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

Percentage Accurate: 92.2% → 91.7%

Time: 2.9s

Precision: binary64

Cost: 452

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

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq 2.7 \cdot 10^{-174}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= z 2.7e-174) (/ (* x y) z) (* y (/ x 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 (z <= 2.7e-174) {
		tmp = (x * y) / z;
	} 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

↓

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 (z <= 2.7d-174) then
        tmp = (x * y) / z
    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;
}

↓

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.7e-174) {
		tmp = (x * y) / z;
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	tmp = 0
	if z <= 2.7e-174:
		tmp = (x * y) / z
	else:
		tmp = y * (x / 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 (z <= 2.7e-174)
		tmp = Float64(Float64(x * y) / z);
	else
		tmp = Float64(y * Float64(x / 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 (z <= 2.7e-174)
		tmp = (x * y) / z;
	else
		tmp = y * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_] := If[LessEqual[z, 2.7e-174], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]

Target

Original	92.2%
Target	91.6%
Herbie	91.7%

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

Derivation

1. Split input into 2 regimes

2. if z < 2.69999999999999988e-174

   1. Initial program 94.4%

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

   if 2.69999999999999988e-174 < z

   1. Initial program 87.0%

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

   2. Simplified 94.6%

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

      [Start] 87.0	\[ \frac{x \cdot y}{z} \]
      associate-*l/ [<=] 94.6	\[ \color{blue}{\frac{x}{z} \cdot y} \]

3. Recombined 2 regimes into one program.

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.7 \cdot 10^{-174}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	92.3%
Cost	320

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

Alternative 2
Accuracy	91.7%
Cost	320

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

Alternative 3
Accuracy	91.8%
Cost	320

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

Reproduce

herbie shell --seed 2023161 
(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))