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

Average Error: 16.6 → 7.7

Time: 23.3s

Precision: binary64

Cost: 3400

Math

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

↓

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

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 INFINITY)
       (/ (+ x (/ z (/ t y))) (+ a (+ 1.0 (/ b (/ t y)))))
       (/ (+ 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 <= ((double) INFINITY)) {
		tmp = (x + (z / (t / y))) / (a + (1.0 + (b / (t / y))));
	} 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 <= Double.POSITIVE_INFINITY) {
		tmp = (x + (z / (t / y))) / (a + (1.0 + (b / (t / y))));
	} 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 <= math.inf:
		tmp = (x + (z / (t / y))) / (a + (1.0 + (b / (t / y))))
	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 <= Inf)
		tmp = Float64(Float64(x + Float64(z / Float64(t / y))) / Float64(a + Float64(1.0 + Float64(b / Float64(t / y)))));
	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 <= Inf)
		tmp = (x + (z / (t / y))) / (a + (1.0 + (b / (t / y))));
	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, Infinity], N[(N[(x + N[(z / N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a + N[(1.0 + N[(b / N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z + N[(t / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]]

Target

Original	16.6
Target	13.4
Herbie	7.7

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

Derivation

1. Split input into 3 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 36.2

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

   3. Simplified 13.4

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

      [Start] 36.2	\[ \frac{y \cdot z}{t \cdot \left(1 + \left(\frac{y \cdot b}{t} + a\right)\right)} \]
      times-frac [=>] 13.4	\[ \color{blue}{\frac{y}{t} \cdot \frac{z}{1 + \left(\frac{y \cdot b}{t} + a\right)}} \]
      associate-+r+ [=>] 13.4	\[ \frac{y}{t} \cdot \frac{z}{\color{blue}{\left(1 + \frac{y \cdot b}{t}\right) + a}} \]
      +-commutative [<=] 13.4	\[ \frac{y}{t} \cdot \frac{z}{\color{blue}{a + \left(1 + \frac{y \cdot b}{t}\right)}} \]
      associate-*l/ [<=] 13.4	\[ \frac{y}{t} \cdot \frac{z}{a + \left(1 + \color{blue}{\frac{y}{t} \cdot b}\right)} \]
      *-commutative [=>] 13.4	\[ \frac{y}{t} \cdot \frac{z}{a + \left(1 + \color{blue}{b \cdot \frac{y}{t}}\right)} \]

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

   1. Initial program 9.0

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

   2. Simplified 9.6

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

      [Start] 9.0	\[ \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-/l* [=>] 10.5	\[ \frac{x + \color{blue}{\frac{y}{\frac{t}{z}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-+l+ [=>] 10.5	\[ \frac{x + \frac{y}{\frac{t}{z}}}{\color{blue}{a + \left(1 + \frac{y \cdot b}{t}\right)}} \]
      *-commutative [=>] 10.5	\[ \frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{\color{blue}{b \cdot y}}{t}\right)} \]
      associate-/l* [=>] 9.6	\[ \frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \color{blue}{\frac{b}{\frac{t}{y}}}\right)} \]

   3. Applied egg-rr 9.8

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

   4. Applied egg-rr 8.0

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

   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 56.6

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

      [Start] 64.0	\[ \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      +-commutative [=>] 64.0	\[ \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-*r/ [<=] 63.7	\[ \frac{\color{blue}{y \cdot \frac{z}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      *-commutative [<=] 63.7	\[ \frac{\color{blue}{\frac{z}{t} \cdot y} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      fma-def [=>] 63.7	\[ \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-+l+ [=>] 63.7	\[ \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\color{blue}{a + \left(1 + \frac{y \cdot b}{t}\right)}} \]
      +-commutative [=>] 63.7	\[ \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{a + \color{blue}{\left(\frac{y \cdot b}{t} + 1\right)}} \]
      associate-*r/ [<=] 56.6	\[ \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{a + \left(\color{blue}{y \cdot \frac{b}{t}} + 1\right)} \]
      *-commutative [<=] 56.6	\[ \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{a + \left(\color{blue}{\frac{b}{t} \cdot y} + 1\right)} \]
      fma-def [=>] 56.6	\[ \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{a + \color{blue}{\mathsf{fma}\left(\frac{b}{t}, y, 1\right)}} \]

   3. Taylor expanded in b around inf 64.0

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

   4. Simplified 56.2

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

      [Start] 64.0	\[ \frac{t \cdot \left(\frac{y \cdot z}{t} + x\right)}{y \cdot b} \]
      times-frac [=>] 64.0	\[ \color{blue}{\frac{t}{y} \cdot \frac{\frac{y \cdot z}{t} + x}{b}} \]
      +-commutative [=>] 64.0	\[ \frac{t}{y} \cdot \frac{\color{blue}{x + \frac{y \cdot z}{t}}}{b} \]
      associate-/l* [=>] 56.2	\[ \frac{t}{y} \cdot \frac{x + \color{blue}{\frac{y}{\frac{t}{z}}}}{b} \]

   5. Taylor expanded in t around 0 5.0

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

   6. Taylor expanded in b around 0 5.0

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

   7. Simplified 2.6

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

      [Start] 5.0	\[ \frac{\frac{t \cdot x}{y} + z}{b} \]
      +-commutative [=>] 5.0	\[ \frac{\color{blue}{z + \frac{t \cdot x}{y}}}{b} \]
      associate-/l* [=>] 2.6	\[ \frac{z + \color{blue}{\frac{t}{\frac{y}{x}}}}{b} \]

3. Recombined 3 regimes into one program.

4. Final simplification 7.7

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

Alternatives

Alternative 1
Error	19.3
Cost	1760

\[\begin{array}{l} t_1 := a + \left(1 + b \cdot \frac{y}{t}\right)\\ t_2 := \frac{x}{t_1}\\ t_3 := \frac{z + \frac{t}{\frac{y}{x}}}{b}\\ t_4 := \frac{y}{t} \cdot \frac{z}{t_1}\\ \mathbf{if}\;y \leq -1.2 \cdot 10^{+45}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -0.0017:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -2.4 \cdot 10^{-31}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-47}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+45}:\\ \;\;\;\;\frac{x}{a + \left(1 + y \cdot \frac{b}{t}\right)}\\ \mathbf{elif}\;y \leq 2.46 \cdot 10^{+58}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+141}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+160}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 2
Error	19.5
Cost	1760

\[\begin{array}{l} t_1 := a + \left(1 + b \cdot \frac{y}{t}\right)\\ t_2 := \frac{x}{t_1}\\ t_3 := \frac{z + \frac{t}{\frac{y}{x}}}{b}\\ \mathbf{if}\;y \leq -4.3 \cdot 10^{+40}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -0.0096:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -3.1 \cdot 10^{-31}:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{t_1}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-47}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{+45}:\\ \;\;\;\;\frac{x}{a + \left(1 + y \cdot \frac{b}{t}\right)}\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{+84}:\\ \;\;\;\;\frac{y \cdot z}{t \cdot \left(\frac{y \cdot b}{t} + \left(a + 1\right)\right)}\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{+141}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+160}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 3
Error	28.8
Cost	1364

\[\begin{array}{l} t_1 := \frac{x + z \cdot \frac{y}{t}}{a}\\ t_2 := \frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{if}\;a \leq -3.9 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.05 \cdot 10^{-245}:\\ \;\;\;\;\frac{x}{1 + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{-28}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 7800000:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{elif}\;a \leq 3 \cdot 10^{+60}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	22.5
Cost	1364

\[\begin{array}{l} t_1 := \frac{x}{a + \left(1 + b \cdot \frac{y}{t}\right)}\\ t_2 := \frac{z + \frac{t}{\frac{y}{x}}}{b}\\ \mathbf{if}\;y \leq -5.4 \cdot 10^{+45}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -0.0065:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -4.1 \cdot 10^{-32}:\\ \;\;\;\;\frac{y \cdot z}{t + y \cdot b}\\ \mathbf{elif}\;y \leq -3.5 \cdot 10^{-76}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+162}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	22.4
Cost	1364

\[\begin{array}{l} t_1 := \frac{z + \frac{t}{\frac{y}{x}}}{b}\\ \mathbf{if}\;y \leq -1.35 \cdot 10^{+42}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -0.0096:\\ \;\;\;\;\frac{x}{a + \left(1 + b \cdot \frac{y}{t}\right)}\\ \mathbf{elif}\;y \leq -7.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{y \cdot z}{t + y \cdot b}\\ \mathbf{elif}\;y \leq -1.75 \cdot 10^{-74}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+160}:\\ \;\;\;\;\frac{x}{a + \left(1 + \frac{b}{\frac{t}{y}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	20.1
Cost	1364

\[\begin{array}{l} t_1 := \frac{z + \frac{t}{\frac{y}{x}}}{b}\\ \mathbf{if}\;y \leq -3 \cdot 10^{+38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -0.00152:\\ \;\;\;\;\frac{x}{a + \left(1 + b \cdot \frac{y}{t}\right)}\\ \mathbf{elif}\;y \leq -3.1 \cdot 10^{-31}:\\ \;\;\;\;\frac{y \cdot z}{t + y \cdot b}\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-47}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a + 1}\\ \mathbf{elif}\;y \leq 6.3 \cdot 10^{+162}:\\ \;\;\;\;\frac{x}{a + \left(1 + y \cdot \frac{b}{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	19.8
Cost	1364

\[\begin{array}{l} t_1 := \frac{z + \frac{t}{\frac{y}{x}}}{b}\\ \mathbf{if}\;y \leq -3.1 \cdot 10^{+44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -0.0068:\\ \;\;\;\;\frac{x}{a + \left(1 + b \cdot \frac{y}{t}\right)}\\ \mathbf{elif}\;y \leq -3 \cdot 10^{-31}:\\ \;\;\;\;\frac{y \cdot z}{t + y \cdot b}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-48}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+160}:\\ \;\;\;\;\frac{x}{a + \left(1 + y \cdot \frac{b}{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	12.2
Cost	1353

\[\begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{-191} \lor \neg \left(t \leq 5.6 \cdot 10^{-221}\right):\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{a + \left(1 + \frac{b}{\frac{t}{y}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t}{y \cdot b} + \frac{z}{b}\\ \end{array} \]

Alternative 9
Error	12.1
Cost	1352

\[\begin{array}{l} t_1 := a + \left(1 + \frac{b}{\frac{t}{y}}\right)\\ \mathbf{if}\;t \leq -4 \cdot 10^{-188}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{t_1}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-221}:\\ \;\;\;\;\frac{x \cdot t}{y \cdot b} + \frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{t_1}\\ \end{array} \]

Alternative 10
Error	28.1
Cost	1236

\[\begin{array}{l} t_1 := \frac{x}{1 + \frac{y \cdot b}{t}}\\ t_2 := \frac{x + z \cdot \frac{y}{t}}{a}\\ \mathbf{if}\;a \leq -3.8 \cdot 10^{+15}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1.35 \cdot 10^{-243}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.9 \cdot 10^{-297}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 14.3:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2000000000000:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 11
Error	28.8
Cost	1236

\[\begin{array}{l} t_1 := \frac{x + z \cdot \frac{y}{t}}{a}\\ t_2 := \frac{z + \frac{t}{\frac{y}{x}}}{b}\\ \mathbf{if}\;a \leq -3.9 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.45 \cdot 10^{-245}:\\ \;\;\;\;\frac{x}{1 + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;a \leq 2.9 \cdot 10^{-29}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 19500000:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{+55}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	37.2
Cost	588

\[\begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq -8.5 \cdot 10^{-198}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 5.6 \cdot 10^{+62}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

Alternative 13
Error	28.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-31}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{+33}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 14
Error	36.6
Cost	456

\[\begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

Alternative 15
Error	50.3
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023067 
(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))))