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

Average Error: 6.2 → 2.4

Time: 3.8s

Precision: binary64

Cost: 1100

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ t_1 := y \cdot \frac{x}{z}\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-138}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-180}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 10^{+217}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x y) z)) (t_1 (* y (/ x z))))
   (if (<= (* x y) -2e-138)
     t_0
     (if (<= (* x y) 5e-180) t_1 (if (<= (* x y) 1e+217) t_0 t_1)))))

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 t_1 = y * (x / z);
	double tmp;
	if ((x * y) <= -2e-138) {
		tmp = t_0;
	} else if ((x * y) <= 5e-180) {
		tmp = t_1;
	} else if ((x * y) <= 1e+217) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x * y) / z
    t_1 = y * (x / z)
    if ((x * y) <= (-2d-138)) then
        tmp = t_0
    else if ((x * y) <= 5d-180) then
        tmp = t_1
    else if ((x * y) <= 1d+217) then
        tmp = t_0
    else
        tmp = t_1
    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 t_0 = (x * y) / z;
	double t_1 = y * (x / z);
	double tmp;
	if ((x * y) <= -2e-138) {
		tmp = t_0;
	} else if ((x * y) <= 5e-180) {
		tmp = t_1;
	} else if ((x * y) <= 1e+217) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = (x * y) / z
	t_1 = y * (x / z)
	tmp = 0
	if (x * y) <= -2e-138:
		tmp = t_0
	elif (x * y) <= 5e-180:
		tmp = t_1
	elif (x * y) <= 1e+217:
		tmp = t_0
	else:
		tmp = t_1
	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)
	t_1 = Float64(y * Float64(x / z))
	tmp = 0.0
	if (Float64(x * y) <= -2e-138)
		tmp = t_0;
	elseif (Float64(x * y) <= 5e-180)
		tmp = t_1;
	elseif (Float64(x * y) <= 1e+217)
		tmp = t_0;
	else
		tmp = t_1;
	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;
	t_1 = y * (x / z);
	tmp = 0.0;
	if ((x * y) <= -2e-138)
		tmp = t_0;
	elseif ((x * y) <= 5e-180)
		tmp = t_1;
	elseif ((x * y) <= 1e+217)
		tmp = t_0;
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2e-138], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], 5e-180], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 1e+217], t$95$0, t$95$1]]]]]

Target

Original	6.2
Target	6.1
Herbie	2.4

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2.00000000000000013e-138 or 5.0000000000000001e-180 < (*.f64 x y) < 9.9999999999999996e216

   1. Initial program 3.0

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

   if -2.00000000000000013e-138 < (*.f64 x y) < 5.0000000000000001e-180 or 9.9999999999999996e216 < (*.f64 x y)

   1. Initial program 11.1

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

   2. Applied egg-rr 1.4

      \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-180}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq 10^{+217}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.4
Cost	584

\[\begin{array}{l} t_0 := \frac{x}{\frac{z}{y}}\\ \mathbf{if}\;x \leq -1 \cdot 10^{+85}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.3758223994969893 \cdot 10^{-126}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	6.2
Cost	584

\[\begin{array}{l} t_0 := \frac{x}{\frac{z}{y}}\\ \mathbf{if}\;x \leq -1 \cdot 10^{+90}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.3758223994969893 \cdot 10^{-126}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	6.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+90}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \leq 1.3758223994969893 \cdot 10^{-126}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternative 4
Error	6.7
Cost	320

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

Reproduce

herbie shell --seed 2022297 
(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))