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

Average Error: 6.1 → 1.1

Time: 2.4s

Precision: binary64

Cost: 1872

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+289}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;t_0 \leq -5 \cdot 10^{-246}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;t_0 \leq 10^{+288}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x y) z)))
   (if (<= t_0 -5e+289)
     (* y (/ x z))
     (if (<= t_0 -5e-246)
       t_0
       (if (<= t_0 0.0)
         (* x (/ y z))
         (if (<= t_0 1e+288) t_0 (/ x (/ z y))))))))

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 (t_0 <= -5e+289) {
		tmp = y * (x / z);
	} else if (t_0 <= -5e-246) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = x * (y / z);
	} else if (t_0 <= 1e+288) {
		tmp = t_0;
	} else {
		tmp = x / (z / y);
	}
	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 (t_0 <= (-5d+289)) then
        tmp = y * (x / z)
    else if (t_0 <= (-5d-246)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = x * (y / z)
    else if (t_0 <= 1d+288) then
        tmp = t_0
    else
        tmp = x / (z / y)
    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 (t_0 <= -5e+289) {
		tmp = y * (x / z);
	} else if (t_0 <= -5e-246) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = x * (y / z);
	} else if (t_0 <= 1e+288) {
		tmp = t_0;
	} else {
		tmp = x / (z / y);
	}
	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 t_0 <= -5e+289:
		tmp = y * (x / z)
	elif t_0 <= -5e-246:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = x * (y / z)
	elif t_0 <= 1e+288:
		tmp = t_0
	else:
		tmp = x / (z / y)
	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 (t_0 <= -5e+289)
		tmp = Float64(y * Float64(x / z));
	elseif (t_0 <= -5e-246)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(x * Float64(y / z));
	elseif (t_0 <= 1e+288)
		tmp = t_0;
	else
		tmp = Float64(x / Float64(z / y));
	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 (t_0 <= -5e+289)
		tmp = y * (x / z);
	elseif (t_0 <= -5e-246)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = x * (y / z);
	elseif (t_0 <= 1e+288)
		tmp = t_0;
	else
		tmp = x / (z / y);
	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[t$95$0, -5e+289], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -5e-246], t$95$0, If[LessEqual[t$95$0, 0.0], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+288], t$95$0, N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	6.1
Target	6.1
Herbie	1.1

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x y) z) < -5.00000000000000031e289

   1. Initial program 51.1

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

   2. Simplified 4.4

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

      [Start] 51.1	\[ \frac{x \cdot y}{z} \]
      associate-*l/ [<=] 4.4	\[ \color{blue}{\frac{x}{z} \cdot y} \]

   if -5.00000000000000031e289 < (/.f64 (*.f64 x y) z) < -4.9999999999999997e-246 or 0.0 < (/.f64 (*.f64 x y) z) < 1e288

   1. Initial program 0.5

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

   if -4.9999999999999997e-246 < (/.f64 (*.f64 x y) z) < 0.0

   1. Initial program 10.0

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

   2. Simplified 1.7

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

      [Start] 10.0	\[ \frac{x \cdot y}{z} \]
      associate-*r/ [<=] 1.7	\[ \color{blue}{x \cdot \frac{y}{z}} \]

   if 1e288 < (/.f64 (*.f64 x y) z)

   1. Initial program 49.6

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

   2. Simplified 5.5

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

      [Start] 49.6	\[ \frac{x \cdot y}{z} \]
      associate-/l* [=>] 5.5	\[ \color{blue}{\frac{x}{\frac{z}{y}}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{z} \leq -5 \cdot 10^{+289}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \leq -5 \cdot 10^{-246}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \leq 0:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \leq 10^{+288}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternatives

Alternative 1
Error	5.9
Cost	717

\[\begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-147} \lor \neg \left(y \leq -6.7 \cdot 10^{-282}\right) \land y \leq 2.5 \cdot 10^{+144}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 2
Error	6.0
Cost	717

\[\begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{-203}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-280} \lor \neg \left(z \leq 10^{-98}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternative 3
Error	5.9
Cost	716

\[\begin{array}{l} \mathbf{if}\;z \leq -2.06 \cdot 10^{-197}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-286}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;z \leq 10^{-98}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 4
Error	6.1
Cost	320

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

Reproduce

herbie shell --seed 2022364 
(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))