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

Average Error: 5.9 → 2.1

Time: 2.0s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-123}:\\ \;\;\;\;{\left(\frac{z}{x \cdot y}\right)}^{-1}\\ \mathbf{elif}\;x \cdot y \leq 10^{-276}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) (- INFINITY))
   (/ y (/ z x))
   (if (<= (* x y) -1e-123)
     (pow (/ z (* x y)) -1.0)
     (if (<= (* x y) 1e-276) (/ x (/ z y)) (/ (* 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) <= -((double) INFINITY)) {
		tmp = y / (z / x);
	} else if ((x * y) <= -1e-123) {
		tmp = pow((z / (x * y)), -1.0);
	} else if ((x * y) <= 1e-276) {
		tmp = x / (z / y);
	} else {
		tmp = (x * y) / z;
	}
	return tmp;
}

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) <= -Double.POSITIVE_INFINITY) {
		tmp = y / (z / x);
	} else if ((x * y) <= -1e-123) {
		tmp = Math.pow((z / (x * y)), -1.0);
	} else if ((x * y) <= 1e-276) {
		tmp = x / (z / y);
	} 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) <= -math.inf:
		tmp = y / (z / x)
	elif (x * y) <= -1e-123:
		tmp = math.pow((z / (x * y)), -1.0)
	elif (x * y) <= 1e-276:
		tmp = x / (z / y)
	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) <= Float64(-Inf))
		tmp = Float64(y / Float64(z / x));
	elseif (Float64(x * y) <= -1e-123)
		tmp = Float64(z / Float64(x * y)) ^ -1.0;
	elseif (Float64(x * y) <= 1e-276)
		tmp = Float64(x / Float64(z / y));
	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) <= -Inf)
		tmp = y / (z / x);
	elseif ((x * y) <= -1e-123)
		tmp = (z / (x * y)) ^ -1.0;
	elseif ((x * y) <= 1e-276)
		tmp = x / (z / y);
	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], (-Infinity)], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -1e-123], N[Power[N[(z / N[(x * y), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e-276], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	5.9
Target	5.9
Herbie	2.1

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

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -inf.0

   1. Initial program 64.0

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

   2. Applied egg-rr 64.0

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

   3. Applied egg-rr 0.2

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

   if -inf.0 < (*.f64 x y) < -1.0000000000000001e-123

   1. Initial program 0.2

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

   2. Applied egg-rr 0.5

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

   if -1.0000000000000001e-123 < (*.f64 x y) < 1e-276

   1. Initial program 9.0

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

   2. Applied egg-rr 9.5

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

   3. Applied egg-rr 0.9

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

   if 1e-276 < (*.f64 x y)

   1. Initial program 4.2

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

   2. Applied egg-rr 4.6

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

   3. Applied egg-rr 7.5

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

   4. Taylor expanded in x around 0 4.2

      \[\leadsto \color{blue}{\frac{y \cdot x}{z}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-123}:\\ \;\;\;\;{\left(\frac{z}{x \cdot y}\right)}^{-1}\\ \mathbf{elif}\;x \cdot y \leq 10^{-276}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022165 
(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))