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

Average Error: 6.2 → 0.7

Time: 3.7s

Precision: binary64

Cost: 1361

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+269}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{1}{z}\right)\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-166} \lor \neg \left(x \cdot y \leq 0\right) \land x \cdot y \leq 2 \cdot 10^{+131}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) -1e+269)
   (* y (* x (/ 1.0 z)))
   (if (or (<= (* x y) -1e-166)
           (and (not (<= (* x y) 0.0)) (<= (* x y) 2e+131)))
     (/ (* x y) 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+269) {
		tmp = y * (x * (1.0 / z));
	} else if (((x * y) <= -1e-166) || (!((x * y) <= 0.0) && ((x * y) <= 2e+131))) {
		tmp = (x * y) / 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+269)) then
        tmp = y * (x * (1.0d0 / z))
    else if (((x * y) <= (-1d-166)) .or. (.not. ((x * y) <= 0.0d0)) .and. ((x * y) <= 2d+131)) then
        tmp = (x * y) / 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+269) {
		tmp = y * (x * (1.0 / z));
	} else if (((x * y) <= -1e-166) || (!((x * y) <= 0.0) && ((x * y) <= 2e+131))) {
		tmp = (x * y) / 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+269:
		tmp = y * (x * (1.0 / z))
	elif ((x * y) <= -1e-166) or (not ((x * y) <= 0.0) and ((x * y) <= 2e+131)):
		tmp = (x * y) / 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+269)
		tmp = Float64(y * Float64(x * Float64(1.0 / z)));
	elseif ((Float64(x * y) <= -1e-166) || (!(Float64(x * y) <= 0.0) && (Float64(x * y) <= 2e+131)))
		tmp = Float64(Float64(x * y) / z);
	else
		tmp = Float64(x * Float64(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+269)
		tmp = y * (x * (1.0 / z));
	elseif (((x * y) <= -1e-166) || (~(((x * y) <= 0.0)) && ((x * y) <= 2e+131)))
		tmp = (x * y) / 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[LessEqual[N[(x * y), $MachinePrecision], -1e+269], N[(y * N[(x * N[(1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(x * y), $MachinePrecision], -1e-166], And[N[Not[LessEqual[N[(x * y), $MachinePrecision], 0.0]], $MachinePrecision], LessEqual[N[(x * y), $MachinePrecision], 2e+131]]], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]]]

Target

Original	6.2
Target	6.3
Herbie	0.7

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1e269

   1. Initial program 46.1

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

   2. Applied egg-rr 0.3

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

   if -1e269 < (*.f64 x y) < -1.00000000000000004e-166 or -0.0 < (*.f64 x y) < 1.9999999999999998e131

   1. Initial program 0.4

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

   if -1.00000000000000004e-166 < (*.f64 x y) < -0.0 or 1.9999999999999998e131 < (*.f64 x y)

   1. Initial program 12.8

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

   2. Simplified 1.4

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

      [Start] 12.8	\[ \frac{x \cdot y}{z} \]
      associate-*r/ [<=] 1.4	\[ \color{blue}{x \cdot \frac{y}{z}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+269}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{1}{z}\right)\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-166} \lor \neg \left(x \cdot y \leq 0\right) \land x \cdot y \leq 2 \cdot 10^{+131}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.1
Cost	1361

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+120}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-166} \lor \neg \left(x \cdot y \leq 0\right) \land x \cdot y \leq 2 \cdot 10^{+131}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternative 2
Error	6.2
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-240} \lor \neg \left(y \leq 6.5 \cdot 10^{+197}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternative 3
Error	6.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-273}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+200}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 4
Error	6.2
Cost	320

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

Reproduce

herbie shell --seed 2023039 
(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))