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

Percentage Accurate: 100.0% → 100.0%

Time: 5.4s

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* 99.8%

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

   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: 70.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+87} \lor \neg \left(x \leq -2 \cdot 10^{+37}\right) \land \left(x \leq -8.8 \cdot 10^{-5} \lor \neg \left(x \leq 4.4 \cdot 10^{-32}\right)\right):\\ \;\;\;\;y \cdot \frac{x}{2}\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -7.2e+87)
         (and (not (<= x -2e+37)) (or (<= x -8.8e-5) (not (<= x 4.4e-32)))))
   (* y (/ x 2.0))
   (* -0.125 z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -7.2e+87) || (!(x <= -2e+37) && ((x <= -8.8e-5) || !(x <= 4.4e-32)))) {
		tmp = y * (x / 2.0);
	} 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 <= (-7.2d+87)) .or. (.not. (x <= (-2d+37))) .and. (x <= (-8.8d-5)) .or. (.not. (x <= 4.4d-32))) then
        tmp = y * (x / 2.0d0)
    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 <= -7.2e+87) || (!(x <= -2e+37) && ((x <= -8.8e-5) || !(x <= 4.4e-32)))) {
		tmp = y * (x / 2.0);
	} else {
		tmp = -0.125 * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -7.2e+87) or (not (x <= -2e+37) and ((x <= -8.8e-5) or not (x <= 4.4e-32))):
		tmp = y * (x / 2.0)
	else:
		tmp = -0.125 * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -7.2e+87) || (!(x <= -2e+37) && ((x <= -8.8e-5) || !(x <= 4.4e-32))))
		tmp = Float64(y * Float64(x / 2.0));
	else
		tmp = Float64(-0.125 * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -7.2e+87) || (~((x <= -2e+37)) && ((x <= -8.8e-5) || ~((x <= 4.4e-32)))))
		tmp = y * (x / 2.0);
	else
		tmp = -0.125 * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -7.2e+87], And[N[Not[LessEqual[x, -2e+37]], $MachinePrecision], Or[LessEqual[x, -8.8e-5], N[Not[LessEqual[x, 4.4e-32]], $MachinePrecision]]]], N[(y * N[(x / 2.0), $MachinePrecision]), $MachinePrecision], N[(-0.125 * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.19999999999999988e87 or -1.99999999999999991e37 < x < -8.7999999999999998e-5 or 4.4e-32 < x

   1. Initial program 100.0%

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

      1. associate-/l* 99.8%

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

      2. frac-sub 83.7%

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

   3. Applied egg-rr 83.7%

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

      1. *-commutative 83.7%

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

      2. associate-*l/ 83.7%

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

      3. metadata-eval 83.7%

         \[\leadsto \frac{x \cdot 8 - z \cdot \frac{2}{y}}{\frac{\color{blue}{16}}{y}} \]

   5. Simplified 83.7%

      \[\leadsto \color{blue}{\frac{x \cdot 8 - z \cdot \frac{2}{y}}{\frac{16}{y}}} \]

   6. Taylor expanded in x around inf 70.2%

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

      1. associate-*r* 70.2%

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

      2. metadata-eval 70.2%

         \[\leadsto \left(\color{blue}{\frac{8}{16}} \cdot y\right) \cdot x \]

      3. associate-/r/ 70.1%

         \[\leadsto \color{blue}{\frac{8}{\frac{16}{y}}} \cdot x \]

      4. *-commutative 70.1%

         \[\leadsto \color{blue}{x \cdot \frac{8}{\frac{16}{y}}} \]

      5. associate-*r/ 69.1%

         \[\leadsto \color{blue}{\frac{x \cdot 8}{\frac{16}{y}}} \]

      6. associate-/r/ 69.4%

         \[\leadsto \color{blue}{\frac{x \cdot 8}{16} \cdot y} \]

      7. *-commutative 69.4%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot 8}{16}} \]

      8. associate-/l* 70.2%

         \[\leadsto y \cdot \color{blue}{\frac{x}{\frac{16}{8}}} \]

      9. metadata-eval 70.2%

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

   8. Simplified 70.2%

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

   if -7.19999999999999988e87 < x < -1.99999999999999991e37 or -8.7999999999999998e-5 < x < 4.4e-32

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 69.1%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+87} \lor \neg \left(x \leq -2 \cdot 10^{+37}\right) \land \left(x \leq -8.8 \cdot 10^{-5} \lor \neg \left(x \leq 4.4 \cdot 10^{-32}\right)\right):\\ \;\;\;\;y \cdot \frac{x}{2}\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot z\\ \end{array} \]

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: 49.9% 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 51.7%

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

3. Final simplification 51.7%

   \[\leadsto -0.125 \cdot z \]

Reproduce

herbie shell --seed 2023274 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, D"
  :precision binary64
  (- (/ (* x y) 2.0) (/ z 8.0)))