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

Average Error: 6.4 → 0.3

Time: 2.9s

Precision: binary64

Cost: 1362

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty \lor \neg \left(x \cdot y \leq -2 \cdot 10^{-226}\right) \land \left(x \cdot y \leq 2 \cdot 10^{-316} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{+188}\right)\right):\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* x y) (- INFINITY))
         (and (not (<= (* x y) -2e-226))
              (or (<= (* x y) 2e-316) (not (<= (* x y) 5e+188)))))
   (/ y (/ z x))
   (/ (* 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) <= -((double) INFINITY)) || (!((x * y) <= -2e-226) && (((x * y) <= 2e-316) || !((x * y) <= 5e+188)))) {
		tmp = y / (z / x);
	} else {
		tmp = (x * y) / 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) || (!((x * y) <= -2e-226) && (((x * y) <= 2e-316) || !((x * y) <= 5e+188)))) {
		tmp = y / (z / x);
	} 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) <= -math.inf) or (not ((x * y) <= -2e-226) and (((x * y) <= 2e-316) or not ((x * y) <= 5e+188))):
		tmp = y / (z / x)
	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) <= Float64(-Inf)) || (!(Float64(x * y) <= -2e-226) && ((Float64(x * y) <= 2e-316) || !(Float64(x * y) <= 5e+188))))
		tmp = Float64(y / Float64(z / x));
	else
		tmp = Float64(Float64(x * 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) <= -Inf) || (~(((x * y) <= -2e-226)) && (((x * y) <= 2e-316) || ~(((x * y) <= 5e+188)))))
		tmp = y / (z / x);
	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[Or[LessEqual[N[(x * y), $MachinePrecision], (-Infinity)], And[N[Not[LessEqual[N[(x * y), $MachinePrecision], -2e-226]], $MachinePrecision], Or[LessEqual[N[(x * y), $MachinePrecision], 2e-316], N[Not[LessEqual[N[(x * y), $MachinePrecision], 5e+188]], $MachinePrecision]]]], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]]

Target

Original	6.4
Target	6.0
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -inf.0 or -1.99999999999999984e-226 < (*.f64 x y) < 2.000000017e-316 or 5.0000000000000001e188 < (*.f64 x y)

   1. Initial program 22.4

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

   2. Simplified 0.6

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

      [Start] 22.4	\[ \frac{x \cdot y}{z} \]
      associate-*l/ [<=] 0.6	\[ \color{blue}{\frac{x}{z} \cdot y} \]

   3. Applied egg-rr 0.6

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

   if -inf.0 < (*.f64 x y) < -1.99999999999999984e-226 or 2.000000017e-316 < (*.f64 x y) < 5.0000000000000001e188

   1. Initial program 0.3

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty \lor \neg \left(x \cdot y \leq -2 \cdot 10^{-226}\right) \land \left(x \cdot y \leq 2 \cdot 10^{-316} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{+188}\right)\right):\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.0
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-235} \lor \neg \left(y \leq 6.7 \cdot 10^{-146}\right):\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 2
Error	5.8
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-235}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 2.55 \cdot 10^{-145}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 3
Error	5.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+173}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 4
Error	6.1
Cost	320

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

Reproduce

herbie shell --seed 2023054 
(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))