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

Percentage Accurate: 93.0% → 95.2%

Time: 2.3s

Alternatives: 3

Speedup: 0.3×

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: 93.0% 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: 95.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+20}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-204}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+98}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) -2e+20)
   (* y (/ x z))
   (if (<= (* x y) 5e-204)
     (* x (/ y z))
     (if (<= (* x y) 4e+98) (/ (* x y) z) (/ x (/ z y))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((x * y) <= -2e+20) {
		tmp = y * (x / z);
	} else if ((x * y) <= 5e-204) {
		tmp = x * (y / z);
	} else if ((x * y) <= 4e+98) {
		tmp = (x * y) / z;
	} else {
		tmp = x / (z / y);
	}
	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 * y) <= (-2d+20)) then
        tmp = y * (x / z)
    else if ((x * y) <= 5d-204) then
        tmp = x * (y / z)
    else if ((x * y) <= 4d+98) then
        tmp = (x * y) / z
    else
        tmp = x / (z / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((x * y) <= -2e+20) {
		tmp = y * (x / z);
	} else if ((x * y) <= 5e-204) {
		tmp = x * (y / z);
	} else if ((x * y) <= 4e+98) {
		tmp = (x * y) / z;
	} else {
		tmp = x / (z / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (x * y) <= -2e+20:
		tmp = y * (x / z)
	elif (x * y) <= 5e-204:
		tmp = x * (y / z)
	elif (x * y) <= 4e+98:
		tmp = (x * y) / z
	else:
		tmp = x / (z / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(x * y) <= -2e+20)
		tmp = Float64(y * Float64(x / z));
	elseif (Float64(x * y) <= 5e-204)
		tmp = Float64(x * Float64(y / z));
	elseif (Float64(x * y) <= 4e+98)
		tmp = Float64(Float64(x * y) / z);
	else
		tmp = Float64(x / Float64(z / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * y) <= -2e+20)
		tmp = y * (x / z);
	elseif ((x * y) <= 5e-204)
		tmp = x * (y / z);
	elseif ((x * y) <= 4e+98)
		tmp = (x * y) / z;
	else
		tmp = x / (z / y);
	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[N[(x * y), $MachinePrecision], -2e+20], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e-204], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 4e+98], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -2e20

   1. Initial program 95.3%

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

      1. associate-*l/ 93.8%

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

   3. Simplified 93.8%

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

   if -2e20 < (*.f64 x y) < 5.0000000000000002e-204

   1. Initial program 85.1%

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

      1. associate-*r/ 97.6%

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

   3. Simplified 97.6%

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

   if 5.0000000000000002e-204 < (*.f64 x y) < 3.99999999999999999e98

   1. Initial program 99.7%

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

   if 3.99999999999999999e98 < (*.f64 x y)

   1. Initial program 87.5%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+20}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-204}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+98}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 2: 92.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-208}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 4.4e-208) (* x (/ y z)) (* y (/ x z))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.4e-208) {
		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 <= 4.4d-208) 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 <= 4.4e-208) {
		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 <= 4.4e-208:
		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 <= 4.4e-208)
		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 <= 4.4e-208)
		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, 4.4e-208], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.4000000000000001e-208

   1. Initial program 91.7%

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

      1. associate-*r/ 94.7%

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

   3. Simplified 94.7%

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

   if 4.4000000000000001e-208 < y

   1. Initial program 91.0%

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

      1. associate-*l/ 94.5%

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

   3. Simplified 94.5%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-208}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 3: 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 91.4%

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

   1. associate-*r/ 95.4%

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

3. Simplified 95.4%

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

4. Final simplification 95.4%

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

Developer target: 92.3% 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 2023200 
(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))