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

Average Accuracy: 73.8% → 90.6%

Time: 28.5s

Precision: binary64

Cost: 5712

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{1 + \left(a + \frac{y}{\frac{t}{b}}\right)}\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-291}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{\frac{t}{\frac{b}{x}} - \frac{t}{\frac{b \cdot b}{z + z \cdot a}}}{y}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+293}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

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 (/ (+ x (/ (* y z) t)) (+ (/ (* y b) t) (+ a 1.0)))))
   (if (<= t_1 (- INFINITY))
     (* (/ y t) (/ z (+ 1.0 (+ a (/ y (/ t b))))))
     (if (<= t_1 -2e-291)
       t_1
       (if (<= t_1 0.0)
         (+ (/ z b) (/ (- (/ t (/ b x)) (/ t (/ (* b b) (+ z (* z a))))) y))
         (if (<= t_1 2e+293) t_1 (/ 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 = (x + ((y * z) / t)) / (((y * b) / t) + (a + 1.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (y / t) * (z / (1.0 + (a + (y / (t / b)))));
	} else if (t_1 <= -2e-291) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (z / b) + (((t / (b / x)) - (t / ((b * b) / (z + (z * a))))) / y);
	} else if (t_1 <= 2e+293) {
		tmp = t_1;
	} else {
		tmp = z / 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 / (1.0 + (a + (y / (t / b)))));
	} else if (t_1 <= -2e-291) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (z / b) + (((t / (b / x)) - (t / ((b * b) / (z + (z * a))))) / y);
	} else if (t_1 <= 2e+293) {
		tmp = t_1;
	} else {
		tmp = z / 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 / (1.0 + (a + (y / (t / b)))))
	elif t_1 <= -2e-291:
		tmp = t_1
	elif t_1 <= 0.0:
		tmp = (z / b) + (((t / (b / x)) - (t / ((b * b) / (z + (z * a))))) / y)
	elif t_1 <= 2e+293:
		tmp = t_1
	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(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(1.0 + Float64(a + Float64(y / Float64(t / b))))));
	elseif (t_1 <= -2e-291)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(z / b) + Float64(Float64(Float64(t / Float64(b / x)) - Float64(t / Float64(Float64(b * b) / Float64(z + Float64(z * a))))) / y));
	elseif (t_1 <= 2e+293)
		tmp = t_1;
	else
		tmp = Float64(z / 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 / (1.0 + (a + (y / (t / b)))));
	elseif (t_1 <= -2e-291)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = (z / b) + (((t / (b / x)) - (t / ((b * b) / (z + (z * a))))) / y);
	elseif (t_1 <= 2e+293)
		tmp = t_1;
	else
		tmp = z / 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[(1.0 + N[(a + N[(y / N[(t / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e-291], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(z / b), $MachinePrecision] + N[(N[(N[(t / N[(b / x), $MachinePrecision]), $MachinePrecision] - N[(t / N[(N[(b * b), $MachinePrecision] / N[(z + N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+293], t$95$1, N[(z / b), $MachinePrecision]]]]]]

Target

Original	73.8%
Target	78.9%
Herbie	90.6%

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

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))) < -inf.0

   1. Initial program 0.0%

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

   2. Simplified 37.6%

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

      [Start] 0.0	\[ \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      +-commutative [=>] 0.0	\[ \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      *-commutative [=>] 0.0	\[ \frac{\frac{\color{blue}{z \cdot y}}{t} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-*r/ [<=] 37.6	\[ \frac{\color{blue}{z \cdot \frac{y}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      fma-def [=>] 37.6	\[ \frac{\color{blue}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-+l+ [=>] 37.6	\[ \frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{\color{blue}{a + \left(1 + \frac{y \cdot b}{t}\right)}} \]
      +-commutative [=>] 37.6	\[ \frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + \color{blue}{\left(\frac{y \cdot b}{t} + 1\right)}} \]
      associate-*r/ [<=] 37.6	\[ \frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + \left(\color{blue}{y \cdot \frac{b}{t}} + 1\right)} \]
      fma-def [=>] 37.6	\[ \frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + \color{blue}{\mathsf{fma}\left(y, \frac{b}{t}, 1\right)}} \]

   3. Taylor expanded in z around inf 39.5%

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

   4. Simplified 71.9%

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

      [Start] 39.5	\[ \frac{y \cdot z}{t \cdot \left(1 + \left(\frac{y \cdot b}{t} + a\right)\right)} \]
      times-frac [=>] 73.8	\[ \color{blue}{\frac{y}{t} \cdot \frac{z}{1 + \left(\frac{y \cdot b}{t} + a\right)}} \]
      associate-/l* [=>] 71.9	\[ \frac{y}{t} \cdot \frac{z}{1 + \left(\color{blue}{\frac{y}{\frac{t}{b}}} + a\right)} \]
      +-commutative [=>] 71.9	\[ \frac{y}{t} \cdot \frac{z}{1 + \color{blue}{\left(a + \frac{y}{\frac{t}{b}}\right)}} \]

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -1.99999999999999992e-291 or 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 1.9999999999999998e293

   1. Initial program 99.4%

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

   if -1.99999999999999992e-291 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 0.0

   1. Initial program 56.0%

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

   2. Simplified 64.2%

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

      [Start] 56.0	\[ \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      *-commutative [=>] 56.0	\[ \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-/l* [=>] 50.2	\[ \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-*l/ [<=] 64.2	\[ \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}} \]

   3. Taylor expanded in y around -inf 62.4%

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

   4. Simplified 69.3%

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

      [Start] 62.4	\[ -1 \cdot \frac{-1 \cdot \frac{t \cdot x}{b} - -1 \cdot \frac{t \cdot \left(\left(1 + a\right) \cdot z\right)}{{b}^{2}}}{y} + \frac{z}{b} \]
      +-commutative [=>] 62.4	\[ \color{blue}{\frac{z}{b} + -1 \cdot \frac{-1 \cdot \frac{t \cdot x}{b} - -1 \cdot \frac{t \cdot \left(\left(1 + a\right) \cdot z\right)}{{b}^{2}}}{y}} \]
      associate-*r/ [=>] 62.4	\[ \frac{z}{b} + \color{blue}{\frac{-1 \cdot \left(-1 \cdot \frac{t \cdot x}{b} - -1 \cdot \frac{t \cdot \left(\left(1 + a\right) \cdot z\right)}{{b}^{2}}\right)}{y}} \]
      distribute-lft-out-- [=>] 62.4	\[ \frac{z}{b} + \frac{-1 \cdot \color{blue}{\left(-1 \cdot \left(\frac{t \cdot x}{b} - \frac{t \cdot \left(\left(1 + a\right) \cdot z\right)}{{b}^{2}}\right)\right)}}{y} \]
      associate-*r* [=>] 62.4	\[ \frac{z}{b} + \frac{\color{blue}{\left(-1 \cdot -1\right) \cdot \left(\frac{t \cdot x}{b} - \frac{t \cdot \left(\left(1 + a\right) \cdot z\right)}{{b}^{2}}\right)}}{y} \]
      metadata-eval [=>] 62.4	\[ \frac{z}{b} + \frac{\color{blue}{1} \cdot \left(\frac{t \cdot x}{b} - \frac{t \cdot \left(\left(1 + a\right) \cdot z\right)}{{b}^{2}}\right)}{y} \]
      *-lft-identity [=>] 62.4	\[ \frac{z}{b} + \frac{\color{blue}{\frac{t \cdot x}{b} - \frac{t \cdot \left(\left(1 + a\right) \cdot z\right)}{{b}^{2}}}}{y} \]

   if 1.9999999999999998e293 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t)))

   1. Initial program 2.0%

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

   2. Simplified 19.2%

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

      [Start] 2.0	\[ \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      +-commutative [=>] 2.0	\[ \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-*l/ [<=] 11.9	\[ \frac{\color{blue}{\frac{y}{t} \cdot z} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      fma-def [=>] 11.9	\[ \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-+l+ [=>] 11.9	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\color{blue}{a + \left(1 + \frac{y \cdot b}{t}\right)}} \]
      +-commutative [=>] 11.9	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{a + \color{blue}{\left(\frac{y \cdot b}{t} + 1\right)}} \]
      associate-*l/ [<=] 19.2	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{a + \left(\color{blue}{\frac{y}{t} \cdot b} + 1\right)} \]
      fma-def [=>] 19.2	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{a + \color{blue}{\mathsf{fma}\left(\frac{y}{t}, b, 1\right)}} \]

   3. Taylor expanded in y around inf 82.6%

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -\infty:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{1 + \left(a + \frac{y}{\frac{t}{b}}\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -2 \cdot 10^{-291}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{\frac{t}{\frac{b}{x}} - \frac{t}{\frac{b \cdot b}{z + z \cdot a}}}{y}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	89.9%
Cost	5712

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{1 + \left(a + \frac{y}{\frac{t}{b}}\right)}\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-291}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+293}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 2
Accuracy	80.2%
Cost	1616

\[\begin{array}{l} t_1 := \frac{x + y \cdot \frac{z}{t}}{b \cdot \frac{y}{t} + \left(a + 1\right)}\\ \mathbf{if}\;t \leq -2 \cdot 10^{-13}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -5.4 \cdot 10^{-182}:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{elif}\;t \leq -1.12 \cdot 10^{-206}:\\ \;\;\;\;\frac{y \cdot z}{y \cdot b + t \cdot \left(a + 1\right)}\\ \mathbf{elif}\;t \leq 6.3 \cdot 10^{-145}:\\ \;\;\;\;\frac{z}{b} + \frac{x \cdot t}{y \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	80.3%
Cost	1616

\[\begin{array}{l} t_1 := x + y \cdot \frac{z}{t}\\ \mathbf{if}\;t \leq -2 \cdot 10^{-13}:\\ \;\;\;\;\frac{t_1}{\frac{b}{\frac{t}{y}} + \left(a + 1\right)}\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-182}:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{elif}\;t \leq -2.7 \cdot 10^{-206}:\\ \;\;\;\;\frac{y \cdot z}{y \cdot b + t \cdot \left(a + 1\right)}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-145}:\\ \;\;\;\;\frac{z}{b} + \frac{x \cdot t}{y \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{b \cdot \frac{y}{t} + \left(a + 1\right)}\\ \end{array} \]

Alternative 4
Accuracy	80.0%
Cost	1616

\[\begin{array}{l} t_1 := b \cdot \frac{y}{t} + \left(a + 1\right)\\ \mathbf{if}\;t \leq -2 \cdot 10^{-13}:\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{t_1}\\ \mathbf{elif}\;t \leq -3.3 \cdot 10^{-182}:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{elif}\;t \leq -3.2 \cdot 10^{-206}:\\ \;\;\;\;\frac{y \cdot z}{y \cdot b + t \cdot \left(a + 1\right)}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{-145}:\\ \;\;\;\;\frac{z}{b} + \frac{x \cdot t}{y \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{t_1}\\ \end{array} \]

Alternative 5
Accuracy	80.1%
Cost	1616

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-13}:\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{\frac{b}{\frac{t}{y}} + \left(a + 1\right)}\\ \mathbf{elif}\;t \leq -3 \cdot 10^{-182}:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-206}:\\ \;\;\;\;\frac{y \cdot z}{y \cdot b + t \cdot \left(a + 1\right)}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-146}:\\ \;\;\;\;\frac{z}{b} + \frac{x \cdot t}{y \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{b \cdot \frac{y}{t} + \left(a + 1\right)}\\ \end{array} \]

Alternative 6
Accuracy	80.1%
Cost	1616

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-13}:\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{\frac{b}{\frac{t}{y}} + \left(a + 1\right)}\\ \mathbf{elif}\;t \leq -8.2 \cdot 10^{-182}:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{elif}\;t \leq -4.1 \cdot 10^{-206}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y}{\frac{t}{b}} + \left(a + 1\right)}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-145}:\\ \;\;\;\;\frac{z}{b} + \frac{x \cdot t}{y \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{b \cdot \frac{y}{t} + \left(a + 1\right)}\\ \end{array} \]

Alternative 7
Accuracy	53.7%
Cost	1500

\[\begin{array}{l} t_1 := \frac{x + \frac{z}{\frac{t}{y}}}{a}\\ t_2 := \frac{x}{1 + \frac{y \cdot b}{t}}\\ \mathbf{if}\;a \leq -4 \cdot 10^{+66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -9 \cdot 10^{-81}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -5.9 \cdot 10^{-233}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1.9 \cdot 10^{-299}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 4.4 \cdot 10^{-145}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{-63}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 3900000:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Accuracy	54.2%
Cost	1496

\[\begin{array}{l} t_1 := \frac{z}{b} + \frac{x \cdot t}{y \cdot b}\\ t_2 := x + \frac{y \cdot z}{t}\\ \mathbf{if}\;a \leq -4 \cdot 10^{+66}:\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{a}\\ \mathbf{elif}\;a \leq -1.05 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -3.6 \cdot 10^{-150}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -7 \cdot 10^{-299}:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{-142}:\\ \;\;\;\;\frac{x}{1 + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;a \leq 350000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{a}\\ \end{array} \]

Alternative 9
Accuracy	54.9%
Cost	1496

\[\begin{array}{l} t_1 := \frac{z}{b} + \frac{x \cdot t}{y \cdot b}\\ t_2 := \frac{x}{1 + \left(a + b \cdot \frac{y}{t}\right)}\\ \mathbf{if}\;a \leq -1.32 \cdot 10^{+65}:\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{a}\\ \mathbf{elif}\;a \leq -7.8 \cdot 10^{-112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -2.3 \cdot 10^{-231}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -6.4 \cdot 10^{-301}:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{elif}\;a \leq 3.2 \cdot 10^{-144}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 18000000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a}\\ \end{array} \]

Alternative 10
Accuracy	55.0%
Cost	1368

\[\begin{array}{l} t_1 := \frac{z + \frac{x \cdot t}{y}}{b}\\ t_2 := \frac{x + \frac{z}{\frac{t}{y}}}{a}\\ \mathbf{if}\;a \leq -4.3 \cdot 10^{+63}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -6 \cdot 10^{-57}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -2.8 \cdot 10^{-152}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \mathbf{elif}\;a \leq -1.8 \cdot 10^{-301}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 4 \cdot 10^{-142}:\\ \;\;\;\;\frac{x}{1 + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;a \leq 8600000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 11
Accuracy	54.4%
Cost	1368

\[\begin{array}{l} t_1 := \frac{z + \frac{x \cdot t}{y}}{b}\\ t_2 := x + \frac{y \cdot z}{t}\\ \mathbf{if}\;a \leq -2.35 \cdot 10^{+64}:\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{a}\\ \mathbf{elif}\;a \leq -1.06 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -6.2 \cdot 10^{-152}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -7 \cdot 10^{-301}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{-143}:\\ \;\;\;\;\frac{x}{1 + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;a \leq 360000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{a}\\ \end{array} \]

Alternative 12
Accuracy	68.3%
Cost	1365

\[\begin{array}{l} t_1 := \frac{x + \frac{z}{\frac{t}{y}}}{a + 1}\\ \mathbf{if}\;t \leq -5.2 \cdot 10^{+228}:\\ \;\;\;\;\frac{x}{1 + \left(a + \frac{y}{\frac{t}{b}}\right)}\\ \mathbf{elif}\;t \leq -3.1 \cdot 10^{-13}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.5 \cdot 10^{-182}:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{elif}\;t \leq -5.5 \cdot 10^{-206} \lor \neg \left(t \leq 3.4 \cdot 10^{-98}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b} + \frac{x \cdot t}{y \cdot b}\\ \end{array} \]

Alternative 13
Accuracy	68.3%
Cost	1364

\[\begin{array}{l} t_1 := \frac{x + \frac{z}{\frac{t}{y}}}{a + 1}\\ \mathbf{if}\;t \leq -1.9 \cdot 10^{+228}:\\ \;\;\;\;\frac{x}{1 + \left(a + \frac{y}{\frac{t}{b}}\right)}\\ \mathbf{elif}\;t \leq -6.5 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.5 \cdot 10^{-182}:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{elif}\;t \leq -5.5 \cdot 10^{-206}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-98}:\\ \;\;\;\;\frac{z}{b} + \frac{x \cdot t}{y \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Accuracy	68.3%
Cost	1364

\[\begin{array}{l} t_1 := \frac{x + \frac{z}{\frac{t}{y}}}{a + 1}\\ \mathbf{if}\;t \leq -2.3 \cdot 10^{+228}:\\ \;\;\;\;\frac{x}{1 + \left(a + \frac{b}{\frac{t}{y}}\right)}\\ \mathbf{elif}\;t \leq -2.3 \cdot 10^{-13}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.5 \cdot 10^{-182}:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{elif}\;t \leq -5.5 \cdot 10^{-206}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{-98}:\\ \;\;\;\;\frac{z}{b} + \frac{x \cdot t}{y \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Accuracy	68.6%
Cost	1364

\[\begin{array}{l} t_1 := \frac{x + \frac{z}{\frac{t}{y}}}{a + 1}\\ \mathbf{if}\;t \leq -4.2 \cdot 10^{+228}:\\ \;\;\;\;\frac{x}{1 + \left(a + \frac{b}{\frac{t}{y}}\right)}\\ \mathbf{elif}\;t \leq -8 \cdot 10^{-13}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -6 \cdot 10^{-182}:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{elif}\;t \leq -1.35 \cdot 10^{-206}:\\ \;\;\;\;\frac{y \cdot z}{y \cdot b + t \cdot \left(a + 1\right)}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-98}:\\ \;\;\;\;\frac{z}{b} + \frac{x \cdot t}{y \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Accuracy	41.4%
Cost	984

\[\begin{array}{l} \mathbf{if}\;a \leq -9.6 \cdot 10^{+66}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq -4.9 \cdot 10^{-81}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -1 \cdot 10^{-230}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq -1.15 \cdot 10^{-300}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 4 \cdot 10^{-145}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{-17}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

Alternative 17
Accuracy	65.6%
Cost	969

\[\begin{array}{l} \mathbf{if}\;t \leq -1800000000000 \lor \neg \left(t \leq 2.2 \cdot 10^{-98}\right):\\ \;\;\;\;\frac{x}{1 + \left(a + \frac{y}{\frac{t}{b}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b} + \frac{x \cdot t}{y \cdot b}\\ \end{array} \]

Alternative 18
Accuracy	53.2%
Cost	849

\[\begin{array}{l} t_1 := \frac{x}{a + 1}\\ \mathbf{if}\;t \leq -6.5 \cdot 10^{+153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.76 \cdot 10^{+50}:\\ \;\;\;\;\frac{y}{\frac{a + 1}{\frac{z}{t}}}\\ \mathbf{elif}\;t \leq -48000000000000 \lor \neg \left(t \leq 3 \cdot 10^{-98}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 19
Accuracy	53.6%
Cost	849

\[\begin{array}{l} t_1 := \frac{x}{a + 1}\\ \mathbf{if}\;t \leq -3 \cdot 10^{+141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{+50}:\\ \;\;\;\;\frac{z \cdot \frac{y}{t}}{a + 1}\\ \mathbf{elif}\;t \leq -40000000000000 \lor \neg \left(t \leq 3.4 \cdot 10^{-98}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 20
Accuracy	55.5%
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -1850000000000 \lor \neg \left(t \leq 1.42 \cdot 10^{-98}\right):\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 21
Accuracy	41.7%
Cost	456

\[\begin{array}{l} \mathbf{if}\;a \leq -0.0002:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

Alternative 22
Accuracy	19.8%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023144 
(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))))