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

Average Error: 15.9 → 6.5

Time: 24.6s

Precision: binary64

Cost: 5456

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{a + \left(1 + b \cdot \frac{y}{t}\right)}\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{b}{\frac{t}{y}}\right)}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+298}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\frac{y}{\frac{y \cdot b + t \cdot \left(a + 1\right)}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + \frac{t}{\frac{y}{x}}}{b}\\ \end{array} \]

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)) (+ (/ (* y b) t) (+ a 1.0)))))
   (if (<= t_1 (- INFINITY))
     (* (/ y t) (/ z (+ a (+ 1.0 (* b (/ y t))))))
     (if (<= t_1 0.0)
       (/ (+ x (/ y (/ t z))) (+ a (+ 1.0 (/ b (/ t y)))))
       (if (<= t_1 5e+298)
         t_1
         (if (<= t_1 INFINITY)
           (/ y (/ (+ (* y b) (* t (+ a 1.0))) z))
           (/ (+ z (/ t (/ y x))) 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)) / (((y * b) / t) + (a + 1.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (y / t) * (z / (a + (1.0 + (b * (y / t)))));
	} else if (t_1 <= 0.0) {
		tmp = (x + (y / (t / z))) / (a + (1.0 + (b / (t / y))));
	} else if (t_1 <= 5e+298) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = y / (((y * b) + (t * (a + 1.0))) / z);
	} else {
		tmp = (z + (t / (y / x))) / 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)) / (((y * b) / t) + (a + 1.0));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (y / t) * (z / (a + (1.0 + (b * (y / t)))));
	} else if (t_1 <= 0.0) {
		tmp = (x + (y / (t / z))) / (a + (1.0 + (b / (t / y))));
	} else if (t_1 <= 5e+298) {
		tmp = t_1;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = y / (((y * b) + (t * (a + 1.0))) / z);
	} else {
		tmp = (z + (t / (y / x))) / 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)) / (((y * b) / t) + (a + 1.0))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (y / t) * (z / (a + (1.0 + (b * (y / t)))))
	elif t_1 <= 0.0:
		tmp = (x + (y / (t / z))) / (a + (1.0 + (b / (t / y))))
	elif t_1 <= 5e+298:
		tmp = t_1
	elif t_1 <= math.inf:
		tmp = y / (((y * b) + (t * (a + 1.0))) / z)
	else:
		tmp = (z + (t / (y / x))) / 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(Float64(y * b) / t) + Float64(a + 1.0)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(y / t) * Float64(z / Float64(a + Float64(1.0 + Float64(b * Float64(y / t))))));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(x + Float64(y / Float64(t / z))) / Float64(a + Float64(1.0 + Float64(b / Float64(t / y)))));
	elseif (t_1 <= 5e+298)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = Float64(y / Float64(Float64(Float64(y * b) + Float64(t * Float64(a + 1.0))) / z));
	else
		tmp = Float64(Float64(z + Float64(t / Float64(y / x))) / 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)) / (((y * b) / t) + (a + 1.0));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (y / t) * (z / (a + (1.0 + (b * (y / t)))));
	elseif (t_1 <= 0.0)
		tmp = (x + (y / (t / z))) / (a + (1.0 + (b / (t / y))));
	elseif (t_1 <= 5e+298)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = y / (((y * b) + (t * (a + 1.0))) / z);
	else
		tmp = (z + (t / (y / x))) / 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[(N[(y * b), $MachinePrecision] / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(y / t), $MachinePrecision] * N[(z / N[(a + N[(1.0 + N[(b * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(x + N[(y / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a + N[(1.0 + N[(b / N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+298], t$95$1, If[LessEqual[t$95$1, Infinity], N[(y / N[(N[(N[(y * b), $MachinePrecision] + N[(t * N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(z + N[(t / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]]]]

Target

Original	15.9
Target	13.4
Herbie	6.5

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

Derivation

1. Split input into 5 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 x around 0 41.6

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

   3. Simplified 15.5

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

      [Start] 41.6	\[ \frac{y \cdot z}{t \cdot \left(1 + \left(\frac{y \cdot b}{t} + a\right)\right)} \]
      times-frac [=>] 15.5	\[ \color{blue}{\frac{y}{t} \cdot \frac{z}{1 + \left(\frac{y \cdot b}{t} + a\right)}} \]
      associate-+r+ [=>] 15.5	\[ \frac{y}{t} \cdot \frac{z}{\color{blue}{\left(1 + \frac{y \cdot b}{t}\right) + a}} \]
      +-commutative [<=] 15.5	\[ \frac{y}{t} \cdot \frac{z}{\color{blue}{a + \left(1 + \frac{y \cdot b}{t}\right)}} \]
      associate-*l/ [<=] 15.5	\[ \frac{y}{t} \cdot \frac{z}{a + \left(1 + \color{blue}{\frac{y}{t} \cdot b}\right)} \]
      *-commutative [=>] 15.5	\[ \frac{y}{t} \cdot \frac{z}{a + \left(1 + \color{blue}{b \cdot \frac{y}{t}}\right)} \]

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 0.0

   1. Initial program 10.0

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

   2. Simplified 10.1

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

      [Start] 10.0	\[ \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-/l* [=>] 12.3	\[ \frac{x + \color{blue}{\frac{y}{\frac{t}{z}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-+l+ [=>] 12.3	\[ \frac{x + \frac{y}{\frac{t}{z}}}{\color{blue}{a + \left(1 + \frac{y \cdot b}{t}\right)}} \]
      *-commutative [=>] 12.3	\[ \frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{\color{blue}{b \cdot y}}{t}\right)} \]
      associate-/l* [=>] 10.1	\[ \frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \color{blue}{\frac{b}{\frac{t}{y}}}\right)} \]

   if 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 5.0000000000000003e298

   1. Initial program 0.4

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

   if 5.0000000000000003e298 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 60.2

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

   2. Simplified 39.6

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + \mathsf{fma}\left(\frac{y}{t}, b, a\right)}} \]
      Proof

      [Start] 60.2	\[ \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      +-commutative [=>] 60.2	\[ \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-*l/ [<=] 39.6	\[ \frac{\color{blue}{\frac{y}{t} \cdot z} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      fma-def [=>] 39.6	\[ \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      +-commutative [=>] 39.6	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\color{blue}{\frac{y \cdot b}{t} + \left(a + 1\right)}} \]
      associate-+r+ [=>] 39.6	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\color{blue}{\left(\frac{y \cdot b}{t} + a\right) + 1}} \]
      +-commutative [=>] 39.6	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\color{blue}{1 + \left(\frac{y \cdot b}{t} + a\right)}} \]
      associate-*l/ [<=] 39.6	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + \left(\color{blue}{\frac{y}{t} \cdot b} + a\right)} \]
      fma-def [=>] 39.6	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + \color{blue}{\mathsf{fma}\left(\frac{y}{t}, b, a\right)}} \]

   3. Taylor expanded in z around inf 37.2

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

   4. Simplified 17.5

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

      [Start] 37.2	\[ \frac{y \cdot z}{t \cdot \left(\frac{y \cdot b}{t} + \left(1 + a\right)\right)} \]
      associate-/l* [=>] 16.6	\[ \color{blue}{\frac{y}{\frac{t \cdot \left(\frac{y \cdot b}{t} + \left(1 + a\right)\right)}{z}}} \]
      +-commutative [=>] 16.6	\[ \frac{y}{\frac{t \cdot \color{blue}{\left(\left(1 + a\right) + \frac{y \cdot b}{t}\right)}}{z}} \]
      associate-*r/ [<=] 17.5	\[ \frac{y}{\frac{t \cdot \left(\left(1 + a\right) + \color{blue}{y \cdot \frac{b}{t}}\right)}{z}} \]

   5. Taylor expanded in t around 0 16.6

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

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

   1. Initial program 64.0

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

   2. Simplified 58.6

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

      [Start] 64.0	\[ \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-/l* [=>] 63.7	\[ \frac{x + \color{blue}{\frac{y}{\frac{t}{z}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-+l+ [=>] 63.7	\[ \frac{x + \frac{y}{\frac{t}{z}}}{\color{blue}{a + \left(1 + \frac{y \cdot b}{t}\right)}} \]
      *-commutative [=>] 63.7	\[ \frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{\color{blue}{b \cdot y}}{t}\right)} \]
      associate-/l* [=>] 58.6	\[ \frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \color{blue}{\frac{b}{\frac{t}{y}}}\right)} \]

   3. Taylor expanded in t around 0 17.2

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

   4. Simplified 4.3

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

      [Start] 17.2	\[ \frac{z}{b} + \left(\frac{x}{y \cdot b} - \frac{\left(1 + a\right) \cdot z}{y \cdot {b}^{2}}\right) \cdot t \]
      *-commutative [=>] 17.2	\[ \frac{z}{b} + \color{blue}{t \cdot \left(\frac{x}{y \cdot b} - \frac{\left(1 + a\right) \cdot z}{y \cdot {b}^{2}}\right)} \]
      associate-/r* [=>] 16.3	\[ \frac{z}{b} + t \cdot \left(\color{blue}{\frac{\frac{x}{y}}{b}} - \frac{\left(1 + a\right) \cdot z}{y \cdot {b}^{2}}\right) \]
      times-frac [=>] 4.4	\[ \frac{z}{b} + t \cdot \left(\frac{\frac{x}{y}}{b} - \color{blue}{\frac{1 + a}{y} \cdot \frac{z}{{b}^{2}}}\right) \]
      unpow2 [=>] 4.4	\[ \frac{z}{b} + t \cdot \left(\frac{\frac{x}{y}}{b} - \frac{1 + a}{y} \cdot \frac{z}{\color{blue}{b \cdot b}}\right) \]
      associate-/r* [=>] 4.3	\[ \frac{z}{b} + t \cdot \left(\frac{\frac{x}{y}}{b} - \frac{1 + a}{y} \cdot \color{blue}{\frac{\frac{z}{b}}{b}}\right) \]

   5. Taylor expanded in b around inf 3.9

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

   6. Simplified 1.4

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

      [Start] 3.9	\[ \frac{\frac{t \cdot x}{y} + z}{b} \]
      +-commutative [=>] 3.9	\[ \frac{\color{blue}{z + \frac{t \cdot x}{y}}}{b} \]
      associate-/l* [=>] 1.4	\[ \frac{z + \color{blue}{\frac{t}{\frac{y}{x}}}}{b} \]

3. Recombined 5 regimes into one program.

4. Final simplification 6.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -\infty:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{a + \left(1 + b \cdot \frac{y}{t}\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 0:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{b}{\frac{t}{y}}\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 5 \cdot 10^{+298}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq \infty:\\ \;\;\;\;\frac{y}{\frac{y \cdot b + t \cdot \left(a + 1\right)}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + \frac{t}{\frac{y}{x}}}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.8
Cost	15944

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

Alternative 2
Error	7.8
Cost	3528

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

Alternative 3
Error	15.1
Cost	2012

\[\begin{array}{l} t_1 := \frac{x + y \cdot \frac{z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ t_2 := \frac{z + \frac{t}{\frac{y}{x}}}{b}\\ \mathbf{if}\;t \leq -4.5 \cdot 10^{-165}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -7 \cdot 10^{-216}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -5.6 \cdot 10^{-231}:\\ \;\;\;\;x \cdot \frac{1}{\left(a + 1\right) + b \cdot \frac{y}{t}}\\ \mathbf{elif}\;t \leq -1.2 \cdot 10^{-276}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t \leq 1.36 \cdot 10^{-279}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{-238}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{elif}\;t \leq 10^{-118}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	12.9
Cost	1616

\[\begin{array}{l} t_1 := \frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{b}{\frac{t}{y}}\right)}\\ t_2 := \frac{z + \frac{t}{\frac{y}{x}}}{b}\\ \mathbf{if}\;t \leq -2.9 \cdot 10^{-165}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.36 \cdot 10^{-279}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-238}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-117}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	13.0
Cost	1616

\[\begin{array}{l} t_1 := x + \frac{y}{\frac{t}{z}}\\ t_2 := \frac{z + \frac{t}{\frac{y}{x}}}{b}\\ \mathbf{if}\;t \leq -5.5 \cdot 10^{-166}:\\ \;\;\;\;\frac{t_1}{a + \left(1 + \frac{b}{\frac{t}{y}}\right)}\\ \mathbf{elif}\;t \leq 1.36 \cdot 10^{-279}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 4.7 \cdot 10^{-238}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{-120}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{a + \left(1 + \frac{y}{\frac{t}{b}}\right)}\\ \end{array} \]

Alternative 6
Error	21.6
Cost	1496

\[\begin{array}{l} t_1 := \frac{z + \frac{t}{\frac{y}{x}}}{b}\\ t_2 := \frac{x + \frac{y}{\frac{t}{z}}}{a + 1}\\ \mathbf{if}\;t \leq -7.9 \cdot 10^{+43}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-54}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{1 + b \cdot \frac{y}{t}}\\ \mathbf{elif}\;t \leq -9 \cdot 10^{-85}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{elif}\;t \leq 1.36 \cdot 10^{-279}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{-238}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-117}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	29.3
Cost	1236

\[\begin{array}{l} t_1 := \frac{x}{a + 1}\\ \mathbf{if}\;y \leq -2.4 \cdot 10^{+82}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-131}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-149}:\\ \;\;\;\;y \cdot \frac{z}{t + t \cdot a}\\ \mathbf{elif}\;y \leq 3.75 \cdot 10^{-65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+94}:\\ \;\;\;\;\frac{x}{1 + b \cdot \frac{y}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 8
Error	23.1
Cost	1234

\[\begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{-90} \lor \neg \left(t \leq 1.36 \cdot 10^{-279} \lor \neg \left(t \leq 1.92 \cdot 10^{-239}\right) \land t \leq 2.5 \cdot 10^{-117}\right):\\ \;\;\;\;\frac{x}{a + \left(1 + b \cdot \frac{y}{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + \frac{t}{\frac{y}{x}}}{b}\\ \end{array} \]

Alternative 9
Error	21.2
Cost	1234

\[\begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{-91} \lor \neg \left(t \leq 1.36 \cdot 10^{-279}\right) \land \left(t \leq 1.92 \cdot 10^{-239} \lor \neg \left(t \leq 1.45 \cdot 10^{-117}\right)\right):\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + \frac{t}{\frac{y}{x}}}{b}\\ \end{array} \]

Alternative 10
Error	21.1
Cost	1232

\[\begin{array}{l} t_1 := \frac{z + \frac{t}{\frac{y}{x}}}{b}\\ t_2 := \frac{x + \frac{y}{\frac{t}{z}}}{a + 1}\\ \mathbf{if}\;t \leq -2.3 \cdot 10^{-92}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.36 \cdot 10^{-279}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-237}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-117}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 11
Error	29.0
Cost	1112

\[\begin{array}{l} t_1 := \frac{x}{a + 1}\\ \mathbf{if}\;y \leq -1.65 \cdot 10^{+81}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-131}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-149}:\\ \;\;\;\;y \cdot \frac{z}{t + t \cdot a}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{+68}:\\ \;\;\;\;\frac{\frac{t}{\frac{y}{x}}}{b}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+97}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 12
Error	27.3
Cost	1106

\[\begin{array}{l} \mathbf{if}\;t \leq -3.4 \cdot 10^{-85} \lor \neg \left(t \leq 1.36 \cdot 10^{-279}\right) \land \left(t \leq 1.92 \cdot 10^{-239} \lor \neg \left(t \leq 9.8 \cdot 10^{-55}\right)\right):\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + \frac{t}{\frac{y}{x}}}{b}\\ \end{array} \]

Alternative 13
Error	35.9
Cost	588

\[\begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{+60}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq -2.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

Alternative 14
Error	28.4
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-90} \lor \neg \left(t \leq 2.55 \cdot 10^{-117}\right):\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 15
Error	36.0
Cost	456

\[\begin{array}{l} \mathbf{if}\;a \leq -1:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

Alternative 16
Error	50.5
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023012 
(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))))