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

Average Error: 6.1 → 2.3

Time: 2.1s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{-123}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-222}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq 10^{+150}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

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) -5e-123)
     t_0
     (if (<= (* x y) 5e-222)
       (/ x (/ z y))
       (if (<= (* x y) 1e+150) t_0 (* y (/ x z)))))))

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) <= -5e-123) {
		tmp = t_0;
	} else if ((x * y) <= 5e-222) {
		tmp = x / (z / y);
	} else if ((x * y) <= 1e+150) {
		tmp = t_0;
	} else {
		tmp = y * (x / 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) :: t_0
    real(8) :: tmp
    t_0 = (x * y) / z
    if ((x * y) <= (-5d-123)) then
        tmp = t_0
    else if ((x * y) <= 5d-222) then
        tmp = x / (z / y)
    else if ((x * y) <= 1d+150) then
        tmp = t_0
    else
        tmp = y * (x / 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 t_0 = (x * y) / z;
	double tmp;
	if ((x * y) <= -5e-123) {
		tmp = t_0;
	} else if ((x * y) <= 5e-222) {
		tmp = x / (z / y);
	} else if ((x * y) <= 1e+150) {
		tmp = t_0;
	} else {
		tmp = y * (x / z);
	}
	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) <= -5e-123:
		tmp = t_0
	elif (x * y) <= 5e-222:
		tmp = x / (z / y)
	elif (x * y) <= 1e+150:
		tmp = t_0
	else:
		tmp = y * (x / z)
	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) <= -5e-123)
		tmp = t_0;
	elseif (Float64(x * y) <= 5e-222)
		tmp = Float64(x / Float64(z / y));
	elseif (Float64(x * y) <= 1e+150)
		tmp = t_0;
	else
		tmp = Float64(y * Float64(x / z));
	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) <= -5e-123)
		tmp = t_0;
	elseif ((x * y) <= 5e-222)
		tmp = x / (z / y);
	elseif ((x * y) <= 1e+150)
		tmp = t_0;
	else
		tmp = y * (x / z);
	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], -5e-123], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], 5e-222], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+150], t$95$0, N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.1
Target	6.1
Herbie	2.3

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -5.0000000000000003e-123 or 5.00000000000000008e-222 < (*.f64 x y) < 9.99999999999999981e149

   1. Initial program 3.0

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

   2. Applied egg-rr 3.3

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

   3. Taylor expanded in z around 0 3.0

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

   if -5.0000000000000003e-123 < (*.f64 x y) < 5.00000000000000008e-222

   1. Initial program 8.9

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

   2. Applied egg-rr 8.9

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

   3. Applied egg-rr 1.1

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

   if 9.99999999999999981e149 < (*.f64 x y)

   1. Initial program 17.6

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

   2. Applied egg-rr 17.7

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

   3. Applied egg-rr 1.9

      \[\leadsto \color{blue}{\frac{y}{1} \cdot \frac{x}{z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{-123}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-222}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq 10^{+150}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(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))