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

Average Error: 9.53% → 1.43%

Time: 3.2s

Precision: binary64

Cost: 1361

\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+289}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-207} \lor \neg \left(x \cdot y \leq 2 \cdot 10^{-164}\right) \land x \cdot y \leq 1.45 \cdot 10^{+96}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) -5e+289)
   (* y (/ x z))
   (if (or (<= (* x y) -1e-207)
           (and (not (<= (* x y) 2e-164)) (<= (* x y) 1.45e+96)))
     (/ (* x y) z)
     (* x (/ y z)))))

C

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

↓

double code(double x, double y, double z) {
	double tmp;
	if ((x * y) <= -5e+289) {
		tmp = y * (x / z);
	} else if (((x * y) <= -1e-207) || (!((x * y) <= 2e-164) && ((x * y) <= 1.45e+96))) {
		tmp = (x * y) / z;
	} else {
		tmp = x * (y / z);
	}
	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
    code = (x * y) / z
end 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) <= (-5d+289)) then
        tmp = y * (x / z)
    else if (((x * y) <= (-1d-207)) .or. (.not. ((x * y) <= 2d-164)) .and. ((x * y) <= 1.45d+96)) then
        tmp = (x * y) / z
    else
        tmp = x * (y / z)
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z) {
	double tmp;
	if ((x * y) <= -5e+289) {
		tmp = y * (x / z);
	} else if (((x * y) <= -1e-207) || (!((x * y) <= 2e-164) && ((x * y) <= 1.45e+96))) {
		tmp = (x * y) / z;
	} else {
		tmp = x * (y / z);
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	tmp = 0
	if (x * y) <= -5e+289:
		tmp = y * (x / z)
	elif ((x * y) <= -1e-207) or (not ((x * y) <= 2e-164) and ((x * y) <= 1.45e+96)):
		tmp = (x * y) / z
	else:
		tmp = x * (y / z)
	return tmp

Julia

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

↓

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * y) <= -5e+289)
		tmp = Float64(y * Float64(x / z));
	elseif ((Float64(x * y) <= -1e-207) || (!(Float64(x * y) <= 2e-164) && (Float64(x * y) <= 1.45e+96)))
		tmp = Float64(Float64(x * y) / z);
	else
		tmp = Float64(x * Float64(y / z));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * y) <= -5e+289)
		tmp = y * (x / z);
	elseif (((x * y) <= -1e-207) || (~(((x * y) <= 2e-164)) && ((x * y) <= 1.45e+96)))
		tmp = (x * y) / z;
	else
		tmp = x * (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_] := If[LessEqual[N[(x * y), $MachinePrecision], -5e+289], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(x * y), $MachinePrecision], -1e-207], And[N[Not[LessEqual[N[(x * y), $MachinePrecision], 2e-164]], $MachinePrecision], LessEqual[N[(x * y), $MachinePrecision], 1.45e+96]]], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]]]

Target

Original	9.53%
Target	9.82%
Herbie	1.43%

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -5.00000000000000031e289

   1. Initial program 80.08

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

   2. Simplified 0.41

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

      [Start] 80.08	\[ \frac{x \cdot y}{z} \]
      associate-*l/ [<=] 0.41	\[ \color{blue}{\frac{x}{z} \cdot y} \]

   if -5.00000000000000031e289 < (*.f64 x y) < -9.99999999999999925e-208 or 1.99999999999999992e-164 < (*.f64 x y) < 1.44999999999999989e96

   1. Initial program 0.37

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

   if -9.99999999999999925e-208 < (*.f64 x y) < 1.99999999999999992e-164 or 1.44999999999999989e96 < (*.f64 x y)

   1. Initial program 16.2

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

   2. Simplified 2.95

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

      [Start] 16.2	\[ \frac{x \cdot y}{z} \]
      associate-*r/ [<=] 2.95	\[ \color{blue}{x \cdot \frac{y}{z}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.43

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+289}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-207} \lor \neg \left(x \cdot y \leq 2 \cdot 10^{-164}\right) \land x \cdot y \leq 1.45 \cdot 10^{+96}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.91%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -1.92 \cdot 10^{+115} \lor \neg \left(z \leq 2.1 \cdot 10^{-268}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternative 2
Error	10.28%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{-178}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+149}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 3
Error	9.68%
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+114}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-298}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 4
Error	9.88%
Cost	320

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

Reproduce

herbie shell --seed 2023102 
(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))