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

Percentage Accurate: 92.2% → 96.9%

Time: 2.9s

Alternatives: 4

Speedup: 0.5×

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: 92.2% 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: 96.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{-257}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+293}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x y) z)))
   (if (<= t_0 -4e-257)
     t_0
     (if (<= t_0 0.0) (* y (/ x z)) (if (<= t_0 5e+293) t_0 (* x (/ y z)))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double t_0 = (x * y) / z;
	double tmp;
	if (t_0 <= -4e-257) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = y * (x / z);
	} else if (t_0 <= 5e+293) {
		tmp = t_0;
	} else {
		tmp = x * (y / 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) :: t_0
    real(8) :: tmp
    t_0 = (x * y) / z
    if (t_0 <= (-4d-257)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = y * (x / z)
    else if (t_0 <= 5d+293) then
        tmp = t_0
    else
        tmp = x * (y / z)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double t_0 = (x * y) / z;
	double tmp;
	if (t_0 <= -4e-257) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = y * (x / z);
	} else if (t_0 <= 5e+293) {
		tmp = t_0;
	} else {
		tmp = x * (y / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	t_0 = (x * y) / z
	tmp = 0
	if t_0 <= -4e-257:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = y * (x / z)
	elif t_0 <= 5e+293:
		tmp = t_0
	else:
		tmp = x * (y / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	t_0 = Float64(Float64(x * y) / z)
	tmp = 0.0
	if (t_0 <= -4e-257)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(y * Float64(x / z));
	elseif (t_0 <= 5e+293)
		tmp = t_0;
	else
		tmp = Float64(x * Float64(y / z));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	t_0 = (x * y) / z;
	tmp = 0.0;
	if (t_0 <= -4e-257)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = y * (x / z);
	elseif (t_0 <= 5e+293)
		tmp = t_0;
	else
		tmp = x * (y / 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_] := Block[{t$95$0 = N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-257], t$95$0, If[LessEqual[t$95$0, 0.0], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+293], t$95$0, N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x y) z) < -3.9999999999999999e-257 or -0.0 < (/.f64 (*.f64 x y) z) < 5.00000000000000033e293

   1. Initial program 97.6%

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

   if -3.9999999999999999e-257 < (/.f64 (*.f64 x y) z) < -0.0

   1. Initial program 88.2%

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

      1. associate-*l/ 99.6%

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

   3. Simplified 99.6%

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

   if 5.00000000000000033e293 < (/.f64 (*.f64 x y) z)

   1. Initial program 70.8%

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

      1. associate-*r/ 96.5%

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

   3. Simplified 96.5%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{z} \leq -4 \cdot 10^{-257}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \leq 0:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternative 2: 92.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.78 \cdot 10^{-261} \lor \neg \left(y \leq 1.35 \cdot 10^{+141}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.78e-261) (not (<= y 1.35e+141)))
   (* y (/ x z))
   (* x (/ y z))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.78e-261) || !(y <= 1.35e+141)) {
		tmp = y * (x / z);
	} else {
		tmp = x * (y / 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 <= (-1.78d-261)) .or. (.not. (y <= 1.35d+141))) then
        tmp = y * (x / z)
    else
        tmp = x * (y / 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 <= -1.78e-261) || !(y <= 1.35e+141)) {
		tmp = y * (x / z);
	} else {
		tmp = x * (y / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (y <= -1.78e-261) or not (y <= 1.35e+141):
		tmp = y * (x / z)
	else:
		tmp = x * (y / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.78e-261) || !(y <= 1.35e+141))
		tmp = Float64(y * Float64(x / z));
	else
		tmp = Float64(x * Float64(y / z));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.78e-261) || ~((y <= 1.35e+141)))
		tmp = y * (x / z);
	else
		tmp = x * (y / 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[Or[LessEqual[y, -1.78e-261], N[Not[LessEqual[y, 1.35e+141]], $MachinePrecision]], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.77999999999999995e-261 or 1.35e141 < y

   1. Initial program 92.1%

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

      1. associate-*l/ 91.2%

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

   3. Simplified 91.2%

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

   if -1.77999999999999995e-261 < y < 1.35e141

   1. Initial program 93.6%

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

      1. associate-*r/ 92.4%

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

   3. Simplified 92.4%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.78 \cdot 10^{-261} \lor \neg \left(y \leq 1.35 \cdot 10^{+141}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternative 3: 92.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-259}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+140}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

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-259)
   (/ y (/ z x))
   (if (<= y 2.9e+140) (* x (/ y z)) (* y (/ x z)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.5e-259) {
		tmp = y / (z / x);
	} else if (y <= 2.9e+140) {
		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-259)) then
        tmp = y / (z / x)
    else if (y <= 2.9d+140) 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-259) {
		tmp = y / (z / x);
	} else if (y <= 2.9e+140) {
		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-259:
		tmp = y / (z / x)
	elif y <= 2.9e+140:
		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-259)
		tmp = Float64(y / Float64(z / x));
	elseif (y <= 2.9e+140)
		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-259)
		tmp = y / (z / x);
	elseif (y <= 2.9e+140)
		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-259], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.9e+140], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.49999999999999974e-259

   1. Initial program 91.2%

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

      1. associate-*r/ 93.8%

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

   3. Simplified 93.8%

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

      1. *-commutative 93.8%

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

      2. associate-*l/ 91.2%

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

      3. associate-/l* 91.6%

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

   5. Applied egg-rr 91.6%

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

   if -4.49999999999999974e-259 < y < 2.8999999999999999e140

   1. Initial program 93.6%

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

      1. associate-*r/ 92.4%

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

   3. Simplified 92.4%

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

   if 2.8999999999999999e140 < y

   1. Initial program 94.5%

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

      1. associate-*l/ 88.4%

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

   3. Simplified 88.4%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-259}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+140}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 4: 92.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.7%

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

   1. associate-*r/ 92.6%

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

3. Simplified 92.6%

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

4. Final simplification 92.6%

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

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 2023196 
(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))