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

Average Error: 6.1 → 1.4

Time: 2.1s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+245}:\\ \;\;\;\;{\left(\frac{\frac{z}{y}}{x}\right)}^{-1}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 0:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+153}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x y) z)))
   (if (<= (* x y) -2e+245)
     (pow (/ (/ z y) x) -1.0)
     (if (<= (* x y) -1e-50)
       t_0
       (if (<= (* x y) 0.0)
         (* y (/ x z))
         (if (<= (* x y) 5e+153) t_0 (/ y (/ z x))))))))

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 tmp;
	if ((x * y) <= -2e+245) {
		tmp = pow(((z / y) / x), -1.0);
	} else if ((x * y) <= -1e-50) {
		tmp = t_0;
	} else if ((x * y) <= 0.0) {
		tmp = y * (x / z);
	} else if ((x * y) <= 5e+153) {
		tmp = t_0;
	} else {
		tmp = y / (z / x);
	}
	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) :: tmp
    t_0 = (x * y) / z
    if ((x * y) <= (-2d+245)) then
        tmp = ((z / y) / x) ** (-1.0d0)
    else if ((x * y) <= (-1d-50)) then
        tmp = t_0
    else if ((x * y) <= 0.0d0) then
        tmp = y * (x / z)
    else if ((x * y) <= 5d+153) then
        tmp = t_0
    else
        tmp = y / (z / x)
    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 tmp;
	if ((x * y) <= -2e+245) {
		tmp = Math.pow(((z / y) / x), -1.0);
	} else if ((x * y) <= -1e-50) {
		tmp = t_0;
	} else if ((x * y) <= 0.0) {
		tmp = y * (x / z);
	} else if ((x * y) <= 5e+153) {
		tmp = t_0;
	} else {
		tmp = y / (z / x);
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = (x * y) / z
	tmp = 0
	if (x * y) <= -2e+245:
		tmp = math.pow(((z / y) / x), -1.0)
	elif (x * y) <= -1e-50:
		tmp = t_0
	elif (x * y) <= 0.0:
		tmp = y * (x / z)
	elif (x * y) <= 5e+153:
		tmp = t_0
	else:
		tmp = y / (z / x)
	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)
	tmp = 0.0
	if (Float64(x * y) <= -2e+245)
		tmp = Float64(Float64(z / y) / x) ^ -1.0;
	elseif (Float64(x * y) <= -1e-50)
		tmp = t_0;
	elseif (Float64(x * y) <= 0.0)
		tmp = Float64(y * Float64(x / z));
	elseif (Float64(x * y) <= 5e+153)
		tmp = t_0;
	else
		tmp = Float64(y / Float64(z / x));
	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;
	tmp = 0.0;
	if ((x * y) <= -2e+245)
		tmp = ((z / y) / x) ^ -1.0;
	elseif ((x * y) <= -1e-50)
		tmp = t_0;
	elseif ((x * y) <= 0.0)
		tmp = y * (x / z);
	elseif ((x * y) <= 5e+153)
		tmp = t_0;
	else
		tmp = y / (z / x);
	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]}, If[LessEqual[N[(x * y), $MachinePrecision], -2e+245], N[Power[N[(N[(z / y), $MachinePrecision] / x), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -1e-50], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], 0.0], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+153], t$95$0, N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.1
Target	6.1
Herbie	1.4

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

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -2.00000000000000009e245

   1. Initial program 35.4

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

   2. Applied egg-rr 35.5

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

   3. Taylor expanded in z around 0 35.5

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

   4. Simplified 0.6

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

   if -2.00000000000000009e245 < (*.f64 x y) < -1.00000000000000001e-50 or 0.0 < (*.f64 x y) < 5.00000000000000018e153

   1. Initial program 0.5

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

   if -1.00000000000000001e-50 < (*.f64 x y) < 0.0

   1. Initial program 8.9

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

   2. Applied egg-rr 9.3

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

   3. Applied egg-rr 2.9

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

   4. Applied egg-rr 2.9

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

   if 5.00000000000000018e153 < (*.f64 x y)

   1. Initial program 17.0

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

   2. Applied egg-rr 17.1

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

   3. Applied egg-rr 2.3

      \[\leadsto \color{blue}{\frac{y}{\frac{z}{x}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+245}:\\ \;\;\;\;{\left(\frac{\frac{z}{y}}{x}\right)}^{-1}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-50}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 0:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022156 
(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))