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

Average Error: 6.3 → 1.9

Time: 3.1s

Precision: binary64

Cost: 1100

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-176}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-167}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x y) z)))
   (if (<= (* x y) (- INFINITY))
     (/ y (/ z x))
     (if (<= (* x y) -5e-176)
       t_0
       (if (<= (* x y) 2e-167) (* y (/ x z)) t_0)))))

C

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

↓

double code(double x, double y, double z) {
	double t_0 = (x * y) / z;
	double tmp;
	if ((x * y) <= -((double) INFINITY)) {
		tmp = y / (z / x);
	} else if ((x * y) <= -5e-176) {
		tmp = t_0;
	} else if ((x * y) <= 2e-167) {
		tmp = y * (x / z);
	} else {
		tmp = t_0;
	}
	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 t_0 = (x * y) / z;
	double tmp;
	if ((x * y) <= -Double.POSITIVE_INFINITY) {
		tmp = y / (z / x);
	} else if ((x * y) <= -5e-176) {
		tmp = t_0;
	} else if ((x * y) <= 2e-167) {
		tmp = y * (x / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = (x * y) / z
	tmp = 0
	if (x * y) <= -math.inf:
		tmp = y / (z / x)
	elif (x * y) <= -5e-176:
		tmp = t_0
	elif (x * y) <= 2e-167:
		tmp = y * (x / z)
	else:
		tmp = t_0
	return tmp

Julia

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

↓

function code(x, y, z)
	t_0 = Float64(Float64(x * y) / z)
	tmp = 0.0
	if (Float64(x * y) <= Float64(-Inf))
		tmp = Float64(y / Float64(z / x));
	elseif (Float64(x * y) <= -5e-176)
		tmp = t_0;
	elseif (Float64(x * y) <= 2e-167)
		tmp = Float64(y * Float64(x / z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z)
	t_0 = (x * y) / z;
	tmp = 0.0;
	if ((x * y) <= -Inf)
		tmp = y / (z / x);
	elseif ((x * y) <= -5e-176)
		tmp = t_0;
	elseif ((x * y) <= 2e-167)
		tmp = y * (x / z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], (-Infinity)], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -5e-176], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], 2e-167], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Target

Original	6.3
Target	6.2
Herbie	1.9

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 64.0

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

   2. Applied egg-rr 64.0

      \[\leadsto \color{blue}{\frac{1}{z} \cdot \left(x \cdot y\right)} \]

   3. Applied egg-rr 0.2

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

   if -inf.0 < (*.f64 x y) < -5e-176 or 2e-167 < (*.f64 x y)

   1. Initial program 2.5

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

   if -5e-176 < (*.f64 x y) < 2e-167

   1. Initial program 9.4

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

   2. Applied egg-rr 0.8

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

4. Final simplification 1.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-176}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-167}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.0
Cost	584

\[\begin{array}{l} t_0 := \frac{x}{\frac{z}{y}}\\ \mathbf{if}\;y \leq 3.3681833822607847 \cdot 10^{-299}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 10^{+200}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	6.0
Cost	584

\[\begin{array}{l} t_0 := \frac{x}{\frac{z}{y}}\\ \mathbf{if}\;y \leq 3.3681833822607847 \cdot 10^{-299}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 10^{+200}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	5.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq 3.3681833822607847 \cdot 10^{-299}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 10^{+200}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 4
Error	6.0
Cost	320

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

Reproduce

herbie shell --seed 2022295 
(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))