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

Average Error: 16.9 → 8.1

Time: 27.0s

Precision: binary64

Cost: 5712

Math

\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

↓

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{x}{a - -1} + y \cdot \frac{z}{t \cdot \left(1 + a\right)}\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-286}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{y \cdot z}{y \cdot b + t \cdot a}\\ \mathbf{elif}\;t_1 \leq 10^{+287}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))

↓

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_1 (- INFINITY))
     (+ (/ x (- a -1.0)) (* y (/ z (* t (+ 1.0 a)))))
     (if (<= t_1 -1e-286)
       t_1
       (if (<= t_1 0.0)
         (/ (* y z) (+ (* y b) (* t a)))
         (if (<= t_1 1e+287) t_1 (/ z b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (x / (a - -1.0)) + (y * (z / (t * (1.0 + a))));
	} else if (t_1 <= -1e-286) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (y * z) / ((y * b) + (t * a));
	} else if (t_1 <= 1e+287) {
		tmp = t_1;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
}

↓

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (x / (a - -1.0)) + (y * (z / (t * (1.0 + a))));
	} else if (t_1 <= -1e-286) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (y * z) / ((y * b) + (t * a));
	} else if (t_1 <= 1e+287) {
		tmp = t_1;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))

↓

def code(x, y, z, t, a, b):
	t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (x / (a - -1.0)) + (y * (z / (t * (1.0 + a))))
	elif t_1 <= -1e-286:
		tmp = t_1
	elif t_1 <= 0.0:
		tmp = (y * z) / ((y * b) + (t * a))
	elif t_1 <= 1e+287:
		tmp = t_1
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
end

↓

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(x / Float64(a - -1.0)) + Float64(y * Float64(z / Float64(t * Float64(1.0 + a)))));
	elseif (t_1 <= -1e-286)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(y * z) / Float64(Float64(y * b) + Float64(t * a)));
	elseif (t_1 <= 1e+287)
		tmp = t_1;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
end

↓

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (x / (a - -1.0)) + (y * (z / (t * (1.0 + a))));
	elseif (t_1 <= -1e-286)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = (y * z) / ((y * b) + (t * a));
	elseif (t_1 <= 1e+287)
		tmp = t_1;
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(x / N[(a - -1.0), $MachinePrecision]), $MachinePrecision] + N[(y * N[(z / N[(t * N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1e-286], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(y * z), $MachinePrecision] / N[(N[(y * b), $MachinePrecision] + N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+287], t$95$1, N[(z / b), $MachinePrecision]]]]]]

Target

Original	16.9
Target	12.8
Herbie	8.1

\[\begin{array}{l} \mathbf{if}\;t < -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t < 3.036967103737246 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array} \]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -inf.0

   1. Initial program 64.0

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   2. Taylor expanded in y around 0 39.9

      \[\leadsto \color{blue}{y \cdot \left(\frac{z}{t \cdot \left(1 + a\right)} - \frac{b \cdot x}{t \cdot {\left(1 + a\right)}^{2}}\right) + \frac{x}{1 + a}} \]

   3. Simplified 39.9

      \[\leadsto \color{blue}{\frac{x}{a - -1} + y \cdot \left(\frac{z}{t \cdot \left(a - -1\right)} - \frac{b \cdot x}{t \cdot {\left(a - -1\right)}^{2}}\right)} \]
      Proof

      [Start] 39.9	\[ y \cdot \left(\frac{z}{t \cdot \left(1 + a\right)} - \frac{b \cdot x}{t \cdot {\left(1 + a\right)}^{2}}\right) + \frac{x}{1 + a} \]
      rational_best-simplify-1 [=>] 39.9	\[ \color{blue}{\frac{x}{1 + a} + y \cdot \left(\frac{z}{t \cdot \left(1 + a\right)} - \frac{b \cdot x}{t \cdot {\left(1 + a\right)}^{2}}\right)} \]
      rational_best-simplify-1 [<=] 39.9	\[ \frac{x}{\color{blue}{a + 1}} + y \cdot \left(\frac{z}{t \cdot \left(1 + a\right)} - \frac{b \cdot x}{t \cdot {\left(1 + a\right)}^{2}}\right) \]
      rational_best-simplify-16 [=>] 39.9	\[ \frac{x}{\color{blue}{a - -1}} + y \cdot \left(\frac{z}{t \cdot \left(1 + a\right)} - \frac{b \cdot x}{t \cdot {\left(1 + a\right)}^{2}}\right) \]
      rational_best-simplify-1 [<=] 39.9	\[ \frac{x}{a - -1} + y \cdot \left(\frac{z}{t \cdot \color{blue}{\left(a + 1\right)}} - \frac{b \cdot x}{t \cdot {\left(1 + a\right)}^{2}}\right) \]
      rational_best-simplify-16 [=>] 39.9	\[ \frac{x}{a - -1} + y \cdot \left(\frac{z}{t \cdot \color{blue}{\left(a - -1\right)}} - \frac{b \cdot x}{t \cdot {\left(1 + a\right)}^{2}}\right) \]
      rational_best-simplify-1 [<=] 39.9	\[ \frac{x}{a - -1} + y \cdot \left(\frac{z}{t \cdot \left(a - -1\right)} - \frac{b \cdot x}{t \cdot {\color{blue}{\left(a + 1\right)}}^{2}}\right) \]
      rational_best-simplify-16 [=>] 39.9	\[ \frac{x}{a - -1} + y \cdot \left(\frac{z}{t \cdot \left(a - -1\right)} - \frac{b \cdot x}{t \cdot {\color{blue}{\left(a - -1\right)}}^{2}}\right) \]

   4. Taylor expanded in a around 0 40.8

      \[\leadsto \frac{x}{a - -1} + y \cdot \left(\frac{z}{t \cdot \left(a - -1\right)} - \frac{b \cdot x}{\color{blue}{t + 2 \cdot \left(a \cdot t\right)}}\right) \]

   5. Simplified 40.8

      \[\leadsto \frac{x}{a - -1} + y \cdot \left(\frac{z}{t \cdot \left(a - -1\right)} - \frac{b \cdot x}{\color{blue}{t + t \cdot \left(a \cdot 2\right)}}\right) \]
      Proof

      [Start] 40.8	\[ \frac{x}{a - -1} + y \cdot \left(\frac{z}{t \cdot \left(a - -1\right)} - \frac{b \cdot x}{t + 2 \cdot \left(a \cdot t\right)}\right) \]
      rational_best-simplify-2 [=>] 40.8	\[ \frac{x}{a - -1} + y \cdot \left(\frac{z}{t \cdot \left(a - -1\right)} - \frac{b \cdot x}{t + 2 \cdot \color{blue}{\left(t \cdot a\right)}}\right) \]
      rational_best-simplify-44 [=>] 40.8	\[ \frac{x}{a - -1} + y \cdot \left(\frac{z}{t \cdot \left(a - -1\right)} - \frac{b \cdot x}{t + \color{blue}{t \cdot \left(2 \cdot a\right)}}\right) \]
      rational_best-simplify-2 [=>] 40.8	\[ \frac{x}{a - -1} + y \cdot \left(\frac{z}{t \cdot \left(a - -1\right)} - \frac{b \cdot x}{t + t \cdot \color{blue}{\left(a \cdot 2\right)}}\right) \]

   6. Taylor expanded in z around inf 39.1

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -1.00000000000000005e-286 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 1.0000000000000001e287

   1. Initial program 0.5

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   if -1.00000000000000005e-286 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 27.4

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   2. Taylor expanded in x around 0 29.5

      \[\leadsto \color{blue}{\frac{y \cdot z}{t \cdot \left(1 + \left(\frac{y \cdot b}{t} + a\right)\right)}} \]

   3. Taylor expanded in t around 0 22.6

      \[\leadsto \frac{y \cdot z}{\color{blue}{y \cdot b + t \cdot \left(1 + a\right)}} \]

   4. Taylor expanded in a around inf 22.6

      \[\leadsto \frac{y \cdot z}{y \cdot b + \color{blue}{a \cdot t}} \]

   5. Simplified 22.6

      \[\leadsto \frac{y \cdot z}{y \cdot b + \color{blue}{t \cdot a}} \]
      Proof

      [Start] 22.6	\[ \frac{y \cdot z}{y \cdot b + a \cdot t} \]
      rational_best-simplify-2 [=>] 22.6	\[ \frac{y \cdot z}{y \cdot b + \color{blue}{t \cdot a}} \]

   if 1.0000000000000001e287 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t)))

   1. Initial program 61.4

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   2. Taylor expanded in y around inf 14.8

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

4. Final simplification 8.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -\infty:\\ \;\;\;\;\frac{x}{a - -1} + y \cdot \frac{z}{t \cdot \left(1 + a\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -1 \cdot 10^{-286}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 0:\\ \;\;\;\;\frac{y \cdot z}{y \cdot b + t \cdot a}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 10^{+287}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	8.9
Cost	9092

\[\begin{array}{l} t_1 := \frac{y \cdot b}{t}\\ t_2 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + t_1}\\ t_3 := 1 + \left(t_1 + a\right)\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\frac{x}{a - -1} + \frac{\left(\frac{z}{1 + a} - \frac{b \cdot x}{{\left(1 + a\right)}^{2}}\right) \cdot y}{t}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{-40}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\frac{y \cdot z}{t \cdot t_3} + \frac{x}{t_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 2
Error	8.5
Cost	5196

\[\begin{array}{l} t_1 := \frac{y \cdot b}{t}\\ t_2 := 1 + \left(t_1 + a\right)\\ t_3 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + t_1}\\ t_4 := \frac{y \cdot z}{t \cdot t_2} + \frac{x}{t_2}\\ \mathbf{if}\;t_3 \leq -1 \cdot 10^{-286}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;\frac{y \cdot z}{y \cdot b + t \cdot a}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 3
Error	25.9
Cost	2024

\[\begin{array}{l} t_1 := \frac{x}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_2 := \frac{\frac{y \cdot z}{t} + x}{1 + a}\\ \mathbf{if}\;b \leq -8.8 \cdot 10^{+199}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;b \leq -3.2 \cdot 10^{+173}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -5.6 \cdot 10^{+116}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;b \leq -6.5 \cdot 10^{+60}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -1.2 \cdot 10^{-27}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-12}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 1.34 \cdot 10^{+29}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+57}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 3.25 \cdot 10^{+63}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{+140}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 4
Error	26.0
Cost	2024

\[\begin{array}{l} t_1 := \frac{y \cdot z}{t} + x\\ t_2 := \frac{x}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_3 := \frac{t_1}{1 + a}\\ \mathbf{if}\;b \leq -6.5 \cdot 10^{+203}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;b \leq -5.6 \cdot 10^{+171}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq -3.15 \cdot 10^{+116}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;b \leq -1.28 \cdot 10^{+63}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq -1.2 \cdot 10^{-27}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;b \leq 5.1 \cdot 10^{-12}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{+30}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{+57}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{+61}:\\ \;\;\;\;\frac{t \cdot t_1}{y \cdot b}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+141}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 5
Error	23.7
Cost	1884

\[\begin{array}{l} t_1 := t \cdot \left(1 + a\right)\\ t_2 := \frac{x}{a - -1} + y \cdot \frac{z}{t_1}\\ t_3 := \frac{\frac{y \cdot z}{t} + x}{1 + a}\\ \mathbf{if}\;t \leq -5.6 \cdot 10^{+48}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.46 \cdot 10^{-5}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t \leq -5.2 \cdot 10^{-29}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -2.1 \cdot 10^{-74}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t \leq -8.8 \cdot 10^{-100}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -1.56 \cdot 10^{-271}:\\ \;\;\;\;\frac{y \cdot z}{y \cdot b + t_1}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-166}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	23.9
Cost	1884

\[\begin{array}{l} t_1 := t \cdot \left(1 + a\right)\\ t_2 := \frac{x}{a - -1}\\ t_3 := \frac{\frac{y \cdot z}{t} + x}{1 + a}\\ \mathbf{if}\;t \leq -5.3 \cdot 10^{+48}:\\ \;\;\;\;t_2 + y \cdot \frac{z}{t_1}\\ \mathbf{elif}\;t \leq -0.000135:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t \leq -1.65 \cdot 10^{-28}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -3.3 \cdot 10^{-74}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t \leq -2.3 \cdot 10^{-98}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -7 \cdot 10^{-271}:\\ \;\;\;\;\frac{y \cdot z}{y \cdot b + t_1}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-164}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t_2 + \frac{\frac{y \cdot z}{1 + a}}{t}\\ \end{array} \]

Alternative 7
Error	30.2
Cost	1760

\[\begin{array}{l} t_1 := \frac{x}{a - -1}\\ \mathbf{if}\;t \leq -3.55 \cdot 10^{+53}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -0.00034:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t \leq -1.22 \cdot 10^{-26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-76}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t \leq -4.3 \cdot 10^{-102}:\\ \;\;\;\;\frac{y \cdot z}{t \cdot \left(a - -1\right)}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-164}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{a} + \frac{y \cdot z}{t \cdot a}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+115}:\\ \;\;\;\;t_1 + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	24.0
Cost	1364

\[\begin{array}{l} t_1 := \frac{x}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t \leq -6.2 \cdot 10^{-29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4.6 \cdot 10^{-76}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t \leq -7 \cdot 10^{-83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.5 \cdot 10^{-271}:\\ \;\;\;\;\frac{y \cdot z}{y \cdot b + t \cdot \left(1 + a\right)}\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-164}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot z}{t} + x}{1 + a}\\ \end{array} \]

Alternative 9
Error	30.0
Cost	1236

\[\begin{array}{l} t_1 := \frac{x}{a - -1}\\ \mathbf{if}\;t \leq -3.55 \cdot 10^{+53}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-6}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t \leq -2.3 \cdot 10^{-27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.15 \cdot 10^{-76}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t \leq -2.4 \cdot 10^{-101}:\\ \;\;\;\;\frac{y \cdot z}{t \cdot \left(a - -1\right)}\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{-164}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	26.2
Cost	1232

\[\begin{array}{l} t_1 := \frac{x}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t \leq -3.8 \cdot 10^{-27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3.6 \cdot 10^{-75}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t \leq -2.05 \cdot 10^{-109}:\\ \;\;\;\;\frac{y \cdot z}{t \cdot \left(a - -1\right)}\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-164}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	29.8
Cost	848

\[\begin{array}{l} t_1 := \frac{x}{a - -1}\\ \mathbf{if}\;t \leq -3.55 \cdot 10^{+53}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -8 \cdot 10^{-5}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t \leq -9.2 \cdot 10^{-29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{-164}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	36.5
Cost	720

\[\begin{array}{l} \mathbf{if}\;a \leq -8.2 \cdot 10^{+51}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq -4.4 \cdot 10^{-147}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{-39}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{+86}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

Alternative 13
Error	37.0
Cost	456

\[\begin{array}{l} \mathbf{if}\;a \leq -0.66:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq 0.118:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

Alternative 14
Error	51.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023092 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (if (< t -1.3659085366310088e-271) (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b))))) (if (< t 3.036967103737246e-130) (/ z b) (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b)))))))

  (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))