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

Average Error: 6.1 → 0.9

Time: 3.2s

Precision: binary64

Cost: 1872

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;t_0 \leq -1 \cdot 10^{-298}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;t_0 \leq 10^{+260}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x y) z)))
   (if (<= t_0 (- INFINITY))
     (* x (/ y z))
     (if (<= t_0 -1e-298)
       t_0
       (if (<= t_0 0.0)
         (* y (/ x z))
         (if (<= t_0 1e+260) t_0 (/ y (/ z x))))))))

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 (t_0 <= -((double) INFINITY)) {
		tmp = x * (y / z);
	} else if (t_0 <= -1e-298) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = y * (x / z);
	} else if (t_0 <= 1e+260) {
		tmp = t_0;
	} else {
		tmp = y / (z / x);
	}
	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 (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = x * (y / z);
	} else if (t_0 <= -1e-298) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = y * (x / z);
	} else if (t_0 <= 1e+260) {
		tmp = t_0;
	} else {
		tmp = y / (z / x);
	}
	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 t_0 <= -math.inf:
		tmp = x * (y / z)
	elif t_0 <= -1e-298:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = y * (x / z)
	elif t_0 <= 1e+260:
		tmp = t_0
	else:
		tmp = y / (z / x)
	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 (t_0 <= Float64(-Inf))
		tmp = Float64(x * Float64(y / z));
	elseif (t_0 <= -1e-298)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(y * Float64(x / z));
	elseif (t_0 <= 1e+260)
		tmp = t_0;
	else
		tmp = Float64(y / Float64(z / x));
	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 (t_0 <= -Inf)
		tmp = x * (y / z);
	elseif (t_0 <= -1e-298)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = y * (x / z);
	elseif (t_0 <= 1e+260)
		tmp = t_0;
	else
		tmp = y / (z / x);
	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[t$95$0, (-Infinity)], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -1e-298], t$95$0, If[LessEqual[t$95$0, 0.0], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+260], t$95$0, N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	6.1
Target	6.2
Herbie	0.9

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

Derivation

1. Split input into 4 regimes

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

   1. Initial program 64.0

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

   2. Simplified 0.3

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

      [Start] 64.0	\[ \frac{x \cdot y}{z} \]
      associate-*r/ [<=] 0.3	\[ \color{blue}{x \cdot \frac{y}{z}} \]

   if -inf.0 < (/.f64 (*.f64 x y) z) < -9.99999999999999912e-299 or -0.0 < (/.f64 (*.f64 x y) z) < 1.00000000000000007e260

   1. Initial program 0.5

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

   if -9.99999999999999912e-299 < (/.f64 (*.f64 x y) z) < -0.0

   1. Initial program 11.5

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

   2. Simplified 0.5

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

      [Start] 11.5	\[ \frac{x \cdot y}{z} \]
      associate-*l/ [<=] 0.5	\[ \color{blue}{\frac{x}{z} \cdot y} \]

   if 1.00000000000000007e260 < (/.f64 (*.f64 x y) z)

   1. Initial program 35.6

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

   2. Simplified 9.7

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

      [Start] 35.6	\[ \frac{x \cdot y}{z} \]
      associate-*l/ [<=] 9.7	\[ \color{blue}{\frac{x}{z} \cdot y} \]

   3. Applied egg-rr 8.9

      \[\leadsto \color{blue}{\frac{y}{\frac{z}{x}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{z} \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \leq -1 \cdot 10^{-298}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \leq 0:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \leq 10^{+260}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.3
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-174} \lor \neg \left(y \leq 7.2 \cdot 10^{-159}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternative 2
Error	6.5
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-172} \lor \neg \left(y \leq 4.8 \cdot 10^{-155}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 3
Error	6.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{-177}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-156}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 4
Error	6.3
Cost	320

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

Reproduce

herbie shell --seed 2023083 
(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))