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

Percentage Accurate: 92.5% → 99.7%

Time: 2.3s

Alternatives: 2

Speedup: 0.2×

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: 92.5% 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: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := x \cdot \frac{y}{z}\\ t_1 := \frac{x \cdot y}{z}\\ \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+289}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-285}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-219}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+307}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

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))) (t_1 (/ (* x y) z)))
   (if (<= (* x y) -1e+289)
     t_0
     (if (<= (* x y) -2e-285)
       t_1
       (if (<= (* x y) 5e-219)
         (/ x (/ z y))
         (if (<= (* x y) 2e+307) t_1 t_0))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double t_0 = x * (y / z);
	double t_1 = (x * y) / z;
	double tmp;
	if ((x * y) <= -1e+289) {
		tmp = t_0;
	} else if ((x * y) <= -2e-285) {
		tmp = t_1;
	} else if ((x * y) <= 5e-219) {
		tmp = x / (z / y);
	} else if ((x * y) <= 2e+307) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = x * (y / z)
    t_1 = (x * y) / z
    if ((x * y) <= (-1d+289)) then
        tmp = t_0
    else if ((x * y) <= (-2d-285)) then
        tmp = t_1
    else if ((x * y) <= 5d-219) then
        tmp = x / (z / y)
    else if ((x * y) <= 2d+307) then
        tmp = t_1
    else
        tmp = t_0
    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 t_1 = (x * y) / z;
	double tmp;
	if ((x * y) <= -1e+289) {
		tmp = t_0;
	} else if ((x * y) <= -2e-285) {
		tmp = t_1;
	} else if ((x * y) <= 5e-219) {
		tmp = x / (z / y);
	} else if ((x * y) <= 2e+307) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	t_0 = x * (y / z)
	t_1 = (x * y) / z
	tmp = 0
	if (x * y) <= -1e+289:
		tmp = t_0
	elif (x * y) <= -2e-285:
		tmp = t_1
	elif (x * y) <= 5e-219:
		tmp = x / (z / y)
	elif (x * y) <= 2e+307:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	t_0 = Float64(x * Float64(y / z))
	t_1 = Float64(Float64(x * y) / z)
	tmp = 0.0
	if (Float64(x * y) <= -1e+289)
		tmp = t_0;
	elseif (Float64(x * y) <= -2e-285)
		tmp = t_1;
	elseif (Float64(x * y) <= 5e-219)
		tmp = Float64(x / Float64(z / y));
	elseif (Float64(x * y) <= 2e+307)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	t_0 = x * (y / z);
	t_1 = (x * y) / z;
	tmp = 0.0;
	if ((x * y) <= -1e+289)
		tmp = t_0;
	elseif ((x * y) <= -2e-285)
		tmp = t_1;
	elseif ((x * y) <= 5e-219)
		tmp = x / (z / y);
	elseif ((x * y) <= 2e+307)
		tmp = t_1;
	else
		tmp = t_0;
	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[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -1e+289], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], -2e-285], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 5e-219], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e+307], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1.0000000000000001e289 or 1.99999999999999997e307 < (*.f64 x y)

   1. Initial program 66.4%

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

      1. associate-*r/ 99.9%

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

      2. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   if -1.0000000000000001e289 < (*.f64 x y) < -2.00000000000000015e-285 or 5.0000000000000002e-219 < (*.f64 x y) < 1.99999999999999997e307

   1. Initial program 99.7%

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

   if -2.00000000000000015e-285 < (*.f64 x y) < 5.0000000000000002e-219

   1. Initial program 76.0%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+289}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-285}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-219}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternative 2: 92.3% 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 91.0%

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

   1. associate-*r/ 91.3%

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

   2. *-commutative 91.3%

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

3. Simplified 91.3%

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

4. Final simplification 91.3%

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

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 2023309 
(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))