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

Average Error: 16.6 → 5.8

Time: 25.6s

Precision: binary64

Cost: 8388

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{t \cdot \left(a + \mathsf{fma}\left(y, \frac{b}{t}, 1\right)\right)}{z}}\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 10^{-284}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{a + \left(1 + \frac{b}{\frac{t}{y}}\right)}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+284}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \end{array} \]

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 (<= t_1 -2e-139)
       t_1
       (if (<= t_1 1e-284)
         (/ (+ x (* y (/ z t))) (+ a (+ 1.0 (/ b (/ t y)))))
         (if (<= t_1 2e+284)
           t_1
           (if (<= t_1 INFINITY)
             (* (/ y t) (/ z (+ (+ a 1.0) (/ y (/ t b)))))
             (+ (/ 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 <= -2e-139) {
		tmp = t_1;
	} else if (t_1 <= 1e-284) {
		tmp = (x + (y * (z / t))) / (a + (1.0 + (b / (t / y))));
	} else if (t_1 <= 2e+284) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (y / t) * (z / ((a + 1.0) + (y / (t / b))));
	} 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 <= -2e-139)
		tmp = t_1;
	elseif (t_1 <= 1e-284)
		tmp = Float64(Float64(x + Float64(y * Float64(z / t))) / Float64(a + Float64(1.0 + Float64(b / Float64(t / y)))));
	elseif (t_1 <= 2e+284)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(y / t) * Float64(z / Float64(Float64(a + 1.0) + Float64(y / Float64(t / b)))));
	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[LessEqual[t$95$1, -2e-139], t$95$1, If[LessEqual[t$95$1, 1e-284], N[(N[(x + N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a + N[(1.0 + N[(b / N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+284], t$95$1, If[LessEqual[t$95$1, Infinity], N[(N[(y / t), $MachinePrecision] * N[(z / N[(N[(a + 1.0), $MachinePrecision] + N[(y / N[(t / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z / b), $MachinePrecision] + N[(N[(t / y), $MachinePrecision] * N[(x / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Target

Original	16.6
Target	13.0
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 42.0

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

      [Start] 64.0	\[ \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-/l* [=>] 42.0	\[ \frac{x + \color{blue}{\frac{y}{\frac{t}{z}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-+l+ [=>] 42.0	\[ \frac{x + \frac{y}{\frac{t}{z}}}{\color{blue}{a + \left(1 + \frac{y \cdot b}{t}\right)}} \]
      associate-/l* [=>] 42.0	\[ \frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \color{blue}{\frac{y}{\frac{t}{b}}}\right)} \]

   3. Taylor expanded in x around 0 38.4

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

   4. Simplified 14.0

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

      [Start] 38.4	\[ \frac{y \cdot z}{t \cdot \left(\frac{y \cdot b}{t} + \left(1 + a\right)\right)} \]
      associate-/l* [=>] 11.9	\[ \color{blue}{\frac{y}{\frac{t \cdot \left(\frac{y \cdot b}{t} + \left(1 + a\right)\right)}{z}}} \]
      associate-+r+ [=>] 11.9	\[ \frac{y}{\frac{t \cdot \color{blue}{\left(\left(\frac{y \cdot b}{t} + 1\right) + a\right)}}{z}} \]
      associate-*r/ [<=] 14.0	\[ \frac{y}{\frac{t \cdot \left(\left(\color{blue}{y \cdot \frac{b}{t}} + 1\right) + a\right)}{z}} \]
      fma-udef [<=] 14.0	\[ \frac{y}{\frac{t \cdot \left(\color{blue}{\mathsf{fma}\left(y, \frac{b}{t}, 1\right)} + a\right)}{z}} \]
      +-commutative [<=] 14.0	\[ \frac{y}{\frac{t \cdot \color{blue}{\left(a + \mathsf{fma}\left(y, \frac{b}{t}, 1\right)\right)}}{z}} \]

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -2.00000000000000006e-139 or 1.00000000000000004e-284 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 2.00000000000000016e284

   1. Initial program 0.3

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

   if -2.00000000000000006e-139 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 1.00000000000000004e-284

   1. Initial program 20.7

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

   2. Simplified 14.6

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

      [Start] 20.7	\[ \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-/l* [=>] 20.5	\[ \frac{x + \color{blue}{\frac{y}{\frac{t}{z}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-+l+ [=>] 20.5	\[ \frac{x + \frac{y}{\frac{t}{z}}}{\color{blue}{a + \left(1 + \frac{y \cdot b}{t}\right)}} \]
      *-commutative [=>] 20.5	\[ \frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{\color{blue}{b \cdot y}}{t}\right)} \]
      associate-/l* [=>] 14.6	\[ \frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \color{blue}{\frac{b}{\frac{t}{y}}}\right)} \]

   3. Applied egg-rr 14.6

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

   if 2.00000000000000016e284 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 55.8

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

   2. Simplified 34.6

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

      [Start] 55.8	\[ \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      +-commutative [=>] 55.8	\[ \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-*l/ [<=] 34.6	\[ \frac{\color{blue}{\frac{y}{t} \cdot z} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      fma-def [=>] 34.6	\[ \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      +-commutative [=>] 34.6	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\color{blue}{\frac{y \cdot b}{t} + \left(a + 1\right)}} \]
      associate-+r+ [=>] 34.6	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\color{blue}{\left(\frac{y \cdot b}{t} + a\right) + 1}} \]
      +-commutative [=>] 34.6	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\color{blue}{1 + \left(\frac{y \cdot b}{t} + a\right)}} \]
      associate-*l/ [<=] 34.6	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + \left(\color{blue}{\frac{y}{t} \cdot b} + a\right)} \]
      fma-def [=>] 34.6	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + \color{blue}{\mathsf{fma}\left(\frac{y}{t}, b, a\right)}} \]

   3. Taylor expanded in z around inf 43.8

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

   4. Simplified 22.9

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

      [Start] 43.8	\[ \frac{y \cdot z}{t \cdot \left(\frac{y \cdot b}{t} + \left(1 + a\right)\right)} \]
      times-frac [=>] 21.8	\[ \color{blue}{\frac{y}{t} \cdot \frac{z}{\frac{y \cdot b}{t} + \left(1 + a\right)}} \]
      +-commutative [=>] 21.8	\[ \frac{y}{t} \cdot \frac{z}{\color{blue}{\left(1 + a\right) + \frac{y \cdot b}{t}}} \]
      associate-/l* [=>] 22.9	\[ \frac{y}{t} \cdot \frac{z}{\left(1 + a\right) + \color{blue}{\frac{y}{\frac{t}{b}}}} \]

   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. Simplified 57.4

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{a + \mathsf{fma}\left(\frac{b}{t}, y, 1\right)}} \]
      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/ [<=] 57.4	\[ \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{a + \left(\color{blue}{y \cdot \frac{b}{t}} + 1\right)} \]
      *-commutative [<=] 57.4	\[ \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{a + \left(\color{blue}{\frac{b}{t} \cdot y} + 1\right)} \]
      fma-def [=>] 57.4	\[ \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{a + \color{blue}{\mathsf{fma}\left(\frac{b}{t}, y, 1\right)}} \]

   3. Taylor expanded in t around 0 17.1

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

   4. Simplified 17.1

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

      [Start] 17.1	\[ \frac{z}{b} + \left(\frac{x}{y \cdot b} - \frac{\left(1 + a\right) \cdot z}{y \cdot {b}^{2}}\right) \cdot t \]
      *-commutative [=>] 17.1	\[ \frac{z}{b} + \color{blue}{t \cdot \left(\frac{x}{y \cdot b} - \frac{\left(1 + a\right) \cdot z}{y \cdot {b}^{2}}\right)} \]
      *-commutative [=>] 17.1	\[ \frac{z}{b} + t \cdot \left(\frac{x}{y \cdot b} - \frac{\color{blue}{z \cdot \left(1 + a\right)}}{y \cdot {b}^{2}}\right) \]
      unpow2 [=>] 17.1	\[ \frac{z}{b} + t \cdot \left(\frac{x}{y \cdot b} - \frac{z \cdot \left(1 + a\right)}{y \cdot \color{blue}{\left(b \cdot b\right)}}\right) \]

   5. Taylor expanded in x around inf 4.6

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

   6. Simplified 2.9

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

      [Start] 4.6	\[ \frac{z}{b} + \frac{t \cdot x}{y \cdot b} \]
      times-frac [=>] 2.9	\[ \frac{z}{b} + \color{blue}{\frac{t}{y} \cdot \frac{x}{b}} \]
      *-commutative [=>] 2.9	\[ \frac{z}{b} + \color{blue}{\frac{x}{b} \cdot \frac{t}{y}} \]

3. Recombined 5 regimes into one program.

4. Final simplification 5.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{t \cdot \left(a + \mathsf{fma}\left(y, \frac{b}{t}, 1\right)\right)}{z}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -2 \cdot 10^{-139}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 10^{-284}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{a + \left(1 + \frac{b}{\frac{t}{y}}\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 2 \cdot 10^{+284}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq \infty:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	5.9
Cost	6740

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ t_2 := \frac{y}{t} \cdot \frac{z}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 10^{-284}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{a + \left(1 + \frac{b}{\frac{t}{y}}\right)}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+284}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \end{array} \]

Alternative 2
Error	36.5
Cost	2292

\[\begin{array}{l} t_1 := \frac{x}{1 + b \cdot \frac{y}{t}}\\ t_2 := \frac{x}{a + 1}\\ t_3 := \frac{y}{t} \cdot \frac{z}{a + 1}\\ \mathbf{if}\;z \leq -5.3 \cdot 10^{+281}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{+207}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{+93}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.75 \cdot 10^{+70}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;z \leq -980:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{-107}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2.95 \cdot 10^{-136}:\\ \;\;\;\;t \cdot \frac{x}{y \cdot b}\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{-238}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{-231}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{+117}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+161}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+201}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 3
Error	35.7
Cost	1896

\[\begin{array}{l} t_1 := \frac{y \cdot z}{t \cdot \left(a + 1\right)}\\ t_2 := \frac{y}{t} \cdot \frac{z}{a + 1}\\ t_3 := \frac{x}{a + 1}\\ \mathbf{if}\;x \leq -3.6 \cdot 10^{-86}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{-119}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-194}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -5.7 \cdot 10^{-273}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-273}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{-244}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-184}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-161}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-86}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4.15 \cdot 10^{+139}:\\ \;\;\;\;\frac{x}{1 + b \cdot \frac{y}{t}}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 4
Error	34.3
Cost	1764

\[\begin{array}{l} t_1 := \frac{y}{t} \cdot \frac{z}{a + 1}\\ t_2 := \frac{x}{a + 1}\\ \mathbf{if}\;x \leq -5.2 \cdot 10^{-85}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-120}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;x \leq -8.1 \cdot 10^{-196}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -9.8 \cdot 10^{-272}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{-288}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-245}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-188}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-161}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-121}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	15.1
Cost	1616

\[\begin{array}{l} t_1 := \frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{y}{\frac{t}{b}}\right)}\\ t_2 := \frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{if}\;b \leq -1.2 \cdot 10^{+229}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -5.4 \cdot 10^{-239}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-206}:\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{a + 1}\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{+160}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	28.3
Cost	1496

\[\begin{array}{l} t_1 := \frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ t_2 := \frac{x + z \cdot \frac{y}{t}}{a}\\ t_3 := x + \frac{y \cdot z}{t}\\ \mathbf{if}\;a \leq -1.8 \cdot 10^{+64}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -9.2 \cdot 10^{-35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -2.2 \cdot 10^{-124}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 2.15 \cdot 10^{-227}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{-44}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 7.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{a + \left(1 + b \cdot \frac{y}{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	28.3
Cost	1496

\[\begin{array}{l} t_1 := \frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ t_2 := \frac{x + z \cdot \frac{y}{t}}{a}\\ t_3 := x + \frac{y \cdot z}{t}\\ \mathbf{if}\;a \leq -1.8 \cdot 10^{+63}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -8 \cdot 10^{-35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -4.8 \cdot 10^{-123}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{-225}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 8.5 \cdot 10^{-45}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 1.65 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{a + \left(1 + \frac{b}{\frac{t}{y}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	28.4
Cost	1368

\[\begin{array}{l} t_1 := \frac{x + z \cdot \frac{y}{t}}{a}\\ t_2 := x + \frac{y \cdot z}{t}\\ \mathbf{if}\;a \leq -8 \cdot 10^{+35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -2.65 \cdot 10^{-33}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -6 \cdot 10^{-124}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{-174}:\\ \;\;\;\;\frac{x}{1 + b \cdot \frac{y}{t}}\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{-44}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	28.9
Cost	1368

\[\begin{array}{l} t_1 := \frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ t_2 := \frac{x + z \cdot \frac{y}{t}}{a}\\ t_3 := x + \frac{y \cdot z}{t}\\ \mathbf{if}\;a \leq -5 \cdot 10^{+61}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1.1 \cdot 10^{-33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -3.4 \cdot 10^{-123}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{-229}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{-44}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Error	12.3
Cost	1353

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-188} \lor \neg \left(t \leq 4.4 \cdot 10^{-95}\right):\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{a + \left(1 + \frac{b}{\frac{t}{y}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \end{array} \]

Alternative 11
Error	12.3
Cost	1352

\[\begin{array}{l} t_1 := a + \left(1 + \frac{b}{\frac{t}{y}}\right)\\ \mathbf{if}\;t \leq -5.35 \cdot 10^{-188}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{t_1}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-93}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{t_1}\\ \end{array} \]

Alternative 12
Error	34.6
Cost	1108

\[\begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ \mathbf{if}\;a \leq -1.35 \cdot 10^{+93}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq -3 \cdot 10^{-33}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -1.14 \cdot 10^{-150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -2.55 \cdot 10^{-225}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{-44}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a + 1}\\ \end{array} \]

Alternative 13
Error	20.5
Cost	1100

\[\begin{array}{l} t_1 := \frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{if}\;b \leq -5.9 \cdot 10^{+227}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -1.15 \cdot 10^{+23}:\\ \;\;\;\;\frac{x}{a + \left(1 + \frac{b}{\frac{t}{y}}\right)}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{+86}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	20.5
Cost	1100

\[\begin{array}{l} t_1 := \frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{if}\;b \leq -2.8 \cdot 10^{+228}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -1.5 \cdot 10^{+20}:\\ \;\;\;\;\frac{x}{a + \left(1 + \frac{b}{\frac{t}{y}}\right)}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{+85}:\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	37.6
Cost	980

\[\begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{+93}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{-306}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 2.7 \cdot 10^{-93}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{-46}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 0.82:\\ \;\;\;\;x - x \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

Alternative 16
Error	37.6
Cost	852

\[\begin{array}{l} \mathbf{if}\;a \leq -3.6 \cdot 10^{+93}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{-306}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-98}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{-46}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

Alternative 17
Error	28.3
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{-90} \lor \neg \left(t \leq 3.3 \cdot 10^{-94}\right):\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 18
Error	37.3
Cost	456

\[\begin{array}{l} \mathbf{if}\;a \leq -1350000000000:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

Alternative 19
Error	51.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023073 
(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))))