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

Percentage Accurate: 100.0% → 100.0%

Time: 2.8s

Alternatives: 3

Speedup: 1.8×

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, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. div-inv N/A

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

   9. lower-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. lift-/.f64 N/A

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

   12. clear-num N/A

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

   13. associate-/r/ N/A

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

   14. distribute-lft-neg-in N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

   19. lower-*.f64 N/A

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

   20. metadata-eval N/A

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

   21. metadata-eval 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 78.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot y\right) \cdot 0.5\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-91}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-30}:\\ \;\;\;\;z \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* x y) 0.5)))
   (if (<= (* x y) -2e-91) t_0 (if (<= (* x y) 2e-30) (* z -0.125) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x * y) * 0.5;
	double tmp;
	if ((x * y) <= -2e-91) {
		tmp = t_0;
	} else if ((x * y) <= 2e-30) {
		tmp = z * -0.125;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (x * y) * 0.5d0
    if ((x * y) <= (-2d-91)) then
        tmp = t_0
    else if ((x * y) <= 2d-30) then
        tmp = z * (-0.125d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x * y) * 0.5;
	double tmp;
	if ((x * y) <= -2e-91) {
		tmp = t_0;
	} else if ((x * y) <= 2e-30) {
		tmp = z * -0.125;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x * y) * 0.5
	tmp = 0
	if (x * y) <= -2e-91:
		tmp = t_0
	elif (x * y) <= 2e-30:
		tmp = z * -0.125
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * y) * 0.5)
	tmp = 0.0
	if (Float64(x * y) <= -2e-91)
		tmp = t_0;
	elseif (Float64(x * y) <= 2e-30)
		tmp = Float64(z * -0.125);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x * y) * 0.5;
	tmp = 0.0;
	if ((x * y) <= -2e-91)
		tmp = t_0;
	elseif ((x * y) <= 2e-30)
		tmp = z * -0.125;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * y), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2e-91], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], 2e-30], N[(z * -0.125), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2.00000000000000004e-91 or 2e-30 < (*.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 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot z} \]
   4. Step-by-step derivation

      1. lower-*.f64 16.4

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

   5. Applied rewrites 16.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(x \cdot y\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 85.2

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

   8. Applied rewrites 85.2%

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

   if -2.00000000000000004e-91 < (*.f64 x y) < 2e-30

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot z} \]
   4. Step-by-step derivation

      1. lower-*.f64 88.2

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

   5. Applied rewrites 88.2%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-91}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 0.5\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-30}:\\ \;\;\;\;z \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.8% accurate, 5.2× 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

   \[\leadsto \color{blue}{\frac{-1}{8} \cdot z} \]
4. Step-by-step derivation

   1. lower-*.f64 45.6

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

5. Applied rewrites 45.6%

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

6. Final simplification 45.6%

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

Reproduce

herbie shell --seed 2024331 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, D"
  :precision binary64
  (- (/ (* x y) 2.0) (/ z 8.0)))