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

Average Accuracy: 90.0% → 98.9%

Time: 4.0s

Precision: binary64

Cost: 1361

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+177}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-299} \lor \neg \left(x \cdot y \leq 2 \cdot 10^{-175}\right) \land x \cdot y \leq 10^{+111}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) -5e+177)
   (/ x (/ z y))
   (if (or (<= (* x y) -5e-299)
           (and (not (<= (* x y) 2e-175)) (<= (* x y) 1e+111)))
     (/ (* x y) z)
     (/ 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 tmp;
	if ((x * y) <= -5e+177) {
		tmp = x / (z / y);
	} else if (((x * y) <= -5e-299) || (!((x * y) <= 2e-175) && ((x * y) <= 1e+111))) {
		tmp = (x * y) / z;
	} 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) :: tmp
    if ((x * y) <= (-5d+177)) then
        tmp = x / (z / y)
    else if (((x * y) <= (-5d-299)) .or. (.not. ((x * y) <= 2d-175)) .and. ((x * y) <= 1d+111)) then
        tmp = (x * y) / z
    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 tmp;
	if ((x * y) <= -5e+177) {
		tmp = x / (z / y);
	} else if (((x * y) <= -5e-299) || (!((x * y) <= 2e-175) && ((x * y) <= 1e+111))) {
		tmp = (x * y) / z;
	} else {
		tmp = y / (z / x);
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	tmp = 0
	if (x * y) <= -5e+177:
		tmp = x / (z / y)
	elif ((x * y) <= -5e-299) or (not ((x * y) <= 2e-175) and ((x * y) <= 1e+111)):
		tmp = (x * y) / z
	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)
	tmp = 0.0
	if (Float64(x * y) <= -5e+177)
		tmp = Float64(x / Float64(z / y));
	elseif ((Float64(x * y) <= -5e-299) || (!(Float64(x * y) <= 2e-175) && (Float64(x * y) <= 1e+111)))
		tmp = Float64(Float64(x * y) / z);
	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)
	tmp = 0.0;
	if ((x * y) <= -5e+177)
		tmp = x / (z / y);
	elseif (((x * y) <= -5e-299) || (~(((x * y) <= 2e-175)) && ((x * y) <= 1e+111)))
		tmp = (x * y) / z;
	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_] := If[LessEqual[N[(x * y), $MachinePrecision], -5e+177], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(x * y), $MachinePrecision], -5e-299], And[N[Not[LessEqual[N[(x * y), $MachinePrecision], 2e-175]], $MachinePrecision], LessEqual[N[(x * y), $MachinePrecision], 1e+111]]], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]]]

Target

Original	90.0%
Target	90.1%
Herbie	98.9%

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -5.0000000000000003e177

   1. Initial program 61.1%

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

   2. Simplified 97.9%

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

      [Start] 61.1	\[ \frac{x \cdot y}{z} \]
      associate-/l* [=>] 97.9	\[ \color{blue}{\frac{x}{\frac{z}{y}}} \]

   if -5.0000000000000003e177 < (*.f64 x y) < -4.99999999999999956e-299 or 2e-175 < (*.f64 x y) < 9.99999999999999957e110

   1. Initial program 99.6%

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

   if -4.99999999999999956e-299 < (*.f64 x y) < 2e-175 or 9.99999999999999957e110 < (*.f64 x y)

   1. Initial program 78.7%

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

   2. Simplified 97.9%

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

      [Start] 78.7	\[ \frac{x \cdot y}{z} \]
      associate-*l/ [<=] 97.9	\[ \color{blue}{\frac{x}{z} \cdot y} \]

   3. Applied egg-rr 97.9%

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

      [Start] 97.9	\[ \frac{x}{z} \cdot y \]
      *-commutative [=>] 97.9	\[ \color{blue}{y \cdot \frac{x}{z}} \]
      clear-num [=>] 97.6	\[ y \cdot \color{blue}{\frac{1}{\frac{z}{x}}} \]
      un-div-inv [=>] 97.9	\[ \color{blue}{\frac{y}{\frac{z}{x}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+177}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-299} \lor \neg \left(x \cdot y \leq 2 \cdot 10^{-175}\right) \land x \cdot y \leq 10^{+111}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	90.5%
Cost	320

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

Reproduce

herbie shell --seed 2023131 
(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))