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

Average Error: 16.5 → 4.6

Time: 29.2s

Precision: binary64

Cost: 6740

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ t_2 := \frac{y}{\frac{t \cdot \left(\left(a + 1\right) + y \cdot \frac{b}{t}\right)}{z}}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-317}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+304}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

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)) (+ (/ (* y b) t) (+ a 1.0))))
        (t_2 (/ y (/ (* t (+ (+ a 1.0) (* y (/ b t)))) z))))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 -2e-317)
       t_1
       (if (<= t_1 0.0)
         (+ (/ z b) (* (/ t y) (/ x b)))
         (if (<= t_1 4e+304) t_1 (if (<= t_1 INFINITY) t_2 (/ 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)) / (((y * b) / t) + (a + 1.0));
	double t_2 = y / ((t * ((a + 1.0) + (y * (b / t)))) / z);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= -2e-317) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (z / b) + ((t / y) * (x / b));
	} else if (t_1 <= 4e+304) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} 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)) / (((y * b) / t) + (a + 1.0));
	double t_2 = y / ((t * ((a + 1.0) + (y * (b / t)))) / z);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_1 <= -2e-317) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (z / b) + ((t / y) * (x / b));
	} else if (t_1 <= 4e+304) {
		tmp = t_1;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} 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)) / (((y * b) / t) + (a + 1.0))
	t_2 = y / ((t * ((a + 1.0) + (y * (b / t)))) / z)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = t_2
	elif t_1 <= -2e-317:
		tmp = t_1
	elif t_1 <= 0.0:
		tmp = (z / b) + ((t / y) * (x / b))
	elif t_1 <= 4e+304:
		tmp = t_1
	elif t_1 <= math.inf:
		tmp = t_2
	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(Float64(y * b) / t) + Float64(a + 1.0)))
	t_2 = Float64(y / Float64(Float64(t * Float64(Float64(a + 1.0) + Float64(y * Float64(b / t)))) / z))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= -2e-317)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(z / b) + Float64(Float64(t / y) * Float64(x / b)));
	elseif (t_1 <= 4e+304)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = t_2;
	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)) / (((y * b) / t) + (a + 1.0));
	t_2 = y / ((t * ((a + 1.0) + (y * (b / t)))) / z);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = t_2;
	elseif (t_1 <= -2e-317)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = (z / b) + ((t / y) * (x / b));
	elseif (t_1 <= 4e+304)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = t_2;
	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[(N[(y * b), $MachinePrecision] / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y / N[(N[(t * N[(N[(a + 1.0), $MachinePrecision] + N[(y * N[(b / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, -2e-317], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(z / b), $MachinePrecision] + N[(N[(t / y), $MachinePrecision] * N[(x / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+304], t$95$1, If[LessEqual[t$95$1, Infinity], t$95$2, N[(z / b), $MachinePrecision]]]]]]]]

Target

Original	16.5
Target	12.7
Herbie	4.6

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

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 or 3.9999999999999998e304 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 63.6

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

   2. Simplified 41.4

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

      [Start] 63.6	\[ \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      +-commutative [=>] 63.6	\[ \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-*l/ [<=] 41.4	\[ \frac{\color{blue}{\frac{y}{t} \cdot z} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      fma-def [=>] 41.4	\[ \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      +-commutative [=>] 41.4	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\color{blue}{\frac{y \cdot b}{t} + \left(a + 1\right)}} \]
      associate-+r+ [=>] 41.4	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\color{blue}{\left(\frac{y \cdot b}{t} + a\right) + 1}} \]
      +-commutative [=>] 41.4	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\color{blue}{1 + \left(\frac{y \cdot b}{t} + a\right)}} \]
      associate-*l/ [<=] 41.4	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + \left(\color{blue}{\frac{y}{t} \cdot b} + a\right)} \]
      fma-def [=>] 41.4	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + \color{blue}{\mathsf{fma}\left(\frac{y}{t}, b, a\right)}} \]

   3. Taylor expanded in z around inf 38.4

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

   4. Simplified 14.8

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

      [Start] 38.4	\[ \frac{y \cdot z}{t \cdot \left(\frac{y \cdot b}{t} + \left(1 + a\right)\right)} \]
      associate-/l* [=>] 13.4	\[ \color{blue}{\frac{y}{\frac{t \cdot \left(\frac{y \cdot b}{t} + \left(1 + a\right)\right)}{z}}} \]
      +-commutative [=>] 13.4	\[ \frac{y}{\frac{t \cdot \color{blue}{\left(\left(1 + a\right) + \frac{y \cdot b}{t}\right)}}{z}} \]
      associate-*r/ [<=] 14.8	\[ \frac{y}{\frac{t \cdot \left(\left(1 + a\right) + \color{blue}{y \cdot \frac{b}{t}}\right)}{z}} \]

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -1.99999997e-317 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 3.9999999999999998e304

   1. Initial program 0.5

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

   if -1.99999997e-317 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 30.6

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

   2. Simplified 20.2

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

      [Start] 30.6	\[ \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      +-commutative [=>] 30.6	\[ \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-*r/ [<=] 30.2	\[ \frac{\color{blue}{y \cdot \frac{z}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      *-commutative [<=] 30.2	\[ \frac{\color{blue}{\frac{z}{t} \cdot y} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      fma-def [=>] 30.2	\[ \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-+l+ [=>] 30.2	\[ \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\color{blue}{a + \left(1 + \frac{y \cdot b}{t}\right)}} \]
      +-commutative [=>] 30.2	\[ \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{a + \color{blue}{\left(\frac{y \cdot b}{t} + 1\right)}} \]
      associate-*r/ [<=] 20.2	\[ \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{a + \left(\color{blue}{y \cdot \frac{b}{t}} + 1\right)} \]
      *-commutative [<=] 20.2	\[ \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{a + \left(\color{blue}{\frac{b}{t} \cdot y} + 1\right)} \]
      fma-def [=>] 20.2	\[ \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{a + \color{blue}{\mathsf{fma}\left(\frac{b}{t}, y, 1\right)}} \]

   3. Taylor expanded in b around inf 32.6

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

   4. Simplified 21.1

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

      [Start] 32.6	\[ \frac{t \cdot \left(\frac{y \cdot z}{t} + x\right)}{y \cdot b} \]
      times-frac [=>] 20.4	\[ \color{blue}{\frac{t}{y} \cdot \frac{\frac{y \cdot z}{t} + x}{b}} \]
      +-commutative [=>] 20.4	\[ \frac{t}{y} \cdot \frac{\color{blue}{x + \frac{y \cdot z}{t}}}{b} \]
      associate-/l* [=>] 21.1	\[ \frac{t}{y} \cdot \frac{x + \color{blue}{\frac{y}{\frac{t}{z}}}}{b} \]

   5. Taylor expanded in t around 0 25.7

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

   6. Simplified 17.4

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

      [Start] 25.7	\[ \frac{t \cdot x}{y \cdot b} + \frac{z}{b} \]
      +-commutative [=>] 25.7	\[ \color{blue}{\frac{z}{b} + \frac{t \cdot x}{y \cdot b}} \]
      times-frac [=>] 17.4	\[ \frac{z}{b} + \color{blue}{\frac{t}{y} \cdot \frac{x}{b}} \]

   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. Taylor expanded in y around inf 2.7

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

4. Final simplification 4.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{t \cdot \left(\left(a + 1\right) + y \cdot \frac{b}{t}\right)}{z}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -2 \cdot 10^{-317}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 4 \cdot 10^{+304}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq \infty:\\ \;\;\;\;\frac{y}{\frac{t \cdot \left(\left(a + 1\right) + y \cdot \frac{b}{t}\right)}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	23.4
Cost	1756

\[\begin{array}{l} t_1 := \frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ t_2 := b \cdot \frac{y}{t}\\ t_3 := a + \left(1 + t_2\right)\\ t_4 := \frac{y}{t} \cdot \frac{z}{t_3}\\ \mathbf{if}\;y \leq -9.5 \cdot 10^{+189}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.7 \cdot 10^{+128}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y \leq -1.34 \cdot 10^{-21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-180}:\\ \;\;\;\;\frac{x}{t_3}\\ \mathbf{elif}\;y \leq -4.9 \cdot 10^{-186}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-202}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{a + \left(-1 + \left(t_2 + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	23.4
Cost	1756

\[\begin{array}{l} t_1 := \frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ t_2 := b \cdot \frac{y}{t}\\ t_3 := a + \left(1 + t_2\right)\\ \mathbf{if}\;y \leq -1 \cdot 10^{+190}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.7 \cdot 10^{+128}:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{t_3}\\ \mathbf{elif}\;y \leq -3.9 \cdot 10^{-19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-180}:\\ \;\;\;\;\frac{x}{t_3}\\ \mathbf{elif}\;y \leq -4.9 \cdot 10^{-186}:\\ \;\;\;\;\frac{y \cdot z}{t \cdot \left(\frac{y \cdot b}{t} + \left(a + 1\right)\right)}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-203}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{a + \left(-1 + \left(t_2 + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	23.3
Cost	1756

\[\begin{array}{l} t_1 := \frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ t_2 := b \cdot \frac{y}{t}\\ \mathbf{if}\;y \leq -9.5 \cdot 10^{+189}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.7 \cdot 10^{+128}:\\ \;\;\;\;\frac{-z}{\frac{t}{y} \cdot \left(-1 - \left(a + t_2\right)\right)}\\ \mathbf{elif}\;y \leq -3.55 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-180}:\\ \;\;\;\;\frac{x}{a + \left(1 + t_2\right)}\\ \mathbf{elif}\;y \leq -4.9 \cdot 10^{-186}:\\ \;\;\;\;\frac{y \cdot z}{t \cdot \left(\frac{y \cdot b}{t} + \left(a + 1\right)\right)}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-203}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{x}{a + \left(-1 + \left(t_2 + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	13.3
Cost	1748

\[\begin{array}{l} t_1 := \frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{b}{\frac{t}{y}}\right)}\\ \mathbf{if}\;t \leq -7.4 \cdot 10^{-16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -5.9 \cdot 10^{-64}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t \leq -2.6 \cdot 10^{-123}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{-169}:\\ \;\;\;\;\frac{y \cdot z}{t \cdot \left(\frac{y \cdot b}{t} + \left(a + 1\right)\right)}\\ \mathbf{elif}\;t \leq 2.85 \cdot 10^{-89}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	13.2
Cost	1748

\[\begin{array}{l} t_1 := x + \frac{y}{\frac{t}{z}}\\ t_2 := \frac{t_1}{a + \left(1 + \frac{b}{\frac{t}{y}}\right)}\\ \mathbf{if}\;t \leq -7.4 \cdot 10^{-16}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -5.9 \cdot 10^{-64}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t \leq -1.18 \cdot 10^{-123}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -5.4 \cdot 10^{-172}:\\ \;\;\;\;\frac{y \cdot z}{t \cdot \left(\frac{y \cdot b}{t} + \left(a + 1\right)\right)}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-104}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{a + \left(1 + \frac{y}{\frac{t}{b}}\right)}\\ \end{array} \]

Alternative 6
Error	32.8
Cost	1633

\[\begin{array}{l} t_1 := \frac{y \cdot b}{t}\\ t_2 := \frac{x}{a + t_1}\\ t_3 := \frac{x}{1 + t_1}\\ \mathbf{if}\;a \leq -5.4 \cdot 10^{+39}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -6.6 \cdot 10^{-92}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-129}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 3.55 \cdot 10^{-97}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{-32}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{+44} \lor \neg \left(a \leq 8.8 \cdot 10^{+113}\right) \land a \leq 2.3 \cdot 10^{+159}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	31.4
Cost	1633

\[\begin{array}{l} t_1 := \frac{y \cdot b}{t}\\ t_2 := \frac{x}{1 + t_1}\\ \mathbf{if}\;a \leq -5.3 \cdot 10^{+39}:\\ \;\;\;\;\frac{x}{a + t_1}\\ \mathbf{elif}\;a \leq -9.2 \cdot 10^{-92}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 3 \cdot 10^{-128}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 3.9 \cdot 10^{-97}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 3.1 \cdot 10^{-37}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 2.05 \cdot 10^{+43} \lor \neg \left(a \leq 9 \cdot 10^{+112}\right) \land a \leq 2.3 \cdot 10^{+159}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a}\\ \end{array} \]

Alternative 8
Error	31.5
Cost	1632

\[\begin{array}{l} t_1 := \frac{y \cdot b}{t}\\ t_2 := \frac{x}{1 + t_1}\\ \mathbf{if}\;a \leq -8.5 \cdot 10^{+39}:\\ \;\;\;\;\frac{x}{a + t_1}\\ \mathbf{elif}\;a \leq -2.1 \cdot 10^{-92}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{-129}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{-97}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 1.5 \cdot 10^{-33}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 3.9 \cdot 10^{+45}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{+114}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a}\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{+159}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a}\\ \end{array} \]

Alternative 9
Error	27.4
Cost	1236

\[\begin{array}{l} t_1 := \frac{x}{a + 1}\\ \mathbf{if}\;t \leq -1.26 \cdot 10^{+89}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 11800:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{+46}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.7 \cdot 10^{+67}:\\ \;\;\;\;\frac{x}{1 + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+87}:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	23.9
Cost	1233

\[\begin{array}{l} t_1 := \frac{x}{a + \left(1 + b \cdot \frac{y}{t}\right)}\\ \mathbf{if}\;t \leq -1.4 \cdot 10^{+82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -5.3 \cdot 10^{+22}:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{a + 1}\\ \mathbf{elif}\;t \leq -9 \cdot 10^{-16} \lor \neg \left(t \leq 3.7 \cdot 10^{-88}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \end{array} \]

Alternative 11
Error	24.4
Cost	1233

\[\begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{+82}:\\ \;\;\;\;\frac{x}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;t \leq -5.5 \cdot 10^{+22}:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{a + 1}\\ \mathbf{elif}\;t \leq -1.2 \cdot 10^{-15} \lor \neg \left(t \leq 4.7 \cdot 10^{-88}\right):\\ \;\;\;\;\frac{x}{a + \left(1 + b \cdot \frac{y}{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \end{array} \]

Alternative 12
Error	22.0
Cost	1232

\[\begin{array}{l} t_1 := \frac{x + z \cdot \frac{y}{t}}{a + 1}\\ \mathbf{if}\;t \leq -1.8 \cdot 10^{+156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{x}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;t \leq -1.55 \cdot 10^{-15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-88}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a + \left(1 + b \cdot \frac{y}{t}\right)}\\ \end{array} \]

Alternative 13
Error	21.9
Cost	1232

\[\begin{array}{l} t_1 := \frac{x + z \cdot \frac{y}{t}}{a + 1}\\ t_2 := b \cdot \frac{y}{t}\\ \mathbf{if}\;t \leq -1.6 \cdot 10^{+156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.2 \cdot 10^{+83}:\\ \;\;\;\;\frac{x}{a + \left(-1 + \left(t_2 + 2\right)\right)}\\ \mathbf{elif}\;t \leq -1.4 \cdot 10^{-15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a + \left(1 + t_2\right)}\\ \end{array} \]

Alternative 14
Error	29.1
Cost	849

\[\begin{array}{l} t_1 := \frac{x}{a + 1}\\ \mathbf{if}\;t \leq -3.1 \cdot 10^{+83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -5.6 \cdot 10^{+19}:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{a + 1}\\ \mathbf{elif}\;t \leq -8.5 \cdot 10^{-14} \lor \neg \left(t \leq 3.4 \cdot 10^{-65}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 15
Error	38.2
Cost	721

\[\begin{array}{l} \mathbf{if}\;a \leq -1.65 \cdot 10^{+67}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq 6.4 \cdot 10^{+50} \lor \neg \left(a \leq 1.15 \cdot 10^{+113}\right) \land a \leq 2.3 \cdot 10^{+159}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

Alternative 16
Error	28.8
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-25}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 6500000000000:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 17
Error	47.7
Cost	192

\[\frac{x}{a} \]

Reproduce

herbie shell --seed 2023016 
(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))))