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

Average Error: 16.9 → 5.9

Time: 21.0s

Precision: binary64

Cost: 8388

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{t \cdot \left(a + \mathsf{fma}\left(y, \frac{b}{t}, 1\right)\right)}{z}}\\ \mathbf{elif}\;t_1 \leq -5 \cdot 10^{-301} \lor \neg \left(t_1 \leq 0\right) \land t_1 \leq 10^{+305}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \end{array} \]

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 (+ a (fma y (/ b t) 1.0))) z))
     (if (or (<= t_1 -5e-301) (and (not (<= t_1 0.0)) (<= t_1 1e+305)))
       t_1
       (+ (/ z b) (* (/ 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 * (a + fma(y, (b / t), 1.0))) / z);
	} else if ((t_1 <= -5e-301) || (!(t_1 <= 0.0) && (t_1 <= 1e+305))) {
		tmp = t_1;
	} else {
		tmp = (z / b) + ((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(y / Float64(Float64(t * Float64(a + fma(y, Float64(b / t), 1.0))) / z));
	elseif ((t_1 <= -5e-301) || (!(t_1 <= 0.0) && (t_1 <= 1e+305)))
		tmp = t_1;
	else
		tmp = Float64(Float64(z / b) + Float64(Float64(t / y) * Float64(x / b)));
	end
	return 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[(y / N[(N[(t * N[(a + N[(y * N[(b / t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$1, -5e-301], And[N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision], LessEqual[t$95$1, 1e+305]]], t$95$1, N[(N[(z / b), $MachinePrecision] + N[(N[(t / y), $MachinePrecision] * N[(x / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	16.9
Target	13.5
Herbie	5.9

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

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. Simplified 39.5

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

      [Start] 64.0	\[ \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-/l* [=>] 39.5	\[ \frac{x + \color{blue}{\frac{y}{\frac{t}{z}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-+l+ [=>] 39.5	\[ \frac{x + \frac{y}{\frac{t}{z}}}{\color{blue}{a + \left(1 + \frac{y \cdot b}{t}\right)}} \]
      associate-/l* [=>] 39.5	\[ \frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \color{blue}{\frac{y}{\frac{t}{b}}}\right)} \]

   3. Taylor expanded in x around 0 40.4

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

   4. Simplified 16.6

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

      [Start] 40.4	\[ \frac{y \cdot z}{t \cdot \left(\frac{y \cdot b}{t} + \left(1 + a\right)\right)} \]
      associate-/l* [=>] 15.6	\[ \color{blue}{\frac{y}{\frac{t \cdot \left(\frac{y \cdot b}{t} + \left(1 + a\right)\right)}{z}}} \]
      associate-+r+ [=>] 15.6	\[ \frac{y}{\frac{t \cdot \color{blue}{\left(\left(\frac{y \cdot b}{t} + 1\right) + a\right)}}{z}} \]
      associate-*r/ [<=] 16.6	\[ \frac{y}{\frac{t \cdot \left(\left(\color{blue}{y \cdot \frac{b}{t}} + 1\right) + a\right)}{z}} \]
      fma-udef [<=] 16.6	\[ \frac{y}{\frac{t \cdot \left(\color{blue}{\mathsf{fma}\left(y, \frac{b}{t}, 1\right)} + a\right)}{z}} \]
      +-commutative [<=] 16.6	\[ \frac{y}{\frac{t \cdot \color{blue}{\left(a + \mathsf{fma}\left(y, \frac{b}{t}, 1\right)\right)}}{z}} \]

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -5.00000000000000013e-301 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 9.9999999999999994e304

   1. Initial program 0.5

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

   if -5.00000000000000013e-301 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -0.0 or 9.9999999999999994e304 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t)))

   1. Initial program 45.2

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

   2. Simplified 34.7

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

      [Start] 45.2	\[ \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      +-commutative [=>] 45.2	\[ \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-*r/ [<=] 41.7	\[ \frac{\color{blue}{y \cdot \frac{z}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      *-commutative [<=] 41.7	\[ \frac{\color{blue}{\frac{z}{t} \cdot y} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      fma-def [=>] 41.7	\[ \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-+l+ [=>] 41.7	\[ \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\color{blue}{a + \left(1 + \frac{y \cdot b}{t}\right)}} \]
      +-commutative [=>] 41.7	\[ \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{a + \color{blue}{\left(\frac{y \cdot b}{t} + 1\right)}} \]
      associate-*r/ [<=] 34.7	\[ \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{a + \left(\color{blue}{y \cdot \frac{b}{t}} + 1\right)} \]
      *-commutative [<=] 34.7	\[ \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{a + \left(\color{blue}{\frac{b}{t} \cdot y} + 1\right)} \]
      fma-def [=>] 34.7	\[ \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{a + \color{blue}{\mathsf{fma}\left(\frac{b}{t}, y, 1\right)}} \]

   3. Taylor expanded in b around inf 47.3

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

   4. Simplified 38.9

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

      [Start] 47.3	\[ \frac{t \cdot \left(\frac{y \cdot z}{t} + x\right)}{y \cdot b} \]
      times-frac [=>] 41.5	\[ \color{blue}{\frac{t}{y} \cdot \frac{\frac{y \cdot z}{t} + x}{b}} \]
      +-commutative [=>] 41.5	\[ \frac{t}{y} \cdot \frac{\color{blue}{x + \frac{y \cdot z}{t}}}{b} \]
      associate-/l* [=>] 38.9	\[ \frac{t}{y} \cdot \frac{x + \color{blue}{\frac{y}{\frac{t}{z}}}}{b} \]

   5. Taylor expanded in t around 0 20.8

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

   6. Simplified 16.1

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

      [Start] 20.8	\[ \frac{t \cdot x}{y \cdot b} + \frac{z}{b} \]
      +-commutative [=>] 20.8	\[ \color{blue}{\frac{z}{b} + \frac{t \cdot x}{y \cdot b}} \]
      times-frac [=>] 16.1	\[ \frac{z}{b} + \color{blue}{\frac{t}{y} \cdot \frac{x}{b}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 5.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{t \cdot \left(a + \mathsf{fma}\left(y, \frac{b}{t}, 1\right)\right)}{z}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -5 \cdot 10^{-301} \lor \neg \left(\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 0\right) \land \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 10^{+305}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.4
Cost	6869

\[\begin{array}{l} t_1 := \frac{y \cdot b}{t} + \left(a + 1\right)\\ t_2 := \frac{x + \frac{y \cdot z}{t}}{t_1}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{t \cdot \left(1 + \left(a + b \cdot \frac{y}{t}\right)\right)}{z}}\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-128}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq -5 \cdot 10^{-301}:\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{t_1}\\ \mathbf{elif}\;t_2 \leq 0 \lor \neg \left(t_2 \leq 10^{+305}\right):\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	6.5
Cost	6869

\[\begin{array}{l} t_1 := \frac{y \cdot b}{t} + \left(a + 1\right)\\ t_2 := \frac{x + \frac{y \cdot z}{t}}{t_1}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{t \cdot \left(1 + \left(a + b \cdot \frac{y}{t}\right)\right)}{z}}\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-128}:\\ \;\;\;\;\frac{y \cdot z}{t \cdot t_1} + \frac{x}{t_1}\\ \mathbf{elif}\;t_2 \leq -5 \cdot 10^{-301}:\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{t_1}\\ \mathbf{elif}\;t_2 \leq 0 \lor \neg \left(t_2 \leq 10^{+305}\right):\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	29.2
Cost	2024

\[\begin{array}{l} t_1 := \frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ t_2 := \frac{x + z \cdot \frac{y}{t}}{a}\\ t_3 := \frac{x}{1 + \frac{y \cdot b}{t}}\\ \mathbf{if}\;a \leq -1.66 \cdot 10^{+133}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1.35 \cdot 10^{-83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -4.8 \cdot 10^{-133}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -1.5 \cdot 10^{-240}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -4.5 \cdot 10^{-290}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 1.02 \cdot 10^{-263}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{-138}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{-125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 8.8 \cdot 10^{-32}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 9.6 \cdot 10^{+53}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	28.7
Cost	1632

\[\begin{array}{l} t_1 := \frac{z + \frac{x \cdot t}{y}}{b}\\ t_2 := \frac{x + z \cdot \frac{y}{t}}{a}\\ t_3 := \frac{x}{1 + \frac{y \cdot b}{t}}\\ \mathbf{if}\;a \leq -1.06 \cdot 10^{+83}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1.42 \cdot 10^{-86}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -4.3 \cdot 10^{-132}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -1.7 \cdot 10^{-240}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.05 \cdot 10^{-297}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{-264}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{-20}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 5.4 \cdot 10^{+54}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	28.6
Cost	1500

\[\begin{array}{l} t_1 := \frac{x}{1 + \frac{y \cdot b}{t}}\\ t_2 := \frac{x + z \cdot \frac{y}{t}}{a}\\ \mathbf{if}\;a \leq -7.6:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1.18 \cdot 10^{-129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.7 \cdot 10^{-240}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -2.7 \cdot 10^{-298}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-259}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{-19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{+53}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	15.7
Cost	1353

\[\begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{+134} \lor \neg \left(b \leq 3.1 \cdot 10^{+99}\right):\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \end{array} \]

Alternative 7
Error	11.7
Cost	1353

\[\begin{array}{l} \mathbf{if}\;t \leq -2.2 \cdot 10^{-144} \lor \neg \left(t \leq 1.4 \cdot 10^{-114}\right):\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{b}{\frac{t}{y}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \end{array} \]

Alternative 8
Error	21.5
Cost	1233

\[\begin{array}{l} t_1 := \frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{if}\;b \leq -3 \cdot 10^{+136}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -7.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{a + \left(1 + \frac{b}{\frac{t}{y}}\right)}\\ \mathbf{elif}\;b \leq -9 \cdot 10^{-85} \lor \neg \left(b \leq 2.05 \cdot 10^{+99}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a + 1}\\ \end{array} \]

Alternative 9
Error	37.8
Cost	1117

\[\begin{array}{l} \mathbf{if}\;a \leq -0.0004:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq -6 \cdot 10^{-297}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{-257}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{-21}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 9.6 \cdot 10^{+106} \lor \neg \left(a \leq 1.2 \cdot 10^{+159}\right) \land a \leq 7.3 \cdot 10^{+208}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

Alternative 10
Error	37.7
Cost	1117

\[\begin{array}{l} \mathbf{if}\;a \leq -0.0004:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq -1.1 \cdot 10^{-297}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{-259}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 8.5 \cdot 10^{-6}:\\ \;\;\;\;x - x \cdot a\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{+106} \lor \neg \left(a \leq 2.7 \cdot 10^{+160}\right) \land a \leq 7.3 \cdot 10^{+208}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

Alternative 11
Error	33.8
Cost	1108

\[\begin{array}{l} t_1 := \frac{x}{a + 1}\\ \mathbf{if}\;b \leq -1.2 \cdot 10^{+37}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;b \leq -8.2 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -9 \cdot 10^{-85}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{+87}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 2.65 \cdot 10^{+243}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{y} \cdot \frac{x}{b}\\ \end{array} \]

Alternative 12
Error	24.1
Cost	969

\[\begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{+73} \lor \neg \left(t \leq 3.8 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{x}{a + \left(1 + \frac{b}{\frac{t}{y}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \end{array} \]

Alternative 13
Error	24.0
Cost	968

\[\begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{+73}:\\ \;\;\;\;\frac{x}{a + \left(1 + \frac{b}{\frac{t}{y}}\right)}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-19}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \end{array} \]

Alternative 14
Error	29.5
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -2.65 \cdot 10^{+68} \lor \neg \left(t \leq 5.05 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 15
Error	37.0
Cost	456

\[\begin{array}{l} \mathbf{if}\;a \leq -0.0004:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

Alternative 16
Error	50.9
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022356 
(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))))