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

Average Accuracy: 92.3% → 95.6%

Time: 2.5s

Precision: binary64

Cost: 1100

• Unsound rule application detected in e-graph. Results may not be sound.

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{-73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 10^{-166}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq 10^{+66}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x y) z)))
   (if (<= (* x y) -4e-73)
     t_0
     (if (<= (* x y) 1e-166)
       (/ x (/ z y))
       (if (<= (* x y) 1e+66) t_0 (* 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 t_0 = (x * y) / z;
	double tmp;
	if ((x * y) <= -4e-73) {
		tmp = t_0;
	} else if ((x * y) <= 1e-166) {
		tmp = x / (z / y);
	} else if ((x * y) <= 1e+66) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (x * y) / z
    if ((x * y) <= (-4d-73)) then
        tmp = t_0
    else if ((x * y) <= 1d-166) then
        tmp = x / (z / y)
    else if ((x * y) <= 1d+66) then
        tmp = t_0
    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 t_0 = (x * y) / z;
	double tmp;
	if ((x * y) <= -4e-73) {
		tmp = t_0;
	} else if ((x * y) <= 1e-166) {
		tmp = x / (z / y);
	} else if ((x * y) <= 1e+66) {
		tmp = t_0;
	} else {
		tmp = x * (y / 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) <= -4e-73:
		tmp = t_0
	elif (x * y) <= 1e-166:
		tmp = x / (z / y)
	elif (x * y) <= 1e+66:
		tmp = t_0
	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)
	t_0 = Float64(Float64(x * y) / z)
	tmp = 0.0
	if (Float64(x * y) <= -4e-73)
		tmp = t_0;
	elseif (Float64(x * y) <= 1e-166)
		tmp = Float64(x / Float64(z / y));
	elseif (Float64(x * y) <= 1e+66)
		tmp = t_0;
	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)
	t_0 = (x * y) / z;
	tmp = 0.0;
	if ((x * y) <= -4e-73)
		tmp = t_0;
	elseif ((x * y) <= 1e-166)
		tmp = x / (z / y);
	elseif ((x * y) <= 1e+66)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -4e-73], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], 1e-166], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+66], t$95$0, N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	92.3%
Target	92.5%
Herbie	95.6%

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -3.99999999999999999e-73 or 1.00000000000000004e-166 < (*.f64 x y) < 9.99999999999999945e65

   1. Initial program 96.5%

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

   if -3.99999999999999999e-73 < (*.f64 x y) < 1.00000000000000004e-166

   1. Initial program 91.6%

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

   2. Simplified 98.7%

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

      [Start] 91.6	\[ \frac{x \cdot y}{z} \]
      associate-/l* [=>] 98.7	\[ \color{blue}{\frac{x}{\frac{z}{y}}} \]

   if 9.99999999999999945e65 < (*.f64 x y)

   1. Initial program 90.6%

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

   2. Simplified 96.1%

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

      [Start] 90.6	\[ \frac{x \cdot y}{z} \]
      associate-*r/ [<=] 96.1	\[ \color{blue}{x \cdot \frac{y}{z}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{-73}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 10^{-166}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq 10^{+66}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	92.2%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-55} \lor \neg \left(x \leq 9.5 \cdot 10^{-202}\right):\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 2
Accuracy	92.2%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-55}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-186}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 3
Accuracy	92.0%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+27}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{-185}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 4
Accuracy	91.9%
Cost	320

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

Reproduce

herbie shell --seed 2023160 
(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))