y) <= -4.15e+40) || (!((x * y) <= -11000000.0) && ((x * y) <= 1.6e-42)))) {
		tmp = (x * y) * 0.5;
	} else {
		tmp = -0.125 * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((x * y) <= -1.6e+142) or not (((x * y) <= -4.15e+40) or (not ((x * y) <= -11000000.0) and ((x * y) <= 1.6e-42))):
		tmp = (x * y) * 0.5
	else:
		tmp = -0.125 * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(x * y) <= -1.6e+142) || !((Float64(x * y) <= -4.15e+40) || (!(Float64(x * y) <= -11000000.0) && (Float64(x * y) <= 1.6e-42))))
		tmp = Float64(Float64(x * y) * 0.5);
	else
		tmp = Float64(-0.125 * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((x * y) <= -1.6e+142) || ~((((x * y) <= -4.15e+40) || (~(((x * y) <= -11000000.0)) && ((x * y) <= 1.6e-42)))))
		tmp = (x * y) * 0.5;
	else
		tmp = -0.125 * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -1.6e+142], N[Not[Or[LessEqual[N[(x * y), $MachinePrecision], -4.15e+40], And[N[Not[LessEqual[N[(x * y), $MachinePrecision], -11000000.0]], $MachinePrecision], LessEqual[N[(x * y), $MachinePrecision], 1.6e-42]]]], $MachinePrecision]], N[(N[(x * y), $MachinePrecision] * 0.5), $MachinePrecision], N[(-0.125 * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.60000000000000003e142 or -4.1499999999999999e40 < (*.f64 x y) < -1.1e7 or 1.60000000000000012e-42 < (*.f64 x y)

   1. Initial program 100.0%

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

      1. clear-num 100.0%

         \[\leadsto \frac{x \cdot y}{2} - \color{blue}{\frac{1}{\frac{8}{z}}} \]

      2. frac-sub 88.4%

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

      3. metadata-eval 88.4%

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

   3. Applied egg-rr 88.4%

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

   4. Taylor expanded in x around inf 82.7%

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

      1. *-commutative 82.7%

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

   6. Simplified 82.7%

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

   if -1.60000000000000003e142 < (*.f64 x y) < -4.1499999999999999e40 or -1.1e7 < (*.f64 x y) < 1.60000000000000012e-42

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 80.3%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1.6 \cdot 10^{+142} \lor \neg \left(x \cdot y \leq -4.15 \cdot 10^{+40} \lor \neg \left(x \cdot y \leq -11000000\right) \land x \cdot y \leq 1.6 \cdot 10^{-42}\right):\\ \;\;\;\;\left(x \cdot y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot z\\ \end{array} \]

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

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. Final simplification 100.0%

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

Alternative 4: 51.6% 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 51.1%

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

3. Final simplification 51.1%

   \[\leadsto -0.125 \cdot z \]

Reproduce

herbie shell --seed 2023308 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, D"
  :precision binary64
  (- (/ (* x y) 2.0) (/ z 8.0)))