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

Average Accuracy: 90.5% → 96.3%

Time: 2.7s

Precision: binary64

Cost: 1101

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+234} \lor \neg \left(x \cdot y \leq -5 \cdot 10^{-101}\right) \land x \cdot y \leq 2 \cdot 10^{-309}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* x y) -1e+234)
         (and (not (<= (* x y) -5e-101)) (<= (* x y) 2e-309)))
   (* y (/ x z))
   (/ (* x y) z)))

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) <= -1e+234) || (!((x * y) <= -5e-101) && ((x * y) <= 2e-309))) {
		tmp = y * (x / z);
	} else {
		tmp = (x * y) / 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) :: tmp
    if (((x * y) <= (-1d+234)) .or. (.not. ((x * y) <= (-5d-101))) .and. ((x * y) <= 2d-309)) then
        tmp = y * (x / z)
    else
        tmp = (x * y) / 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 tmp;
	if (((x * y) <= -1e+234) || (!((x * y) <= -5e-101) && ((x * y) <= 2e-309))) {
		tmp = y * (x / z);
	} else {
		tmp = (x * y) / z;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	tmp = 0
	if ((x * y) <= -1e+234) or (not ((x * y) <= -5e-101) and ((x * y) <= 2e-309)):
		tmp = y * (x / z)
	else:
		tmp = (x * y) / z
	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) <= -1e+234) || (!(Float64(x * y) <= -5e-101) && (Float64(x * y) <= 2e-309)))
		tmp = Float64(y * Float64(x / z));
	else
		tmp = Float64(Float64(x * y) / z);
	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) <= -1e+234) || (~(((x * y) <= -5e-101)) && ((x * y) <= 2e-309)))
		tmp = y * (x / z);
	else
		tmp = (x * y) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]

↓

code[x_, y_, z_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -1e+234], And[N[Not[LessEqual[N[(x * y), $MachinePrecision], -5e-101]], $MachinePrecision], LessEqual[N[(x * y), $MachinePrecision], 2e-309]]], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]]

Target

Original	90.5%
Target	90.8%
Herbie	96.3%

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.00000000000000002e234 or -5.0000000000000001e-101 < (*.f64 x y) < 1.9999999999999988e-309

   1. Initial program 79.5%

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

   2. Simplified 97.2%

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

      [Start] 79.5	\[ \frac{x \cdot y}{z} \]
      associate-*l/ [<=] 97.2	\[ \color{blue}{\frac{x}{z} \cdot y} \]

   if -1.00000000000000002e234 < (*.f64 x y) < -5.0000000000000001e-101 or 1.9999999999999988e-309 < (*.f64 x y)

   1. Initial program 95.9%

      \[\frac{x \cdot y}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+234} \lor \neg \left(x \cdot y \leq -5 \cdot 10^{-101}\right) \land x \cdot y \leq 2 \cdot 10^{-309}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	90.5%
Cost	452

\[\begin{array}{l} \mathbf{if}\;z \leq 6.9 \cdot 10^{-116}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternative 2
Accuracy	90.4%
Cost	452

\[\begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{-119}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 3
Accuracy	90.9%
Cost	320

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

Reproduce

herbie shell --seed 2023151 
(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))