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

Average Error: 6.2 → 0.6

Time: 3.8s

Precision: binary64

Cost: 1360

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ t_1 := \frac{y}{\frac{z}{x}}\\ \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;y \cdot \left(x \cdot \frac{1}{z}\right)\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-291}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-171}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+144}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x y) z)) (t_1 (/ y (/ z x))))
   (if (<= (* x y) (- INFINITY))
     (* y (* x (/ 1.0 z)))
     (if (<= (* x y) -2e-291)
       t_0
       (if (<= (* x y) 2e-171) t_1 (if (<= (* x y) 5e+144) t_0 t_1))))))

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 t_1 = y / (z / x);
	double tmp;
	if ((x * y) <= -((double) INFINITY)) {
		tmp = y * (x * (1.0 / z));
	} else if ((x * y) <= -2e-291) {
		tmp = t_0;
	} else if ((x * y) <= 2e-171) {
		tmp = t_1;
	} else if ((x * y) <= 5e+144) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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 t_1 = y / (z / x);
	double tmp;
	if ((x * y) <= -Double.POSITIVE_INFINITY) {
		tmp = y * (x * (1.0 / z));
	} else if ((x * y) <= -2e-291) {
		tmp = t_0;
	} else if ((x * y) <= 2e-171) {
		tmp = t_1;
	} else if ((x * y) <= 5e+144) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = (x * y) / z
	t_1 = y / (z / x)
	tmp = 0
	if (x * y) <= -math.inf:
		tmp = y * (x * (1.0 / z))
	elif (x * y) <= -2e-291:
		tmp = t_0
	elif (x * y) <= 2e-171:
		tmp = t_1
	elif (x * y) <= 5e+144:
		tmp = t_0
	else:
		tmp = t_1
	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)
	t_1 = Float64(y / Float64(z / x))
	tmp = 0.0
	if (Float64(x * y) <= Float64(-Inf))
		tmp = Float64(y * Float64(x * Float64(1.0 / z)));
	elseif (Float64(x * y) <= -2e-291)
		tmp = t_0;
	elseif (Float64(x * y) <= 2e-171)
		tmp = t_1;
	elseif (Float64(x * y) <= 5e+144)
		tmp = t_0;
	else
		tmp = t_1;
	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;
	t_1 = y / (z / x);
	tmp = 0.0;
	if ((x * y) <= -Inf)
		tmp = y * (x * (1.0 / z));
	elseif ((x * y) <= -2e-291)
		tmp = t_0;
	elseif ((x * y) <= 2e-171)
		tmp = t_1;
	elseif ((x * y) <= 5e+144)
		tmp = t_0;
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], (-Infinity)], N[(y * N[(x * N[(1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -2e-291], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], 2e-171], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 5e+144], t$95$0, t$95$1]]]]]]

Target

Original	6.2
Target	6.3
Herbie	0.6

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 64.0

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

   2. Applied egg-rr 0.3

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

   if -inf.0 < (*.f64 x y) < -1.99999999999999992e-291 or 2e-171 < (*.f64 x y) < 4.9999999999999999e144

   1. Initial program 0.2

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

   if -1.99999999999999992e-291 < (*.f64 x y) < 2e-171 or 4.9999999999999999e144 < (*.f64 x y)

   1. Initial program 13.2

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

   2. Applied egg-rr 1.4

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

   3. Applied egg-rr 1.3

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;y \cdot \left(x \cdot \frac{1}{z}\right)\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-291}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-171}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+144}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	1360

\[\begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ t_1 := \frac{y}{\frac{z}{x}}\\ \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-291}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-171}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+144}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	6.2
Cost	584

\[\begin{array}{l} t_0 := \frac{x}{\frac{z}{y}}\\ \mathbf{if}\;x \leq -2 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.2532605694606044 \cdot 10^{-101}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	6.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+158}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \leq 1.1890679111437375 \cdot 10^{-306}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 4
Error	6.2
Cost	320

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

Reproduce

herbie shell --seed 2022298 
(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))