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

Average Error: 6.2 → 1.1

Time: 3.1s

Precision: binary64

Cost: 1361

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+115}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;x \cdot y \leq -8 \cdot 10^{-118} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{-268}\right) \land x \cdot y \leq 2 \cdot 10^{+187}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) -4e+115)
   (/ y (/ z x))
   (if (or (<= (* x y) -8e-118)
           (and (not (<= (* x y) 5e-268)) (<= (* x y) 2e+187)))
     (/ (* x y) z)
     (/ 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 tmp;
	if ((x * y) <= -4e+115) {
		tmp = y / (z / x);
	} else if (((x * y) <= -8e-118) || (!((x * y) <= 5e-268) && ((x * y) <= 2e+187))) {
		tmp = (x * y) / z;
	} else {
		tmp = x / (z / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * y) / z
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x * y) <= (-4d+115)) then
        tmp = y / (z / x)
    else if (((x * y) <= (-8d-118)) .or. (.not. ((x * y) <= 5d-268)) .and. ((x * y) <= 2d+187)) then
        tmp = (x * y) / z
    else
        tmp = x / (z / y)
    end if
    code = tmp
end function

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) <= -4e+115) {
		tmp = y / (z / x);
	} else if (((x * y) <= -8e-118) || (!((x * y) <= 5e-268) && ((x * y) <= 2e+187))) {
		tmp = (x * y) / z;
	} else {
		tmp = x / (z / y);
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	tmp = 0
	if (x * y) <= -4e+115:
		tmp = y / (z / x)
	elif ((x * y) <= -8e-118) or (not ((x * y) <= 5e-268) and ((x * y) <= 2e+187)):
		tmp = (x * y) / z
	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)
	tmp = 0.0
	if (Float64(x * y) <= -4e+115)
		tmp = Float64(y / Float64(z / x));
	elseif ((Float64(x * y) <= -8e-118) || (!(Float64(x * y) <= 5e-268) && (Float64(x * y) <= 2e+187)))
		tmp = Float64(Float64(x * y) / z);
	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)
	tmp = 0.0;
	if ((x * y) <= -4e+115)
		tmp = y / (z / x);
	elseif (((x * y) <= -8e-118) || (~(((x * y) <= 5e-268)) && ((x * y) <= 2e+187)))
		tmp = (x * y) / z;
	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_] := If[LessEqual[N[(x * y), $MachinePrecision], -4e+115], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(x * y), $MachinePrecision], -8e-118], And[N[Not[LessEqual[N[(x * y), $MachinePrecision], 5e-268]], $MachinePrecision], LessEqual[N[(x * y), $MachinePrecision], 2e+187]]], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision]]]

Target

Original	6.2
Target	6.7
Herbie	1.1

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -4.0000000000000001e115

   1. Initial program 15.6

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

   2. Simplified 3.4

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

      [Start] 15.6	\[ \frac{x \cdot y}{z} \]
      associate-*l/ [<=] 3.4	\[ \color{blue}{\frac{x}{z} \cdot y} \]

   3. Applied egg-rr 3.3

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

   if -4.0000000000000001e115 < (*.f64 x y) < -7.99999999999999988e-118 or 4.9999999999999999e-268 < (*.f64 x y) < 1.99999999999999981e187

   1. Initial program 0.2

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

   if -7.99999999999999988e-118 < (*.f64 x y) < 4.9999999999999999e-268 or 1.99999999999999981e187 < (*.f64 x y)

   1. Initial program 12.3

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

   2. Simplified 1.6

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

      [Start] 12.3	\[ \frac{x \cdot y}{z} \]
      associate-/l* [=>] 1.6	\[ \color{blue}{\frac{x}{\frac{z}{y}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+115}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;x \cdot y \leq -8 \cdot 10^{-118} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{-268}\right) \land x \cdot y \leq 2 \cdot 10^{+187}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.7
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+145} \lor \neg \left(z \leq 1.55 \cdot 10^{-146}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternative 2
Error	6.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+157}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 10^{-146}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 3
Error	6.7
Cost	320

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

Reproduce

herbie shell --seed 2023187 
(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))