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

Average Error: 25.84% → 8.93%

Time: 28.7s

Precision: binary64

Cost: 6868

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ t_2 := \left(a + 1\right) + \frac{y}{\frac{t}{b}}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{x}{t_2} + \frac{y}{t} \cdot \frac{z}{t_2}\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-160}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-318}:\\ \;\;\;\;\left(x + z \cdot \frac{y}{t}\right) \cdot \frac{1}{a + \left(1 + b \cdot \frac{y}{t}\right)}\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{x}{b} \cdot \frac{t}{y}\\ \mathbf{elif}\;t_1 \leq 10^{+307}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

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))))
        (t_2 (+ (+ a 1.0) (/ y (/ t b)))))
   (if (<= t_1 (- INFINITY))
     (+ (/ x t_2) (* (/ y t) (/ z t_2)))
     (if (<= t_1 -1e-160)
       t_1
       (if (<= t_1 -1e-318)
         (* (+ x (* z (/ y t))) (/ 1.0 (+ a (+ 1.0 (* b (/ y t))))))
         (if (<= t_1 0.0)
           (+ (/ z b) (* (/ x b) (/ t y)))
           (if (<= t_1 1e+307) 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 t_2 = (a + 1.0) + (y / (t / b));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (x / t_2) + ((y / t) * (z / t_2));
	} else if (t_1 <= -1e-160) {
		tmp = t_1;
	} else if (t_1 <= -1e-318) {
		tmp = (x + (z * (y / t))) * (1.0 / (a + (1.0 + (b * (y / t)))));
	} else if (t_1 <= 0.0) {
		tmp = (z / b) + ((x / b) * (t / y));
	} else if (t_1 <= 1e+307) {
		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 t_2 = (a + 1.0) + (y / (t / b));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (x / t_2) + ((y / t) * (z / t_2));
	} else if (t_1 <= -1e-160) {
		tmp = t_1;
	} else if (t_1 <= -1e-318) {
		tmp = (x + (z * (y / t))) * (1.0 / (a + (1.0 + (b * (y / t)))));
	} else if (t_1 <= 0.0) {
		tmp = (z / b) + ((x / b) * (t / y));
	} else if (t_1 <= 1e+307) {
		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))
	t_2 = (a + 1.0) + (y / (t / b))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (x / t_2) + ((y / t) * (z / t_2))
	elif t_1 <= -1e-160:
		tmp = t_1
	elif t_1 <= -1e-318:
		tmp = (x + (z * (y / t))) * (1.0 / (a + (1.0 + (b * (y / t)))))
	elif t_1 <= 0.0:
		tmp = (z / b) + ((x / b) * (t / y))
	elif t_1 <= 1e+307:
		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)))
	t_2 = Float64(Float64(a + 1.0) + Float64(y / Float64(t / b)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(x / t_2) + Float64(Float64(y / t) * Float64(z / t_2)));
	elseif (t_1 <= -1e-160)
		tmp = t_1;
	elseif (t_1 <= -1e-318)
		tmp = Float64(Float64(x + Float64(z * Float64(y / t))) * Float64(1.0 / Float64(a + Float64(1.0 + Float64(b * Float64(y / t))))));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(z / b) + Float64(Float64(x / b) * Float64(t / y)));
	elseif (t_1 <= 1e+307)
		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));
	t_2 = (a + 1.0) + (y / (t / b));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (x / t_2) + ((y / t) * (z / t_2));
	elseif (t_1 <= -1e-160)
		tmp = t_1;
	elseif (t_1 <= -1e-318)
		tmp = (x + (z * (y / t))) * (1.0 / (a + (1.0 + (b * (y / t)))));
	elseif (t_1 <= 0.0)
		tmp = (z / b) + ((x / b) * (t / y));
	elseif (t_1 <= 1e+307)
		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]}, Block[{t$95$2 = N[(N[(a + 1.0), $MachinePrecision] + N[(y / N[(t / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(x / t$95$2), $MachinePrecision] + N[(N[(y / t), $MachinePrecision] * N[(z / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1e-160], t$95$1, If[LessEqual[t$95$1, -1e-318], N[(N[(x + N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(a + N[(1.0 + N[(b * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(z / b), $MachinePrecision] + N[(N[(x / b), $MachinePrecision] * N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+307], t$95$1, N[(z / b), $MachinePrecision]]]]]]]]

Target

Original	25.84%
Target	20.36%
Herbie	8.93%

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

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 100

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

   2. Simplified 66.05

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

      [Start] 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-*l/ [<=] 66.06	\[ \frac{\color{blue}{\frac{y}{t} \cdot z} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      fma-def [=>] 66.05	\[ \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      +-commutative [=>] 66.05	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\color{blue}{\frac{y \cdot b}{t} + \left(a + 1\right)}} \]
      associate-+r+ [=>] 66.05	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\color{blue}{\left(\frac{y \cdot b}{t} + a\right) + 1}} \]
      +-commutative [=>] 66.05	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\color{blue}{1 + \left(\frac{y \cdot b}{t} + a\right)}} \]
      associate-*l/ [<=] 66.05	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + \left(\color{blue}{\frac{y}{t} \cdot b} + a\right)} \]
      fma-def [=>] 66.05	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + \color{blue}{\mathsf{fma}\left(\frac{y}{t}, b, a\right)}} \]

   3. Taylor expanded in z around 0 55.5

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

   4. Simplified 25.63

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

      [Start] 55.5	\[ \frac{y \cdot z}{t \cdot \left(\frac{y \cdot b}{t} + \left(1 + a\right)\right)} + \frac{x}{\frac{y \cdot b}{t} + \left(1 + a\right)} \]
      +-commutative [=>] 55.5	\[ \color{blue}{\frac{x}{\frac{y \cdot b}{t} + \left(1 + a\right)} + \frac{y \cdot z}{t \cdot \left(\frac{y \cdot b}{t} + \left(1 + a\right)\right)}} \]
      +-commutative [=>] 55.5	\[ \frac{x}{\color{blue}{\left(1 + a\right) + \frac{y \cdot b}{t}}} + \frac{y \cdot z}{t \cdot \left(\frac{y \cdot b}{t} + \left(1 + a\right)\right)} \]
      associate-/l* [=>] 55.5	\[ \frac{x}{\left(1 + a\right) + \color{blue}{\frac{y}{\frac{t}{b}}}} + \frac{y \cdot z}{t \cdot \left(\frac{y \cdot b}{t} + \left(1 + a\right)\right)} \]
      times-frac [=>] 22.43	\[ \frac{x}{\left(1 + a\right) + \frac{y}{\frac{t}{b}}} + \color{blue}{\frac{y}{t} \cdot \frac{z}{\frac{y \cdot b}{t} + \left(1 + a\right)}} \]
      +-commutative [=>] 22.43	\[ \frac{x}{\left(1 + a\right) + \frac{y}{\frac{t}{b}}} + \frac{y}{t} \cdot \frac{z}{\color{blue}{\left(1 + a\right) + \frac{y \cdot b}{t}}} \]
      associate-/l* [=>] 25.63	\[ \frac{x}{\left(1 + a\right) + \frac{y}{\frac{t}{b}}} + \frac{y}{t} \cdot \frac{z}{\left(1 + a\right) + \color{blue}{\frac{y}{\frac{t}{b}}}} \]

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -9.9999999999999999e-161 or 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 9.99999999999999986e306

   1. Initial program 0.53

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

   if -9.9999999999999999e-161 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -9.9999875e-319

   1. Initial program 2.47

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

   2. Applied egg-rr 56.15

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

   3. Simplified 2.68

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

      [Start] 56.15	\[ \frac{\left(x \cdot x - {\left(\frac{y}{\frac{t}{z}}\right)}^{2}\right) \cdot \frac{1}{a + \left(1 + \frac{b}{t} \cdot y\right)}}{x \cdot x - {\left(\frac{y}{\frac{t}{z}}\right)}^{2}} \cdot \left(x + \frac{y}{\frac{t}{z}}\right) \]
      *-commutative [=>] 56.15	\[ \color{blue}{\left(x + \frac{y}{\frac{t}{z}}\right) \cdot \frac{\left(x \cdot x - {\left(\frac{y}{\frac{t}{z}}\right)}^{2}\right) \cdot \frac{1}{a + \left(1 + \frac{b}{t} \cdot y\right)}}{x \cdot x - {\left(\frac{y}{\frac{t}{z}}\right)}^{2}}} \]
      associate-/l* [<=] 57.06	\[ \left(x + \color{blue}{\frac{y \cdot z}{t}}\right) \cdot \frac{\left(x \cdot x - {\left(\frac{y}{\frac{t}{z}}\right)}^{2}\right) \cdot \frac{1}{a + \left(1 + \frac{b}{t} \cdot y\right)}}{x \cdot x - {\left(\frac{y}{\frac{t}{z}}\right)}^{2}} \]
      associate-*l/ [<=] 56.52	\[ \left(x + \color{blue}{\frac{y}{t} \cdot z}\right) \cdot \frac{\left(x \cdot x - {\left(\frac{y}{\frac{t}{z}}\right)}^{2}\right) \cdot \frac{1}{a + \left(1 + \frac{b}{t} \cdot y\right)}}{x \cdot x - {\left(\frac{y}{\frac{t}{z}}\right)}^{2}} \]
      *-commutative [=>] 56.52	\[ \left(x + \color{blue}{z \cdot \frac{y}{t}}\right) \cdot \frac{\left(x \cdot x - {\left(\frac{y}{\frac{t}{z}}\right)}^{2}\right) \cdot \frac{1}{a + \left(1 + \frac{b}{t} \cdot y\right)}}{x \cdot x - {\left(\frac{y}{\frac{t}{z}}\right)}^{2}} \]
      associate-/l* [=>] 56.81	\[ \left(x + z \cdot \frac{y}{t}\right) \cdot \color{blue}{\frac{x \cdot x - {\left(\frac{y}{\frac{t}{z}}\right)}^{2}}{\frac{x \cdot x - {\left(\frac{y}{\frac{t}{z}}\right)}^{2}}{\frac{1}{a + \left(1 + \frac{b}{t} \cdot y\right)}}}} \]
      associate-/r/ [=>] 45.19	\[ \left(x + z \cdot \frac{y}{t}\right) \cdot \color{blue}{\left(\frac{x \cdot x - {\left(\frac{y}{\frac{t}{z}}\right)}^{2}}{x \cdot x - {\left(\frac{y}{\frac{t}{z}}\right)}^{2}} \cdot \frac{1}{a + \left(1 + \frac{b}{t} \cdot y\right)}\right)} \]
      *-inverses [=>] 9.28	\[ \left(x + z \cdot \frac{y}{t}\right) \cdot \left(\color{blue}{1} \cdot \frac{1}{a + \left(1 + \frac{b}{t} \cdot y\right)}\right) \]
      associate-*l/ [=>] 2.68	\[ \left(x + z \cdot \frac{y}{t}\right) \cdot \left(1 \cdot \frac{1}{a + \left(1 + \color{blue}{\frac{b \cdot y}{t}}\right)}\right) \]
      *-commutative [<=] 2.68	\[ \left(x + z \cdot \frac{y}{t}\right) \cdot \left(1 \cdot \frac{1}{a + \left(1 + \frac{\color{blue}{y \cdot b}}{t}\right)}\right) \]
      associate-*l/ [<=] 2.68	\[ \left(x + z \cdot \frac{y}{t}\right) \cdot \left(1 \cdot \frac{1}{a + \left(1 + \color{blue}{\frac{y}{t} \cdot b}\right)}\right) \]
      *-commutative [=>] 2.68	\[ \left(x + z \cdot \frac{y}{t}\right) \cdot \left(1 \cdot \frac{1}{a + \left(1 + \color{blue}{b \cdot \frac{y}{t}}\right)}\right) \]

   if -9.9999875e-319 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 0.0

   1. Initial program 46.23

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

   2. Simplified 30.72

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

      [Start] 46.23	\[ \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-/l* [=>] 45.74	\[ \frac{x + \color{blue}{\frac{y}{\frac{t}{z}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-+l+ [=>] 45.74	\[ \frac{x + \frac{y}{\frac{t}{z}}}{\color{blue}{a + \left(1 + \frac{y \cdot b}{t}\right)}} \]
      *-commutative [=>] 45.74	\[ \frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{\color{blue}{b \cdot y}}{t}\right)} \]
      associate-/l* [=>] 30.72	\[ \frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \color{blue}{\frac{b}{\frac{t}{y}}}\right)} \]

   3. Taylor expanded in y around -inf 36.96

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

   4. Simplified 36.24

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

      [Start] 36.96	\[ -1 \cdot \frac{-1 \cdot \frac{t \cdot x}{b} - -1 \cdot \frac{t \cdot \left(\left(1 + a\right) \cdot z\right)}{{b}^{2}}}{y} + \frac{z}{b} \]
      +-commutative [=>] 36.96	\[ \color{blue}{\frac{z}{b} + -1 \cdot \frac{-1 \cdot \frac{t \cdot x}{b} - -1 \cdot \frac{t \cdot \left(\left(1 + a\right) \cdot z\right)}{{b}^{2}}}{y}} \]
      associate-*r/ [=>] 36.96	\[ \frac{z}{b} + \color{blue}{\frac{-1 \cdot \left(-1 \cdot \frac{t \cdot x}{b} - -1 \cdot \frac{t \cdot \left(\left(1 + a\right) \cdot z\right)}{{b}^{2}}\right)}{y}} \]

   5. Taylor expanded in b around inf 41.01

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

   6. Simplified 29.25

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

      [Start] 41.01	\[ \frac{z}{b} + \frac{t \cdot x}{y \cdot b} \]
      times-frac [=>] 29.25	\[ \frac{z}{b} + \color{blue}{\frac{t}{y} \cdot \frac{x}{b}} \]
      *-commutative [=>] 29.25	\[ \frac{z}{b} + \color{blue}{\frac{x}{b} \cdot \frac{t}{y}} \]

   if 9.99999999999999986e306 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t)))

   1. Initial program 99.63

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

   2. Simplified 82.38

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

      [Start] 99.63	\[ \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-/l* [=>] 88.93	\[ \frac{x + \color{blue}{\frac{y}{\frac{t}{z}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-+l+ [=>] 88.93	\[ \frac{x + \frac{y}{\frac{t}{z}}}{\color{blue}{a + \left(1 + \frac{y \cdot b}{t}\right)}} \]
      *-commutative [=>] 88.93	\[ \frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{\color{blue}{b \cdot y}}{t}\right)} \]
      associate-/l* [=>] 82.38	\[ \frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \color{blue}{\frac{b}{\frac{t}{y}}}\right)} \]

   3. Taylor expanded in y around inf 18.52

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

4. Final simplification 8.93

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -\infty:\\ \;\;\;\;\frac{x}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}} + \frac{y}{t} \cdot \frac{z}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -1 \cdot 10^{-318}:\\ \;\;\;\;\left(x + z \cdot \frac{y}{t}\right) \cdot \frac{1}{a + \left(1 + b \cdot \frac{y}{t}\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{x}{b} \cdot \frac{t}{y}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 10^{+307}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.18%
Cost	5712

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{a + \left(1 + y \cdot \frac{b}{t}\right)}\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-318}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{x}{b} \cdot \frac{t}{y}\\ \mathbf{elif}\;t_1 \leq 10^{+307}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 2
Error	11.51%
Cost	4556

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-318}:\\ \;\;\;\;\left(x + z \cdot \frac{y}{t}\right) \cdot \frac{1}{a + \left(1 + b \cdot \frac{y}{t}\right)}\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{x}{b} \cdot \frac{t}{y}\\ \mathbf{elif}\;t_1 \leq 10^{+307}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 3
Error	35.16%
Cost	1620

\[\begin{array}{l} t_1 := b \cdot \frac{y}{t}\\ t_2 := \frac{x + z \cdot \frac{y}{t}}{1 + t_1}\\ \mathbf{if}\;b \leq -3.2 \cdot 10^{+188}:\\ \;\;\;\;\frac{z}{b} + \frac{x}{b} \cdot \frac{t}{y}\\ \mathbf{elif}\;b \leq -3 \cdot 10^{+109}:\\ \;\;\;\;x \cdot \frac{1}{\left(a + 1\right) + t_1}\\ \mathbf{elif}\;b \leq -3.6 \cdot 10^{+22}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 7.8 \cdot 10^{-31}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a + 1}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{+70}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{+198}:\\ \;\;\;\;\frac{x}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \end{array} \]

Alternative 4
Error	23.28%
Cost	1616

\[\begin{array}{l} t_1 := \frac{x + y \cdot \frac{z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ t_2 := \frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{if}\;y \leq -1.25 \cdot 10^{+85}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -6 \cdot 10^{-133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-287}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+176}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	23.06%
Cost	1616

\[\begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+135}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -6 \cdot 10^{-133}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{b}{\frac{t}{y}}\right)}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-287}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+175}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \end{array} \]

Alternative 6
Error	34.55%
Cost	1497

\[\begin{array}{l} t_1 := \frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{if}\;b \leq -3 \cdot 10^{+188}:\\ \;\;\;\;\frac{z}{b} + \frac{x}{b} \cdot \frac{t}{y}\\ \mathbf{elif}\;b \leq -5.5 \cdot 10^{+107}:\\ \;\;\;\;\frac{x}{a + \left(1 + y \cdot \frac{b}{t}\right)}\\ \mathbf{elif}\;b \leq -2.1 \cdot 10^{+63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 69000000000:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a + 1}\\ \mathbf{elif}\;b \leq 2.55 \cdot 10^{+73} \lor \neg \left(b \leq 1.52 \cdot 10^{+199}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \end{array} \]

Alternative 7
Error	34.56%
Cost	1497

\[\begin{array}{l} t_1 := \frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{if}\;b \leq -3.7 \cdot 10^{+188}:\\ \;\;\;\;\frac{z}{b} + \frac{x}{b} \cdot \frac{t}{y}\\ \mathbf{elif}\;b \leq -1.15 \cdot 10^{+108}:\\ \;\;\;\;\frac{x}{a + \left(1 + y \cdot \frac{b}{t}\right)}\\ \mathbf{elif}\;b \leq -4.1 \cdot 10^{+61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 4400000000000:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{+73} \lor \neg \left(b \leq 6.5 \cdot 10^{+197}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \end{array} \]

Alternative 8
Error	34.4%
Cost	1497

\[\begin{array}{l} t_1 := \frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{if}\;b \leq -3.5 \cdot 10^{+188}:\\ \;\;\;\;\frac{z}{b} + \frac{x}{b} \cdot \frac{t}{y}\\ \mathbf{elif}\;b \leq -1.36 \cdot 10^{+108}:\\ \;\;\;\;x \cdot \frac{1}{\left(a + 1\right) + b \cdot \frac{y}{t}}\\ \mathbf{elif}\;b \leq -4.5 \cdot 10^{+66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 90000000000000:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{+72} \lor \neg \left(b \leq 6.4 \cdot 10^{+199}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \end{array} \]

Alternative 9
Error	42.54%
Cost	1370

\[\begin{array}{l} t_1 := \frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{if}\;y \leq -390000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{-29}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a}\\ \mathbf{elif}\;y \leq -9.2 \cdot 10^{-38} \lor \neg \left(y \leq 6000000\right) \land \left(y \leq 6.5 \cdot 10^{+42} \lor \neg \left(y \leq 1.02 \cdot 10^{+99}\right)\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a + 1}\\ \end{array} \]

Alternative 10
Error	44.8%
Cost	1104

\[\begin{array}{l} t_1 := \frac{x}{a + 1}\\ \mathbf{if}\;y \leq -7.6 \cdot 10^{+66}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -9.2 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.15 \cdot 10^{-36}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{-57}:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{a + 1}\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+99}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 11
Error	44.86%
Cost	1104

\[\begin{array}{l} t_1 := \frac{x}{a + 1}\\ \mathbf{if}\;y \leq -3.9 \cdot 10^{+62}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -7.5 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{-41}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{-57}:\\ \;\;\;\;\frac{y \cdot z}{t \cdot \left(a + 1\right)}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+99}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 12
Error	46.07%
Cost	972

\[\begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+67}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a}\\ \mathbf{elif}\;y \leq -1.9 \cdot 10^{-132}:\\ \;\;\;\;\frac{x}{1 + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+99}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 13
Error	37.59%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{+73} \lor \neg \left(y \leq 1.3 \cdot 10^{+99}\right):\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a + \left(1 + y \cdot \frac{b}{t}\right)}\\ \end{array} \]

Alternative 14
Error	37.27%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+71} \lor \neg \left(y \leq 2.9 \cdot 10^{+99}\right):\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \end{array} \]

Alternative 15
Error	35.91%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -1.52 \cdot 10^{+78} \lor \neg \left(y \leq 3.1 \cdot 10^{+99}\right):\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \end{array} \]

Alternative 16
Error	61.09%
Cost	720

\[\begin{array}{l} \mathbf{if}\;y \leq -1.22 \cdot 10^{+61}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -6.8 \cdot 10^{-254}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-151}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+93}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 17
Error	61.3%
Cost	720

\[\begin{array}{l} \mathbf{if}\;y \leq -4.05 \cdot 10^{+71}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{-257}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{-151}:\\ \;\;\;\;x - x \cdot a\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+93}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 18
Error	44.41%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+69}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{+99}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 19
Error	57.09%
Cost	456

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

Alternative 20
Error	79.14%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023089 
(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))))