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

Percentage Accurate: 100.0% → 100.0%

Time: 5.9s

Alternatives: 5

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

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{y}{2}, x, \frac{z}{-8}\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (/ y 2.0) x (/ z -8.0)))

C

double code(double x, double y, double z) {
	return fma((y / 2.0), x, (z / -8.0));
}

Julia

function code(x, y, z)
	return fma(Float64(y / 2.0), x, Float64(z / -8.0))
end

Wolfram

code[x_, y_, z_] := N[(N[(y / 2.0), $MachinePrecision] * x + N[(z / -8.0), $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. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. fmm-def N/A

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

   4. fma-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(\left(\frac{y}{2}\right), \color{blue}{x}, \left(\mathsf{neg}\left(\frac{z}{8}\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(y, 2\right), x, \left(\mathsf{neg}\left(\frac{z}{8}\right)\right)\right) \]

   6. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(y, 2\right), x, \left(\frac{z}{\mathsf{neg}\left(8\right)}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(y, 2\right), x, \mathsf{/.f64}\left(z, \left(\mathsf{neg}\left(8\right)\right)\right)\right) \]

   8. metadata-eval 100.0%

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(y, 2\right), x, \mathsf{/.f64}\left(z, -8\right)\right) \]

4. Applied egg-rr 100.0%

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

Alternative 2: 76.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot 0.5\\ \mathbf{if}\;y \cdot x \leq -5 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \cdot x \leq 2 \cdot 10^{-145}:\\ \;\;\;\;z \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* y x) 0.5)))
   (if (<= (* y x) -5e-8) t_0 (if (<= (* y x) 2e-145) (* z -0.125) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (y * x) * 0.5;
	double tmp;
	if ((y * x) <= -5e-8) {
		tmp = t_0;
	} else if ((y * x) <= 2e-145) {
		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 = (y * x) * 0.5d0
    if ((y * x) <= (-5d-8)) then
        tmp = t_0
    else if ((y * x) <= 2d-145) 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 = (y * x) * 0.5;
	double tmp;
	if ((y * x) <= -5e-8) {
		tmp = t_0;
	} else if ((y * x) <= 2e-145) {
		tmp = z * -0.125;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y * x) * 0.5
	tmp = 0
	if (y * x) <= -5e-8:
		tmp = t_0
	elif (y * x) <= 2e-145:
		tmp = z * -0.125
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y * x) * 0.5)
	tmp = 0.0
	if (Float64(y * x) <= -5e-8)
		tmp = t_0;
	elseif (Float64(y * x) <= 2e-145)
		tmp = Float64(z * -0.125);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y * x) * 0.5;
	tmp = 0.0;
	if ((y * x) <= -5e-8)
		tmp = t_0;
	elseif ((y * x) <= 2e-145)
		tmp = z * -0.125;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * x), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[N[(y * x), $MachinePrecision], -5e-8], t$95$0, If[LessEqual[N[(y * x), $MachinePrecision], 2e-145], N[(z * -0.125), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -4.9999999999999998e-8 or 1.99999999999999983e-145 < (*.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

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(x \cdot y\right)}\right) \]

      2. *-lowering-*.f64 78.1%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right) \]

   5. Simplified 78.1%

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

   if -4.9999999999999998e-8 < (*.f64 x y) < 1.99999999999999983e-145

   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. *-lowering-*.f64 84.1%

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \color{blue}{z}\right) \]

   5. Simplified 84.1%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\left(y \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;y \cdot x \leq 2 \cdot 10^{-145}:\\ \;\;\;\;z \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

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. Add Preprocessing

3. Final simplification 100.0%

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

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (- (/ x (/ 2.0 y)) (/ z 8.0)))

C

double code(double x, double y, double z) {
	return (x / (2.0 / y)) - (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 / (2.0d0 / y)) - (z / 8.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (x / (2.0 / y)) - (z / 8.0);
}

Python

def code(x, y, z):
	return (x / (2.0 / y)) - (z / 8.0)

Julia

function code(x, y, z)
	return Float64(Float64(x / Float64(2.0 / y)) - Float64(z / 8.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x / (2.0 / y)) - (z / 8.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(x / N[(2.0 / y), $MachinePrecision]), $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. Step-by-step derivation

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\frac{x \cdot y}{2}\right), \color{blue}{\left(\frac{z}{8}\right)}\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \frac{y}{2}\right), \left(\frac{\color{blue}{z}}{8}\right)\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \frac{1}{\frac{2}{y}}\right), \left(\frac{z}{8}\right)\right) \]

   4. un-div-inv N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\frac{x}{\frac{2}{y}}\right), \left(\frac{\color{blue}{z}}{8}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{2}{y}\right)\right), \left(\frac{\color{blue}{z}}{8}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(2, y\right)\right), \left(\frac{z}{8}\right)\right) \]

   7. /-lowering-/.f64 99.8%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(2, y\right)\right), \mathsf{/.f64}\left(z, \color{blue}{8}\right)\right) \]

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{x}{\frac{2}{y}} - \frac{z}{8}} \]
5. Add Preprocessing

Alternative 5: 51.7% 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

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

   1. *-lowering-*.f64 50.2%

      \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \color{blue}{z}\right) \]

5. Simplified 50.2%

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

6. Final simplification 50.2%

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

Reproduce

herbie shell --seed 2024150 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, D"
  :precision binary64
  (- (/ (* x y) 2.0) (/ z 8.0)))