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

Average Error: 16.1 → 5.8

Time: 34.3s

Precision: binary64

Cost: 9864

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y \cdot b}{t} + \left(a + 1\right)\\ t_2 := \frac{x + \frac{y \cdot z}{t}}{t_1}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{t \cdot \mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}{z}}\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{-315}:\\ \;\;\;\;\frac{x + {\left(\frac{t}{y \cdot z}\right)}^{-1}}{t_1}\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\frac{z}{b} + t \cdot \left(\frac{\frac{x}{b}}{y} + \frac{-1 - a}{\frac{y \cdot \left(b \cdot b\right)}{z}}\right)\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+295}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

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 (+ (/ (* y b) t) (+ a 1.0))) (t_2 (/ (+ x (/ (* y z) t)) t_1)))
   (if (<= t_2 (- INFINITY))
     (/ y (/ (* t (fma y (/ b t) (+ a 1.0))) z))
     (if (<= t_2 -1e-315)
       (/ (+ x (pow (/ t (* y z)) -1.0)) t_1)
       (if (<= t_2 0.0)
         (+ (/ z b) (* t (+ (/ (/ x b) y) (/ (- -1.0 a) (/ (* y (* b b)) z)))))
         (if (<= t_2 2e+295) 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 = ((y * b) / t) + (a + 1.0);
	double t_2 = (x + ((y * z) / t)) / t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = y / ((t * fma(y, (b / t), (a + 1.0))) / z);
	} else if (t_2 <= -1e-315) {
		tmp = (x + pow((t / (y * z)), -1.0)) / t_1;
	} else if (t_2 <= 0.0) {
		tmp = (z / b) + (t * (((x / b) / y) + ((-1.0 - a) / ((y * (b * b)) / z))));
	} else if (t_2 <= 2e+295) {
		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(Float64(y * b) / t) + Float64(a + 1.0))
	t_2 = Float64(Float64(x + Float64(Float64(y * z) / t)) / t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(y / Float64(Float64(t * fma(y, Float64(b / t), Float64(a + 1.0))) / z));
	elseif (t_2 <= -1e-315)
		tmp = Float64(Float64(x + (Float64(t / Float64(y * z)) ^ -1.0)) / t_1);
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(z / b) + Float64(t * Float64(Float64(Float64(x / b) / y) + Float64(Float64(-1.0 - a) / Float64(Float64(y * Float64(b * b)) / z)))));
	elseif (t_2 <= 2e+295)
		tmp = t_2;
	else
		tmp = Float64(z / 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[(N[(y * b), $MachinePrecision] / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(y / N[(N[(t * N[(y * N[(b / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1e-315], N[(N[(x + N[Power[N[(t / N[(y * z), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(z / b), $MachinePrecision] + N[(t * N[(N[(N[(x / b), $MachinePrecision] / y), $MachinePrecision] + N[(N[(-1.0 - a), $MachinePrecision] / N[(N[(y * N[(b * b), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+295], t$95$2, N[(z / b), $MachinePrecision]]]]]]]

Target

Original	16.1
Target	12.9
Herbie	5.8

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

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. Simplified 40.5

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

      [Start] 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-*l/ [<=] 40.5	\[ \frac{\color{blue}{\frac{y}{t} \cdot z} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      fma-def [=>] 40.5	\[ \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      +-commutative [=>] 40.5	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\color{blue}{\frac{y \cdot b}{t} + \left(a + 1\right)}} \]
      associate-+r+ [=>] 40.5	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\color{blue}{\left(\frac{y \cdot b}{t} + a\right) + 1}} \]
      +-commutative [=>] 40.5	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\color{blue}{1 + \left(\frac{y \cdot b}{t} + a\right)}} \]
      associate-*l/ [<=] 40.5	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + \left(\color{blue}{\frac{y}{t} \cdot b} + a\right)} \]
      fma-def [=>] 40.5	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + \color{blue}{\mathsf{fma}\left(\frac{y}{t}, b, a\right)}} \]

   3. Taylor expanded in z around inf 36.6

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

   4. Simplified 20.3

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

      [Start] 36.6	\[ \frac{y \cdot z}{t \cdot \left(\frac{y \cdot b}{t} + \left(1 + a\right)\right)} \]
      times-frac [=>] 18.2	\[ \color{blue}{\frac{y}{t} \cdot \frac{z}{\frac{y \cdot b}{t} + \left(1 + a\right)}} \]
      +-commutative [=>] 18.2	\[ \frac{y}{t} \cdot \frac{z}{\color{blue}{\left(1 + a\right) + \frac{y \cdot b}{t}}} \]
      associate-/l* [=>] 20.3	\[ \frac{y}{t} \cdot \frac{z}{\left(1 + a\right) + \color{blue}{\frac{y}{\frac{t}{b}}}} \]

   5. Applied egg-rr 16.2

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -9.999999985e-316

   1. Initial program 0.5

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

   2. Applied egg-rr 0.6

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

   if -9.999999985e-316 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 29.2

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

   2. Taylor expanded in t around 0 26.3

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

   3. Simplified 19.2

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

      [Start] 26.3	\[ \frac{z}{b} + \left(\frac{x}{y \cdot b} - \frac{\left(1 + a\right) \cdot z}{y \cdot {b}^{2}}\right) \cdot t \]
      *-commutative [=>] 26.3	\[ \frac{z}{b} + \color{blue}{t \cdot \left(\frac{x}{y \cdot b} - \frac{\left(1 + a\right) \cdot z}{y \cdot {b}^{2}}\right)} \]
      *-commutative [=>] 26.3	\[ \frac{z}{b} + t \cdot \left(\frac{x}{\color{blue}{b \cdot y}} - \frac{\left(1 + a\right) \cdot z}{y \cdot {b}^{2}}\right) \]
      associate-/r* [=>] 20.1	\[ \frac{z}{b} + t \cdot \left(\color{blue}{\frac{\frac{x}{b}}{y}} - \frac{\left(1 + a\right) \cdot z}{y \cdot {b}^{2}}\right) \]
      +-commutative [=>] 20.1	\[ \frac{z}{b} + t \cdot \left(\frac{\frac{x}{b}}{y} - \frac{\color{blue}{\left(a + 1\right)} \cdot z}{y \cdot {b}^{2}}\right) \]
      associate-/l* [=>] 19.2	\[ \frac{z}{b} + t \cdot \left(\frac{\frac{x}{b}}{y} - \color{blue}{\frac{a + 1}{\frac{y \cdot {b}^{2}}{z}}}\right) \]
      +-commutative [<=] 19.2	\[ \frac{z}{b} + t \cdot \left(\frac{\frac{x}{b}}{y} - \frac{\color{blue}{1 + a}}{\frac{y \cdot {b}^{2}}{z}}\right) \]
      unpow2 [=>] 19.2	\[ \frac{z}{b} + t \cdot \left(\frac{\frac{x}{b}}{y} - \frac{1 + a}{\frac{y \cdot \color{blue}{\left(b \cdot b\right)}}{z}}\right) \]

   if -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 2e295

   1. Initial program 0.5

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

   if 2e295 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t)))

   1. Initial program 63.2

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

   2. Simplified 51.3

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

      [Start] 63.2	\[ \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      +-commutative [=>] 63.2	\[ \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-*l/ [<=] 56.7	\[ \frac{\color{blue}{\frac{y}{t} \cdot z} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      fma-def [=>] 56.7	\[ \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      +-commutative [=>] 56.7	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\color{blue}{\frac{y \cdot b}{t} + \left(a + 1\right)}} \]
      associate-+r+ [=>] 56.7	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\color{blue}{\left(\frac{y \cdot b}{t} + a\right) + 1}} \]
      +-commutative [=>] 56.7	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\color{blue}{1 + \left(\frac{y \cdot b}{t} + a\right)}} \]
      associate-*l/ [<=] 51.3	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + \left(\color{blue}{\frac{y}{t} \cdot b} + a\right)} \]
      fma-def [=>] 51.3	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + \color{blue}{\mathsf{fma}\left(\frac{y}{t}, b, a\right)}} \]

   3. Taylor expanded in y around inf 12.9

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 5.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{t \cdot \mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}{z}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -1 \cdot 10^{-315}:\\ \;\;\;\;\frac{x + {\left(\frac{t}{y \cdot z}\right)}^{-1}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 0:\\ \;\;\;\;\frac{z}{b} + t \cdot \left(\frac{\frac{x}{b}}{y} + \frac{-1 - a}{\frac{y \cdot \left(b \cdot b\right)}{z}}\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 2 \cdot 10^{+295}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	5.7
Cost	8388

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{t \cdot \mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}{z}}\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-315}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{z}{b} + t \cdot \left(\frac{\frac{x}{b}}{y} + \frac{-1 - a}{\frac{y \cdot \left(b \cdot b\right)}{z}}\right)\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+295}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 2
Error	6.6
Cost	5712

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-315}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{t}{y} \cdot \frac{x + \frac{y}{\frac{t}{z}}}{b}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+295}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 3
Error	5.9
Cost	5712

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-315}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{z}{b} + t \cdot \left(\frac{\frac{x}{b}}{y} + \frac{-1 - a}{\frac{y \cdot \left(b \cdot b\right)}{z}}\right)\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+295}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 4
Error	28.9
Cost	2288

\[\begin{array}{l} t_1 := \frac{x}{1 + \frac{y}{\frac{t}{b}}}\\ t_2 := \frac{z}{b} + \frac{x \cdot t}{y \cdot b}\\ t_3 := x + \frac{y \cdot z}{t}\\ \mathbf{if}\;a \leq -8600000000000:\\ \;\;\;\;\frac{t_3}{a}\\ \mathbf{elif}\;a \leq -2 \cdot 10^{-51}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -2.1 \cdot 10^{-80}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -1.6 \cdot 10^{-94}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -1.15 \cdot 10^{-209}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -2.3 \cdot 10^{-246}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -6.6 \cdot 10^{-297}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 8.5 \cdot 10^{-297}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 3.4 \cdot 10^{-146}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 2.55 \cdot 10^{-17}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 170000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{+50}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a}\\ \end{array} \]

Alternative 5
Error	28.6
Cost	2288

\[\begin{array}{l} t_1 := \frac{z}{b} + \frac{x \cdot t}{y \cdot b}\\ t_2 := x + \frac{y \cdot z}{t}\\ \mathbf{if}\;a \leq -48000000000:\\ \;\;\;\;\frac{t_2}{a}\\ \mathbf{elif}\;a \leq -1.6 \cdot 10^{-51}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -3.3 \cdot 10^{-80}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1.15 \cdot 10^{-91}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -1.4 \cdot 10^{-207}:\\ \;\;\;\;\frac{x}{1 + \frac{y}{\frac{t}{b}}}\\ \mathbf{elif}\;a \leq -4.1 \cdot 10^{-247}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -7 \cdot 10^{-298}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 5 \cdot 10^{-297}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 1.12 \cdot 10^{-148}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 2.25 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 3 \cdot 10^{+17}:\\ \;\;\;\;\frac{x}{\left(a + 1\right) + b \cdot \frac{y}{t}}\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{+54}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a}\\ \end{array} \]

Alternative 6
Error	28.5
Cost	2288

\[\begin{array}{l} t_1 := \frac{z}{b} + \frac{x \cdot t}{y \cdot b}\\ t_2 := x + \frac{y \cdot z}{t}\\ \mathbf{if}\;a \leq -510000000000:\\ \;\;\;\;\frac{t_2}{a}\\ \mathbf{elif}\;a \leq -1.95 \cdot 10^{-51}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -1.12 \cdot 10^{-79}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -9.6 \cdot 10^{-92}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -3.5 \cdot 10^{-209}:\\ \;\;\;\;\frac{x}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \mathbf{elif}\;a \leq -5.5 \cdot 10^{-247}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -7.2 \cdot 10^{-298}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 1.45 \cdot 10^{-297}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{-142}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{-18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.45 \cdot 10^{+19}:\\ \;\;\;\;\frac{x}{\left(a + 1\right) + b \cdot \frac{y}{t}}\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{+54}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a}\\ \end{array} \]

Alternative 7
Error	29.6
Cost	2160

\[\begin{array}{l} t_1 := \frac{x}{1 + \frac{y}{\frac{t}{b}}}\\ t_2 := \frac{x + z \cdot \frac{y}{t}}{a}\\ t_3 := x + \frac{y \cdot z}{t}\\ \mathbf{if}\;a \leq -9600000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1.75 \cdot 10^{-51}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -2.7 \cdot 10^{-79}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -1.8 \cdot 10^{-89}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -3.5 \cdot 10^{-210}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -4.8 \cdot 10^{-246}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -2.2 \cdot 10^{-297}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 5.1 \cdot 10^{-297}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{-148}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 2.7 \cdot 10^{-17}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{+19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{+48}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	29.9
Cost	2160

\[\begin{array}{l} t_1 := \frac{x}{1 + \frac{y}{\frac{t}{b}}}\\ t_2 := x + \frac{y \cdot z}{t}\\ \mathbf{if}\;a \leq -13500000000:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a}\\ \mathbf{elif}\;a \leq -1.7 \cdot 10^{-51}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -2.7 \cdot 10^{-79}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -2.3 \cdot 10^{-88}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -1.8 \cdot 10^{-207}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.2 \cdot 10^{-245}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -2.4 \cdot 10^{-297}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{-296}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 1.45 \cdot 10^{-143}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 4 \cdot 10^{-18}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{+19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{+50}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a}\\ \end{array} \]

Alternative 9
Error	30.0
Cost	2160

\[\begin{array}{l} t_1 := \frac{x}{1 + \frac{y}{\frac{t}{b}}}\\ t_2 := x + \frac{y \cdot z}{t}\\ \mathbf{if}\;a \leq -1250000000000:\\ \;\;\;\;\frac{t_2}{a}\\ \mathbf{elif}\;a \leq -1.7 \cdot 10^{-51}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -3.3 \cdot 10^{-80}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -2.95 \cdot 10^{-92}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -4.8 \cdot 10^{-208}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -7 \cdot 10^{-246}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -4.2 \cdot 10^{-297}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{-296}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{-144}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{+19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.45 \cdot 10^{+48}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a}\\ \end{array} \]

Alternative 10
Error	35.1
Cost	2028

\[\begin{array}{l} t_1 := \frac{x}{1 + \frac{y}{\frac{t}{b}}}\\ t_2 := x + \frac{y \cdot z}{t}\\ \mathbf{if}\;a \leq -2.7 \cdot 10^{+98}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq -1.6 \cdot 10^{-51}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -5 \cdot 10^{-80}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1.3 \cdot 10^{-87}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -1.6 \cdot 10^{-207}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -7.2 \cdot 10^{-246}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -4.9 \cdot 10^{-297}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{-295}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-141}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 1.02 \cdot 10^{+20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 3.3 \cdot 10^{+55}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

Alternative 11
Error	24.6
Cost	1760

\[\begin{array}{l} t_1 := \left(x + \frac{y}{\frac{t}{z}}\right) \cdot \frac{1}{1 + \frac{y \cdot b}{t}}\\ t_2 := \left(a + 1\right) + b \cdot \frac{y}{t}\\ t_3 := \frac{z}{b} + \frac{x \cdot t}{y \cdot b}\\ \mathbf{if}\;b \leq -5.2 \cdot 10^{+140}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq -1.25 \cdot 10^{+103}:\\ \;\;\;\;\frac{x}{t_2}\\ \mathbf{elif}\;b \leq -9.5 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -1.96 \cdot 10^{-19}:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{t_2}\\ \mathbf{elif}\;b \leq -1.12 \cdot 10^{-66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{-77}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{a + 1}\\ \mathbf{elif}\;b \leq 1.16 \cdot 10^{-51}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 12
Error	35.6
Cost	1636

\[\begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ \mathbf{if}\;a \leq -4.8 \cdot 10^{+98}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq -1.8 \cdot 10^{-51}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -8.2 \cdot 10^{-184}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -8.5 \cdot 10^{-246}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -1.1 \cdot 10^{-297}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{-297}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 8.5 \cdot 10^{-146}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 9.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 1.02 \cdot 10^{+22}:\\ \;\;\;\;x \cdot \frac{\frac{t}{y}}{b}\\ \mathbf{elif}\;a \leq 3 \cdot 10^{+51}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

Alternative 13
Error	14.6
Cost	1616

\[\begin{array}{l} t_1 := \frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{b}{\frac{t}{y}}\right)}\\ \mathbf{if}\;y \leq -2.8 \cdot 10^{+159}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+97}:\\ \;\;\;\;\frac{z}{b} + \frac{x \cdot t}{y \cdot b}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+231}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 14
Error	14.4
Cost	1616

\[\begin{array}{l} t_1 := x + \frac{y}{\frac{t}{z}}\\ \mathbf{if}\;y \leq -4.7 \cdot 10^{+159}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+75}:\\ \;\;\;\;\frac{t_1}{a + \left(1 + \frac{b}{\frac{t}{y}}\right)}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+97}:\\ \;\;\;\;\frac{z}{b} + \frac{x \cdot t}{y \cdot b}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+229}:\\ \;\;\;\;\frac{t_1}{a + \left(1 + \frac{y}{\frac{t}{b}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 15
Error	25.4
Cost	1496

\[\begin{array}{l} t_1 := \frac{x + y \cdot \frac{z}{t}}{a + 1}\\ t_2 := \frac{z}{b} + \frac{x \cdot t}{y \cdot b}\\ \mathbf{if}\;y \leq -3.3 \cdot 10^{+87}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-106}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+102}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+193}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 16
Error	23.9
Cost	1496

\[\begin{array}{l} t_1 := \frac{x + y \cdot \frac{z}{t}}{a + 1}\\ t_2 := \frac{z}{b} + \frac{x \cdot t}{y \cdot b}\\ \mathbf{if}\;y \leq -2.95 \cdot 10^{+82}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -3.8 \cdot 10^{-26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{-110}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+47}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a + 1}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+102}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+192}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 17
Error	23.8
Cost	1496

\[\begin{array}{l} t_1 := \frac{x + y \cdot \frac{z}{t}}{a + 1}\\ t_2 := \frac{z}{b} + \frac{x \cdot t}{y \cdot b}\\ \mathbf{if}\;y \leq -1.55 \cdot 10^{+89}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -1.56 \cdot 10^{-24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-106}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 2.65 \cdot 10^{+48}:\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{a + 1}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+101}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+192}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 18
Error	23.7
Cost	1496

\[\begin{array}{l} t_1 := \frac{x + y \cdot \frac{z}{t}}{a + 1}\\ t_2 := \frac{z}{b} + \frac{x \cdot t}{y \cdot b}\\ \mathbf{if}\;y \leq -4.4 \cdot 10^{+92}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -2.4 \cdot 10^{-25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-106}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{+48}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{elif}\;y \leq 3.25 \cdot 10^{+102}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+192}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 19
Error	12.6
Cost	1352

\[\begin{array}{l} \mathbf{if}\;y \leq -2.65 \cdot 10^{+160}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-15}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + b \cdot \frac{y}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{y}{\frac{t}{b}}\right)}\\ \end{array} \]

Alternative 20
Error	37.3
Cost	984

\[\begin{array}{l} \mathbf{if}\;a \leq -2.9 \cdot 10^{+98}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq -1.55 \cdot 10^{-51}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -5.5 \cdot 10^{-179}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 9 \cdot 10^{-284}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{-146}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{+53}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

Alternative 21
Error	29.8
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -96000000 \lor \neg \left(t \leq 1.95 \cdot 10^{+49}\right):\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 22
Error	37.1
Cost	456

\[\begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{-28}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

Alternative 23
Error	51.3
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023046 
(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))))