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

Average Error: 26.23% → 11.96%

Time: 19.7s

Precision: binary64

Cost: 5584

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-166}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-214}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{b}{\frac{t}{y}}\right)}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+294}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \end{array} \]

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 -5e-166)
     t_1
     (if (<= t_1 5e-214)
       (/ (+ x (/ y (/ t z))) (+ a (+ 1.0 (/ b (/ t y)))))
       (if (<= t_1 2e+294)
         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 <= -5e-166) {
		tmp = t_1;
	} else if (t_1 <= 5e-214) {
		tmp = (x + (y / (t / z))) / (a + (1.0 + (b / (t / y))));
	} else if (t_1 <= 2e+294) {
		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;
}

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 <= -5e-166) {
		tmp = t_1;
	} else if (t_1 <= 5e-214) {
		tmp = (x + (y / (t / z))) / (a + (1.0 + (b / (t / y))));
	} else if (t_1 <= 2e+294) {
		tmp = t_1;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (y / t) * (z / ((a + 1.0) + (y / (t / b))));
	} else {
		tmp = (z / b) + ((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 <= -5e-166:
		tmp = t_1
	elif t_1 <= 5e-214:
		tmp = (x + (y / (t / z))) / (a + (1.0 + (b / (t / y))))
	elif t_1 <= 2e+294:
		tmp = t_1
	elif t_1 <= math.inf:
		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 <= -5e-166)
		tmp = t_1;
	elseif (t_1 <= 5e-214)
		tmp = Float64(Float64(x + Float64(y / Float64(t / z))) / Float64(a + Float64(1.0 + Float64(b / Float64(t / y)))));
	elseif (t_1 <= 2e+294)
		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

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 <= -5e-166)
		tmp = t_1;
	elseif (t_1 <= 5e-214)
		tmp = (x + (y / (t / z))) / (a + (1.0 + (b / (t / y))));
	elseif (t_1 <= 2e+294)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = (y / t) * (z / ((a + 1.0) + (y / (t / b))));
	else
		tmp = (z / b) + ((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, -5e-166], t$95$1, If[LessEqual[t$95$1, 5e-214], N[(N[(x + N[(y / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a + N[(1.0 + N[(b / N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+294], 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	26.23%
Target	20.7%
Herbie	11.96%

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

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))) < -5e-166 or 4.9999999999999998e-214 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 2.00000000000000013e294

   1. Initial program 7.59

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

   if -5e-166 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 4.9999999999999998e-214

   1. Initial program 27.53

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

   2. Simplified 20.61

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

      [Start] 27.53	\[ \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-/l* [=>] 27.71	\[ \frac{x + \color{blue}{\frac{y}{\frac{t}{z}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-+l+ [=>] 27.71	\[ \frac{x + \frac{y}{\frac{t}{z}}}{\color{blue}{a + \left(1 + \frac{y \cdot b}{t}\right)}} \]
      *-commutative [=>] 27.71	\[ \frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{\color{blue}{b \cdot y}}{t}\right)} \]
      associate-/l* [=>] 20.61	\[ \frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \color{blue}{\frac{b}{\frac{t}{y}}}\right)} \]

   if 2.00000000000000013e294 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 94.01

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

   2. Simplified 59.53

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

      [Start] 94.01	\[ \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      +-commutative [=>] 94.01	\[ \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-*l/ [<=] 59.55	\[ \frac{\color{blue}{\frac{y}{t} \cdot z} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      fma-def [=>] 59.55	\[ \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      +-commutative [=>] 59.55	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\color{blue}{\frac{y \cdot b}{t} + \left(a + 1\right)}} \]
      associate-+r+ [=>] 59.55	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\color{blue}{\left(\frac{y \cdot b}{t} + a\right) + 1}} \]
      +-commutative [=>] 59.55	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\color{blue}{1 + \left(\frac{y \cdot b}{t} + a\right)}} \]
      associate-*l/ [<=] 59.53	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + \left(\color{blue}{\frac{y}{t} \cdot b} + a\right)} \]
      fma-def [=>] 59.53	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + \color{blue}{\mathsf{fma}\left(\frac{y}{t}, b, a\right)}} \]

   3. Taylor expanded in z around inf 61.97

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

   4. Simplified 32.04

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

      [Start] 61.97	\[ \frac{y \cdot z}{t \cdot \left(\frac{y \cdot b}{t} + \left(1 + a\right)\right)} \]
      times-frac [=>] 28.05	\[ \color{blue}{\frac{y}{t} \cdot \frac{z}{\frac{y \cdot b}{t} + \left(1 + a\right)}} \]
      +-commutative [=>] 28.05	\[ \frac{y}{t} \cdot \frac{z}{\color{blue}{\left(1 + a\right) + \frac{y \cdot b}{t}}} \]
      associate-/l* [=>] 32.04	\[ \frac{y}{t} \cdot \frac{z}{\left(1 + a\right) + \color{blue}{\frac{y}{\frac{t}{b}}}} \]

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

   1. Initial program 100

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

   2. Simplified 88.8

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

      [Start] 100	\[ \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      +-commutative [=>] 100	\[ \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-*r/ [<=] 99.51	\[ \frac{\color{blue}{y \cdot \frac{z}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      *-commutative [<=] 99.51	\[ \frac{\color{blue}{\frac{z}{t} \cdot y} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      fma-def [=>] 99.51	\[ \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-+l+ [=>] 99.51	\[ \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\color{blue}{a + \left(1 + \frac{y \cdot b}{t}\right)}} \]
      +-commutative [=>] 99.51	\[ \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{a + \color{blue}{\left(\frac{y \cdot b}{t} + 1\right)}} \]
      associate-*r/ [<=] 88.8	\[ \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{a + \left(\color{blue}{y \cdot \frac{b}{t}} + 1\right)} \]
      *-commutative [<=] 88.8	\[ \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{a + \left(\color{blue}{\frac{b}{t} \cdot y} + 1\right)} \]
      fma-def [=>] 88.8	\[ \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{a + \color{blue}{\mathsf{fma}\left(\frac{b}{t}, y, 1\right)}} \]

   3. Taylor expanded in b around inf 100

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

   4. Simplified 88.2

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

      [Start] 100	\[ \frac{t \cdot \left(\frac{y \cdot z}{t} + x\right)}{y \cdot b} \]
      times-frac [=>] 100	\[ \color{blue}{\frac{t}{y} \cdot \frac{\frac{y \cdot z}{t} + x}{b}} \]
      +-commutative [=>] 100	\[ \frac{t}{y} \cdot \frac{\color{blue}{x + \frac{y \cdot z}{t}}}{b} \]
      associate-/l* [=>] 88.2	\[ \frac{t}{y} \cdot \frac{x + \color{blue}{\frac{y}{\frac{t}{z}}}}{b} \]

   5. Taylor expanded in t around 0 8.11

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

   6. Simplified 4.46

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

      [Start] 8.11	\[ \frac{t \cdot x}{y \cdot b} + \frac{z}{b} \]
      +-commutative [=>] 8.11	\[ \color{blue}{\frac{z}{b} + \frac{t \cdot x}{y \cdot b}} \]
      times-frac [=>] 4.46	\[ \frac{z}{b} + \color{blue}{\frac{t}{y} \cdot \frac{x}{b}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 11.96

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -5 \cdot 10^{-166}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 5 \cdot 10^{-214}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{b}{\frac{t}{y}}\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 2 \cdot 10^{+294}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq \infty:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	34.17%
Cost	1496

\[\begin{array}{l} t_1 := \frac{x + \frac{z}{\frac{t}{y}}}{1 + y \cdot \frac{b}{t}}\\ \mathbf{if}\;a \leq -4.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a + 1}\\ \mathbf{elif}\;a \leq -4.7 \cdot 10^{-152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.4 \cdot 10^{-176}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 29000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{+115}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a}\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \end{array} \]

Alternative 2
Error	39.11%
Cost	1364

\[\begin{array}{l} t_1 := \frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{if}\;y \leq -5.4 \cdot 10^{-23}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-71}:\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{a}\\ \mathbf{elif}\;y \leq -8.6 \cdot 10^{-88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -6 \cdot 10^{-151}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+121}:\\ \;\;\;\;\frac{x}{a + \left(1 + y \cdot \frac{b}{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	39.05%
Cost	1364

\[\begin{array}{l} t_1 := \frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{if}\;y \leq -1.14 \cdot 10^{-23}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{-71}:\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{a}\\ \mathbf{elif}\;y \leq -2.45 \cdot 10^{-88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-151}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+121}:\\ \;\;\;\;\frac{x}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	25.81%
Cost	1356

\[\begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+117}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a + 1}\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+252}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{b}{\frac{t}{y}}\right)}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+287}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \end{array} \]

Alternative 5
Error	26.69%
Cost	1356

\[\begin{array}{l} t_1 := \frac{y}{\frac{t}{b}}\\ \mathbf{if}\;z \leq -2.7 \cdot 10^{+117}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a + 1}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+251}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + t_1\right)}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+286}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{\left(a + 1\right) + t_1}\\ \end{array} \]

Alternative 6
Error	27%
Cost	1352

\[\begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+175}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{elif}\;b \leq 9.6 \cdot 10^{+140}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \end{array} \]

Alternative 7
Error	41.35%
Cost	1233

\[\begin{array}{l} t_1 := \frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{if}\;y \leq -2.35 \cdot 10^{-24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.55 \cdot 10^{-69}:\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{a}\\ \mathbf{elif}\;y \leq -8.6 \cdot 10^{-88} \lor \neg \left(y \leq 2.75 \cdot 10^{+42}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a + 1}\\ \end{array} \]

Alternative 8
Error	35.05%
Cost	1101

\[\begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{+173}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{elif}\;b \leq -3.4 \cdot 10^{-56} \lor \neg \left(b \leq 1.32 \cdot 10^{+135}\right):\\ \;\;\;\;\frac{x}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a + 1}\\ \end{array} \]

Alternative 9
Error	45.37%
Cost	849

\[\begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-20}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -4.4 \cdot 10^{-85}:\\ \;\;\;\;\frac{y \cdot z}{t \cdot \left(a + 1\right)}\\ \mathbf{elif}\;y \leq -8.6 \cdot 10^{-88} \lor \neg \left(y \leq 8 \cdot 10^{+44}\right):\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a + 1}\\ \end{array} \]

Alternative 10
Error	45.31%
Cost	848

\[\begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-25}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -6.6 \cdot 10^{-86}:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{a + 1}\\ \mathbf{elif}\;y \leq -8.6 \cdot 10^{-88}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+43}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 11
Error	42.52%
Cost	840

\[\begin{array}{l} \mathbf{if}\;a \leq -1150000:\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{a}\\ \mathbf{elif}\;a \leq 38000000:\\ \;\;\;\;\frac{x}{1 + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a}\\ \end{array} \]

Alternative 12
Error	57.6%
Cost	716

\[\begin{array}{l} \mathbf{if}\;a \leq -6800000:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq -1 \cdot 10^{-176}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 1.42 \cdot 10^{+23}:\\ \;\;\;\;x - x \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

Alternative 13
Error	57.1%
Cost	588

\[\begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+20}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq -5.4 \cdot 10^{-191}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 0.0068:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

Alternative 14
Error	45.45%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{-88}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{+42}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 15
Error	57.39%
Cost	456

\[\begin{array}{l} \mathbf{if}\;a \leq -8 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq 0.0068:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

Alternative 16
Error	79.68%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023121 
(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))))