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

Average Error: 6.0 → 0.4

Time: 2.2s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ t_1 := \frac{x}{\frac{z}{y}}\\ \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;y \cdot \left(x \cdot \frac{1}{z}\right)\\ \mathbf{elif}\;x \cdot y \leq -4 \cdot 10^{-242}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 10^{-274}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+161}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x y) z)) (t_1 (/ x (/ z y))))
   (if (<= (* x y) (- INFINITY))
     (* y (* x (/ 1.0 z)))
     (if (<= (* x y) -4e-242)
       t_0
       (if (<= (* x y) 1e-274) t_1 (if (<= (* x y) 5e+161) 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 = x / (z / y);
	double tmp;
	if ((x * y) <= -((double) INFINITY)) {
		tmp = y * (x * (1.0 / z));
	} else if ((x * y) <= -4e-242) {
		tmp = t_0;
	} else if ((x * y) <= 1e-274) {
		tmp = t_1;
	} else if ((x * y) <= 5e+161) {
		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 = x / (z / y);
	double tmp;
	if ((x * y) <= -Double.POSITIVE_INFINITY) {
		tmp = y * (x * (1.0 / z));
	} else if ((x * y) <= -4e-242) {
		tmp = t_0;
	} else if ((x * y) <= 1e-274) {
		tmp = t_1;
	} else if ((x * y) <= 5e+161) {
		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 = x / (z / y)
	tmp = 0
	if (x * y) <= -math.inf:
		tmp = y * (x * (1.0 / z))
	elif (x * y) <= -4e-242:
		tmp = t_0
	elif (x * y) <= 1e-274:
		tmp = t_1
	elif (x * y) <= 5e+161:
		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(x / Float64(z / y))
	tmp = 0.0
	if (Float64(x * y) <= Float64(-Inf))
		tmp = Float64(y * Float64(x * Float64(1.0 / z)));
	elseif (Float64(x * y) <= -4e-242)
		tmp = t_0;
	elseif (Float64(x * y) <= 1e-274)
		tmp = t_1;
	elseif (Float64(x * y) <= 5e+161)
		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 = x / (z / y);
	tmp = 0.0;
	if ((x * y) <= -Inf)
		tmp = y * (x * (1.0 / z));
	elseif ((x * y) <= -4e-242)
		tmp = t_0;
	elseif ((x * y) <= 1e-274)
		tmp = t_1;
	elseif ((x * y) <= 5e+161)
		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[(x / N[(z / y), $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], -4e-242], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], 1e-274], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 5e+161], t$95$0, t$95$1]]]]]]

Target

Original	6.0
Target	6.3
Herbie	0.4

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

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) < -4e-242 or 9.99999999999999966e-275 < (*.f64 x y) < 4.9999999999999997e161

   1. Initial program 0.2

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

   2. Simplified 8.8

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

   3. Taylor expanded in x around 0 0.2

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

   if -4e-242 < (*.f64 x y) < 9.99999999999999966e-275 or 4.9999999999999997e161 < (*.f64 x y)

   1. Initial program 15.4

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

   2. Applied egg-rr 15.4

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

   3. Applied egg-rr 0.8

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;y \cdot \left(x \cdot \frac{1}{z}\right)\\ \mathbf{elif}\;x \cdot y \leq -4 \cdot 10^{-242}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 10^{-274}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+161}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022185 
(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))