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

Average Error: 6.3 → 0.5

Time: 3.1s

Precision: binary64

Cost: 1361

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+305}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-288} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{-314}\right) \land x \cdot y \leq 10^{+127}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) -1e+305)
   (/ y (/ z x))
   (if (or (<= (* x y) -1e-288)
           (and (not (<= (* x y) 5e-314)) (<= (* x y) 1e+127)))
     (/ (* x y) z)
     (* 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) <= -1e+305) {
		tmp = y / (z / x);
	} else if (((x * y) <= -1e-288) || (!((x * y) <= 5e-314) && ((x * y) <= 1e+127))) {
		tmp = (x * y) / z;
	} else {
		tmp = x * (y / z);
	}
	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) <= (-1d+305)) then
        tmp = y / (z / x)
    else if (((x * y) <= (-1d-288)) .or. (.not. ((x * y) <= 5d-314)) .and. ((x * y) <= 1d+127)) then
        tmp = (x * y) / z
    else
        tmp = x * (y / z)
    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) <= -1e+305) {
		tmp = y / (z / x);
	} else if (((x * y) <= -1e-288) || (!((x * y) <= 5e-314) && ((x * y) <= 1e+127))) {
		tmp = (x * y) / z;
	} 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) <= -1e+305:
		tmp = y / (z / x)
	elif ((x * y) <= -1e-288) or (not ((x * y) <= 5e-314) and ((x * y) <= 1e+127)):
		tmp = (x * y) / z
	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) <= -1e+305)
		tmp = Float64(y / Float64(z / x));
	elseif ((Float64(x * y) <= -1e-288) || (!(Float64(x * y) <= 5e-314) && (Float64(x * y) <= 1e+127)))
		tmp = Float64(Float64(x * y) / z);
	else
		tmp = Float64(x * Float64(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) <= -1e+305)
		tmp = y / (z / x);
	elseif (((x * y) <= -1e-288) || (~(((x * y) <= 5e-314)) && ((x * y) <= 1e+127)))
		tmp = (x * y) / z;
	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[LessEqual[N[(x * y), $MachinePrecision], -1e+305], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(x * y), $MachinePrecision], -1e-288], And[N[Not[LessEqual[N[(x * y), $MachinePrecision], 5e-314]], $MachinePrecision], LessEqual[N[(x * y), $MachinePrecision], 1e+127]]], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]]]

Target

Original	6.3
Target	6.4
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 3 regimes

2. if (*.f64 x y) < -9.9999999999999994e304

   1. Initial program 62.4

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

   2. Simplified 0.3

      \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]
      Proof
      (*.f64 (/.f64 x z) y): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 x y) z)): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 0.2

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

   if -9.9999999999999994e304 < (*.f64 x y) < -1.00000000000000006e-288 or 4.99999999982e-314 < (*.f64 x y) < 9.99999999999999955e126

   1. Initial program 0.2

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

   if -1.00000000000000006e-288 < (*.f64 x y) < 4.99999999982e-314 or 9.99999999999999955e126 < (*.f64 x y)

   1. Initial program 17.0

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

   2. Simplified 1.3

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]
      Proof
      (*.f64 (/.f64 x z) y): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 x y) z)): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+305}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-288} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{-314}\right) \land x \cdot y \leq 10^{+127}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.3
Cost	850

\[\begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{-147} \lor \neg \left(z \leq -1.7 \cdot 10^{-275} \lor \neg \left(z \leq 3.05 \cdot 10^{-112}\right) \land z \leq 1.8 \cdot 10^{+110}\right):\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 2
Error	6.2
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{-168} \lor \neg \left(y \leq 7.5 \cdot 10^{+161}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 3
Error	6.3
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq 8.6 \cdot 10^{-178} \lor \neg \left(y \leq 7.4 \cdot 10^{+161}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 4
Error	6.2
Cost	320

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

Reproduce

herbie shell --seed 2022340 
(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))