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

Percentage Accurate: 92.2% → 92.6%

Time: 2.1s

Precision: binary64

Cost: 452

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

Math

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

↓

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-283}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

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

↓

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -7.5e-283) (/ x (/ z y)) (* y (/ x z))))

C

double code(double x, double y, double z) {
	return (x * y) / z;
}

↓

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -7.5e-283) {
		tmp = x / (z / y);
	} else {
		tmp = y * (x / 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

↓

NOTE: x and y should be sorted in increasing order before calling this 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 <= (-7.5d-283)) then
        tmp = x / (z / y)
    else
        tmp = y * (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	return (x * y) / z;
}

↓

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -7.5e-283) {
		tmp = x / (z / y);
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Python

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

↓

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if x <= -7.5e-283:
		tmp = x / (z / y)
	else:
		tmp = y * (x / z)
	return tmp

Julia

function code(x, y, z)
	return Float64(Float64(x * y) / z)
end

↓

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (x <= -7.5e-283)
		tmp = Float64(x / Float64(z / y));
	else
		tmp = Float64(y * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * y) / z;
end

↓

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -7.5e-283)
		tmp = x / (z / y);
	else
		tmp = y * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[x, -7.5e-283], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]

Target

Original	92.2%
Target	92.0%
Herbie	92.6%

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

Derivation

1. Split input into 2 regimes

2. if x < -7.5000000000000001e-283

   1. Initial program 88.7%

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

   2. Simplified 98.3%

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}} \]
      Step-by-step derivation

      [Start] 88.7%	\[ \frac{x \cdot y}{z} \]
      associate-/l* [=>] 98.3%	\[ \color{blue}{\frac{x}{\frac{z}{y}}} \]

   if -7.5000000000000001e-283 < x

   1. Initial program 91.5%

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

   2. Simplified 93.4%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]
      Step-by-step derivation

      [Start] 91.5%	\[ \frac{x \cdot y}{z} \]
      associate-*l/ [<=] 93.4%	\[ \color{blue}{\frac{x}{z} \cdot y} \]

3. Recombined 2 regimes into one program.

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-283}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	92.6%
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-283}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 2
Accuracy	92.8%
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq -1.28 \cdot 10^{-282}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 3
Accuracy	92.1%
Cost	320

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

Reproduce

herbie shell --seed 2023167 
(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))