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

Average Error: 6.5 → 0.5

Time: 3.0s

Precision: binary64

Cost: 1360

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-231}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 0:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+140}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

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) -1e-231)
       t_0
       (if (<= (* x y) 0.0)
         (* x (/ y z))
         (if (<= (* x y) 5e+140) t_0 (/ x (/ z y))))))))

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) <= -1e-231) {
		tmp = t_0;
	} else if ((x * y) <= 0.0) {
		tmp = x * (y / z);
	} else if ((x * y) <= 5e+140) {
		tmp = t_0;
	} else {
		tmp = x / (z / y);
	}
	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) <= -1e-231) {
		tmp = t_0;
	} else if ((x * y) <= 0.0) {
		tmp = x * (y / z);
	} else if ((x * y) <= 5e+140) {
		tmp = t_0;
	} else {
		tmp = x / (z / y);
	}
	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) <= -1e-231:
		tmp = t_0
	elif (x * y) <= 0.0:
		tmp = x * (y / z)
	elif (x * y) <= 5e+140:
		tmp = t_0
	else:
		tmp = x / (z / y)
	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) <= -1e-231)
		tmp = t_0;
	elseif (Float64(x * y) <= 0.0)
		tmp = Float64(x * Float64(y / z));
	elseif (Float64(x * y) <= 5e+140)
		tmp = t_0;
	else
		tmp = Float64(x / Float64(z / y));
	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) <= -1e-231)
		tmp = t_0;
	elseif ((x * y) <= 0.0)
		tmp = x * (y / z);
	elseif ((x * y) <= 5e+140)
		tmp = t_0;
	else
		tmp = x / (z / y);
	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], -1e-231], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], 0.0], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+140], t$95$0, N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	6.5
Target	6.3
Herbie	0.5

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

Derivation

1. Split input into 4 regimes

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

   1. Initial program 64.0

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

   2. Simplified 0.3

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

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

   3. Applied egg-rr 0.2

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

   if -inf.0 < (*.f64 x y) < -9.9999999999999999e-232 or 0.0 < (*.f64 x y) < 5.00000000000000008e140

   1. Initial program 0.4

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

   if -9.9999999999999999e-232 < (*.f64 x y) < 0.0

   1. Initial program 15.7

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

   2. Simplified 0.2

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

      [Start] 15.7	\[ \frac{x \cdot y}{z} \]
      associate-*r/ [<=] 0.2	\[ \color{blue}{x \cdot \frac{y}{z}} \]

   if 5.00000000000000008e140 < (*.f64 x y)

   1. Initial program 17.0

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

   2. Simplified 2.4

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

      [Start] 17.0	\[ \frac{x \cdot y}{z} \]
      associate-/l* [=>] 2.4	\[ \color{blue}{\frac{x}{\frac{z}{y}}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-231}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 0:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+140}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternatives

Alternative 1
Error	5.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{-258}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 2
Error	6.0
Cost	320

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

Reproduce

herbie shell --seed 2023060 
(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))