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

Average Error: 9.75% → 3.27%

Time: 3.1s

Precision: binary64

Cost: 1101

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{-144} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{+158}\right):\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) -5e-309)
   (* (* x y) (/ 1.0 z))
   (if (or (<= (* x y) 4e-144) (not (<= (* x y) 5e+158)))
     (/ y (/ z x))
     (/ (* 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) <= -5e-309) {
		tmp = (x * y) * (1.0 / z);
	} else if (((x * y) <= 4e-144) || !((x * y) <= 5e+158)) {
		tmp = y / (z / x);
	} 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) <= (-5d-309)) then
        tmp = (x * y) * (1.0d0 / z)
    else if (((x * y) <= 4d-144) .or. (.not. ((x * y) <= 5d+158))) then
        tmp = y / (z / x)
    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) <= -5e-309) {
		tmp = (x * y) * (1.0 / z);
	} else if (((x * y) <= 4e-144) || !((x * y) <= 5e+158)) {
		tmp = y / (z / x);
	} 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) <= -5e-309:
		tmp = (x * y) * (1.0 / z)
	elif ((x * y) <= 4e-144) or not ((x * y) <= 5e+158):
		tmp = y / (z / x)
	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) <= -5e-309)
		tmp = Float64(Float64(x * y) * Float64(1.0 / z));
	elseif ((Float64(x * y) <= 4e-144) || !(Float64(x * y) <= 5e+158))
		tmp = Float64(y / Float64(z / x));
	else
		tmp = Float64(Float64(x * 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) <= -5e-309)
		tmp = (x * y) * (1.0 / z);
	elseif (((x * y) <= 4e-144) || ~(((x * y) <= 5e+158)))
		tmp = y / (z / x);
	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], -5e-309], N[(N[(x * y), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(x * y), $MachinePrecision], 4e-144], N[Not[LessEqual[N[(x * y), $MachinePrecision], 5e+158]], $MachinePrecision]], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]]]

Target

Original	9.75%
Target	9.87%
Herbie	3.27%

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -4.9999999999999995e-309

   1. Initial program 5.96

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

   2. Applied egg-rr 6.08

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

   if -4.9999999999999995e-309 < (*.f64 x y) < 3.9999999999999998e-144 or 4.9999999999999996e158 < (*.f64 x y)

   1. Initial program 21.33

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

   2. Simplified 2.01

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

      [Start] 21.33	\[ \frac{x \cdot y}{z} \]
      associate-*l/ [<=] 2.01	\[ \color{blue}{\frac{x}{z} \cdot y} \]

   3. Applied egg-rr 1.8

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

   if 3.9999999999999998e-144 < (*.f64 x y) < 4.9999999999999996e158

   1. Initial program 0.38

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

4. Final simplification 3.27

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{-144} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{+158}\right):\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	3.13%
Cost	1100

\[\begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-234}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-200}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+158}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 2
Error	8.93%
Cost	850

\[\begin{array}{l} \mathbf{if}\;y \leq -7.3 \cdot 10^{-283} \lor \neg \left(y \leq 5.5 \cdot 10^{-251}\right) \land \left(y \leq 9 \cdot 10^{+164} \lor \neg \left(y \leq 1.3 \cdot 10^{+229}\right)\right):\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 3
Error	9.28%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+237}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-72}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 4
Error	9.45%
Cost	320

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

Reproduce

herbie shell --seed 2023089 
(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))