Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, A

Percentage Accurate: 92.4% → 93.2%

Time: 2.5s

Alternatives: 3

Speedup: 0.7×

Specification

Math

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

FPCore

(FPCore (x y z) :precision binary64 (/ (* x y) z))

C

double code(double x, double y, double z) {
	return (x * y) / 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 = (x * y) / z
end function

Java

public static double code(double x, double y, double z) {
	return (x * y) / z;
}

Python

def code(x, y, z):
	return (x * y) / z

Julia

function code(x, y, z)
	return Float64(Float64(x * y) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * y) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]

Initial Program: 92.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (/ (* x y) z))

C

double code(double x, double y, double z) {
	return (x * y) / 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 = (x * y) / z
end function

Java

public static double code(double x, double y, double z) {
	return (x * y) / z;
}

Python

def code(x, y, z):
	return (x * y) / z

Julia

function code(x, y, z)
	return Float64(Float64(x * y) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * y) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]

Alternative 1: 93.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-87}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -2e-87) (/ (* x y) z) (* y (/ x z))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -2e-87) {
		tmp = (x * y) / z;
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 <= (-2d-87)) then
        tmp = (x * y) / z
    else
        tmp = y * (x / z)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2e-87) {
		tmp = (x * y) / z;
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if x <= -2e-87:
		tmp = (x * y) / z
	else:
		tmp = y * (x / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (x <= -2e-87)
		tmp = Float64(Float64(x * y) / z);
	else
		tmp = Float64(y * Float64(x / z));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2e-87)
		tmp = (x * y) / z;
	else
		tmp = y * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[x, -2e-87], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.00000000000000004e-87

   1. Initial program 96.3%

      \[\frac{x \cdot y}{z} \]

   if -2.00000000000000004e-87 < x

   1. Initial program 90.9%

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

      1. associate-/l* 89.9%

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

   3. Simplified 89.9%

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

      1. associate-/r/ 91.4%

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

   5. Applied egg-rr 91.4%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-87}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 2: 92.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-288}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 4.5e-288) (* x (/ y z)) (* y (/ x z))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.5e-288) {
		tmp = x * (y / z);
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 <= 4.5d-288) then
        tmp = x * (y / z)
    else
        tmp = y * (x / z)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.5e-288) {
		tmp = x * (y / z);
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if y <= 4.5e-288:
		tmp = x * (y / z)
	else:
		tmp = y * (x / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (y <= 4.5e-288)
		tmp = Float64(x * Float64(y / z));
	else
		tmp = Float64(y * Float64(x / z));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.5e-288)
		tmp = x * (y / z);
	else
		tmp = y * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 4.5e-288], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.5000000000000002e-288

   1. Initial program 94.0%

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

      1. associate-*r/ 90.7%

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

      2. *-commutative 90.7%

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

   3. Simplified 90.7%

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

   if 4.5000000000000002e-288 < y

   1. Initial program 91.2%

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

      1. associate-/l* 90.1%

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

   3. Simplified 90.1%

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

      1. associate-/r/ 94.1%

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

   5. Applied egg-rr 94.1%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-288}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 3: 91.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ y \cdot \frac{x}{z} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (* y (/ x z)))

C

assert(x < y);
double code(double x, double y, double z) {
	return y * (x / z);
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 / z)
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	return y * (x / z);
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	return y * (x / z)

Julia

x, y = sort([x, y])
function code(x, y, z)
	return Float64(y * Float64(x / z))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y, z)
	tmp = y * (x / z);
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.6%

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

   1. associate-/l* 90.4%

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

3. Simplified 90.4%

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

   1. associate-/r/ 92.2%

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

5. Applied egg-rr 92.2%

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

6. Final simplification 92.2%

   \[\leadsto y \cdot \frac{x}{z} \]

Developer target: 92.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z < -4.262230790519429 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z < 1.7042130660650472 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (< z -4.262230790519429e-138)
   (/ (* x y) z)
   (if (< z 1.7042130660650472e-164) (/ x (/ z y)) (* (/ x z) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z < -4.262230790519429e-138) {
		tmp = (x * y) / z;
	} else if (z < 1.7042130660650472e-164) {
		tmp = x / (z / y);
	} else {
		tmp = (x / z) * y;
	}
	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 (z < (-4.262230790519429d-138)) then
        tmp = (x * y) / z
    else if (z < 1.7042130660650472d-164) then
        tmp = x / (z / y)
    else
        tmp = (x / z) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z < -4.262230790519429e-138) {
		tmp = (x * y) / z;
	} else if (z < 1.7042130660650472e-164) {
		tmp = x / (z / y);
	} else {
		tmp = (x / z) * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z < -4.262230790519429e-138:
		tmp = (x * y) / z
	elif z < 1.7042130660650472e-164:
		tmp = x / (z / y)
	else:
		tmp = (x / z) * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z < -4.262230790519429e-138)
		tmp = Float64(Float64(x * y) / z);
	elseif (z < 1.7042130660650472e-164)
		tmp = Float64(x / Float64(z / y));
	else
		tmp = Float64(Float64(x / z) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z < -4.262230790519429e-138)
		tmp = (x * y) / z;
	elseif (z < 1.7042130660650472e-164)
		tmp = x / (z / y);
	else
		tmp = (x / z) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Less[z, -4.262230790519429e-138], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], If[Less[z, 1.7042130660650472e-164], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]]]

Reproduce

herbie shell --seed 2023279 
(FPCore (x y z)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -4.262230790519429e-138) (/ (* x y) z) (if (< z 1.7042130660650472e-164) (/ x (/ z y)) (* (/ x z) y)))

  (/ (* x y) z))