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

Average Error: 6.3 → 0.8

Time: 2.0s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;y \cdot \left(x \cdot \frac{1}{z}\right)\\ \mathbf{elif}\;t_0 \leq -2.508990892122152 \cdot 10^{-287}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;x \cdot \frac{1}{\frac{z}{y}}\\ \mathbf{elif}\;t_0 \leq 1.9835413246058474 \cdot 10^{+290}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x y) z)))
   (if (<= t_0 (- INFINITY))
     (* y (* x (/ 1.0 z)))
     (if (<= t_0 -2.508990892122152e-287)
       t_0
       (if (<= t_0 0.0)
         (* x (/ 1.0 (/ z y)))
         (if (<= t_0 1.9835413246058474e+290) t_0 (/ y (/ z x))))))))

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 (t_0 <= -((double) INFINITY)) {
		tmp = y * (x * (1.0 / z));
	} else if (t_0 <= -2.508990892122152e-287) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = x * (1.0 / (z / y));
	} else if (t_0 <= 1.9835413246058474e+290) {
		tmp = t_0;
	} else {
		tmp = y / (z / x);
	}
	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 t_0 = (x * y) / z;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = y * (x * (1.0 / z));
	} else if (t_0 <= -2.508990892122152e-287) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = x * (1.0 / (z / y));
	} else if (t_0 <= 1.9835413246058474e+290) {
		tmp = t_0;
	} else {
		tmp = y / (z / x);
	}
	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 t_0 <= -math.inf:
		tmp = y * (x * (1.0 / z))
	elif t_0 <= -2.508990892122152e-287:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = x * (1.0 / (z / y))
	elif t_0 <= 1.9835413246058474e+290:
		tmp = t_0
	else:
		tmp = y / (z / x)
	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 (t_0 <= Float64(-Inf))
		tmp = Float64(y * Float64(x * Float64(1.0 / z)));
	elseif (t_0 <= -2.508990892122152e-287)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(x * Float64(1.0 / Float64(z / y)));
	elseif (t_0 <= 1.9835413246058474e+290)
		tmp = t_0;
	else
		tmp = Float64(y / Float64(z / x));
	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 (t_0 <= -Inf)
		tmp = y * (x * (1.0 / z));
	elseif (t_0 <= -2.508990892122152e-287)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = x * (1.0 / (z / y));
	elseif (t_0 <= 1.9835413246058474e+290)
		tmp = t_0;
	else
		tmp = y / (z / x);
	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[t$95$0, (-Infinity)], N[(y * N[(x * N[(1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -2.508990892122152e-287], t$95$0, If[LessEqual[t$95$0, 0.0], N[(x * N[(1.0 / N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.9835413246058474e+290], t$95$0, N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.3
Target	6.5
Herbie	0.8

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

Derivation

1. Split input into 4 regimes

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

   1. Initial program 64.0

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

   2. Applied egg-rr 0.4

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

   if -inf.0 < (/.f64 (*.f64 x y) z) < -2.5089908921221521e-287 or -0.0 < (/.f64 (*.f64 x y) z) < 1.9835413246058474e290

   1. Initial program 0.6

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

   2. Taylor expanded in x around 0 0.6

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

   3. Simplified 8.5

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

   4. Taylor expanded in y around 0 0.6

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

   if -2.5089908921221521e-287 < (/.f64 (*.f64 x y) z) < -0.0

   1. Initial program 10.3

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

   2. Applied egg-rr 0.9

      \[\leadsto \color{blue}{x \cdot \frac{1}{\frac{z}{y}}} \]

   if 1.9835413246058474e290 < (/.f64 (*.f64 x y) z)

   1. Initial program 50.2

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

   2. Applied egg-rr 50.2

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

   3. Applied egg-rr 3.8

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{z} \leq -\infty:\\ \;\;\;\;y \cdot \left(x \cdot \frac{1}{z}\right)\\ \mathbf{elif}\;\frac{x \cdot y}{z} \leq -2.508990892122152 \cdot 10^{-287}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \leq 0:\\ \;\;\;\;x \cdot \frac{1}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \leq 1.9835413246058474 \cdot 10^{+290}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022152 
(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))