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

Average Error: 9.98% → 0.92%

Time: 3.6s

Precision: binary64

Cost: 1361

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-121} \lor \neg \left(x \cdot y \leq 10^{-188}\right) \land x \cdot y \leq 4 \cdot 10^{+219}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) (- INFINITY))
   (* x (/ y z))
   (if (or (<= (* x y) -1e-121)
           (and (not (<= (* x y) 1e-188)) (<= (* x y) 4e+219)))
     (/ (* x y) z)
     (* y (/ x 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) <= -((double) INFINITY)) {
		tmp = x * (y / z);
	} else if (((x * y) <= -1e-121) || (!((x * y) <= 1e-188) && ((x * y) <= 4e+219))) {
		tmp = (x * y) / z;
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

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) <= -Double.POSITIVE_INFINITY) {
		tmp = x * (y / z);
	} else if (((x * y) <= -1e-121) || (!((x * y) <= 1e-188) && ((x * y) <= 4e+219))) {
		tmp = (x * y) / z;
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	tmp = 0
	if (x * y) <= -math.inf:
		tmp = x * (y / z)
	elif ((x * y) <= -1e-121) or (not ((x * y) <= 1e-188) and ((x * y) <= 4e+219)):
		tmp = (x * y) / z
	else:
		tmp = y * (x / 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) <= Float64(-Inf))
		tmp = Float64(x * Float64(y / z));
	elseif ((Float64(x * y) <= -1e-121) || (!(Float64(x * y) <= 1e-188) && (Float64(x * y) <= 4e+219)))
		tmp = Float64(Float64(x * y) / z);
	else
		tmp = Float64(y * Float64(x / 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) <= -Inf)
		tmp = x * (y / z);
	elseif (((x * y) <= -1e-121) || (~(((x * y) <= 1e-188)) && ((x * y) <= 4e+219)))
		tmp = (x * y) / z;
	else
		tmp = y * (x / 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], (-Infinity)], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(x * y), $MachinePrecision], -1e-121], And[N[Not[LessEqual[N[(x * y), $MachinePrecision], 1e-188]], $MachinePrecision], LessEqual[N[(x * y), $MachinePrecision], 4e+219]]], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Target

Original	9.98%
Target	10.06%
Herbie	0.92%

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -inf.0

   1. Initial program 100

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

   2. Simplified 0.38

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

      [Start] 100	\[ \frac{x \cdot y}{z} \]
      associate-*r/ [<=] 0.38	\[ \color{blue}{x \cdot \frac{y}{z}} \]

   if -inf.0 < (*.f64 x y) < -9.9999999999999998e-122 or 9.9999999999999995e-189 < (*.f64 x y) < 3.99999999999999986e219

   1. Initial program 0.38

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

   if -9.9999999999999998e-122 < (*.f64 x y) < 9.9999999999999995e-189 or 3.99999999999999986e219 < (*.f64 x y)

   1. Initial program 17.82

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

   2. Simplified 1.74

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

      [Start] 17.82	\[ \frac{x \cdot y}{z} \]
      associate-*l/ [<=] 1.74	\[ \color{blue}{\frac{x}{z} \cdot y} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.92

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-121} \lor \neg \left(x \cdot y \leq 10^{-188}\right) \land x \cdot y \leq 4 \cdot 10^{+219}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.22%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{-201} \lor \neg \left(y \leq 5.8 \cdot 10^{+107}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternative 2
Error	9.27%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-200} \lor \neg \left(y \leq 7.8 \cdot 10^{+110}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 3
Error	9.74%
Cost	320

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

Reproduce

herbie shell --seed 2023103 
(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))