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

Average Error: 6.3 → 0.9

Time: 2.3s

Precision: binary64

\[[x, y] = \mathsf{sort}([x, y]) \\]

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{\frac{z}{y}}\\ t_1 := \frac{x \cdot y}{z}\\ \mathbf{if}\;x \cdot y \leq -5.139843644639862 \cdot 10^{+134}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq -4.759939683140499 \cdot 10^{-100}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 5.634535459507377 \cdot 10^{-233}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 2.557453590764515 \cdot 10^{+208}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ x (/ z y))) (t_1 (/ (* x y) z)))
   (if (<= (* x y) -5.139843644639862e+134)
     (* y (/ x z))
     (if (<= (* x y) -4.759939683140499e-100)
       t_1
       (if (<= (* x y) 5.634535459507377e-233)
         t_0
         (if (<= (* x y) 2.557453590764515e+208) t_1 t_0))))))

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 / (z / y);
	double t_1 = (x * y) / z;
	double tmp;
	if ((x * y) <= -5.139843644639862e+134) {
		tmp = y * (x / z);
	} else if ((x * y) <= -4.759939683140499e-100) {
		tmp = t_1;
	} else if ((x * y) <= 5.634535459507377e-233) {
		tmp = t_0;
	} else if ((x * y) <= 2.557453590764515e+208) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	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 / (z / y)
    t_1 = (x * y) / z
    if ((x * y) <= (-5.139843644639862d+134)) then
        tmp = y * (x / z)
    else if ((x * y) <= (-4.759939683140499d-100)) then
        tmp = t_1
    else if ((x * y) <= 5.634535459507377d-233) then
        tmp = t_0
    else if ((x * y) <= 2.557453590764515d+208) then
        tmp = t_1
    else
        tmp = t_0
    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 / (z / y);
	double t_1 = (x * y) / z;
	double tmp;
	if ((x * y) <= -5.139843644639862e+134) {
		tmp = y * (x / z);
	} else if ((x * y) <= -4.759939683140499e-100) {
		tmp = t_1;
	} else if ((x * y) <= 5.634535459507377e-233) {
		tmp = t_0;
	} else if ((x * y) <= 2.557453590764515e+208) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = x / (z / y)
	t_1 = (x * y) / z
	tmp = 0
	if (x * y) <= -5.139843644639862e+134:
		tmp = y * (x / z)
	elif (x * y) <= -4.759939683140499e-100:
		tmp = t_1
	elif (x * y) <= 5.634535459507377e-233:
		tmp = t_0
	elif (x * y) <= 2.557453590764515e+208:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	return Float64(Float64(x * y) / z)
end

↓

function code(x, y, z)
	t_0 = Float64(x / Float64(z / y))
	t_1 = Float64(Float64(x * y) / z)
	tmp = 0.0
	if (Float64(x * y) <= -5.139843644639862e+134)
		tmp = Float64(y * Float64(x / z));
	elseif (Float64(x * y) <= -4.759939683140499e-100)
		tmp = t_1;
	elseif (Float64(x * y) <= 5.634535459507377e-233)
		tmp = t_0;
	elseif (Float64(x * y) <= 2.557453590764515e+208)
		tmp = t_1;
	else
		tmp = t_0;
	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 / (z / y);
	t_1 = (x * y) / z;
	tmp = 0.0;
	if ((x * y) <= -5.139843644639862e+134)
		tmp = y * (x / z);
	elseif ((x * y) <= -4.759939683140499e-100)
		tmp = t_1;
	elseif ((x * y) <= 5.634535459507377e-233)
		tmp = t_0;
	elseif ((x * y) <= 2.557453590764515e+208)
		tmp = t_1;
	else
		tmp = t_0;
	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[(x / N[(z / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -5.139843644639862e+134], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -4.759939683140499e-100], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 5.634535459507377e-233], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], 2.557453590764515e+208], t$95$1, t$95$0]]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.3
Target	6.2
Herbie	0.9

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -5.1398436446398618e134

   1. Initial program 16.2

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

   2. Applied associate-/l*_binary64 3.4

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

   3. Applied associate-/r/_binary64 3.1

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

   if -5.1398436446398618e134 < (*.f64 x y) < -4.7599396831404991e-100 or 5.63453545950737703e-233 < (*.f64 x y) < 2.5574535907645148e208

   1. Initial program 0.2

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

   if -4.7599396831404991e-100 < (*.f64 x y) < 5.63453545950737703e-233 or 2.5574535907645148e208 < (*.f64 x y)

   1. Initial program 12.0

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

   2. Applied associate-/l*_binary64 1.3

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5.139843644639862 \cdot 10^{+134}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq -4.759939683140499 \cdot 10^{-100}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 5.634535459507377 \cdot 10^{-233}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq 2.557453590764515 \cdot 10^{+208}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(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))