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

Average Accuracy: 74.4% → 90.3%

Time: 33.9s

Precision: binary64

Cost: 5712

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ t_2 := \left(a + 1\right) + \frac{y}{\frac{t}{b}}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+151}:\\ \;\;\;\;\frac{x}{t_2} + \frac{y}{t} \cdot \frac{z}{t_2}\\ \mathbf{elif}\;t_1 \leq -5 \cdot 10^{-309}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{t}{y} \cdot \frac{x + \frac{y}{\frac{t}{z}}}{b}\\ \mathbf{elif}\;t_1 \leq 10^{+308}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

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 -5e+151)
     (+ (/ x t_2) (* (/ y t) (/ z t_2)))
     (if (<= t_1 -5e-309)
       t_1
       (if (<= t_1 0.0)
         (* (/ t y) (/ (+ x (/ y (/ t z))) b))
         (if (<= t_1 1e+308) 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 <= -5e+151) {
		tmp = (x / t_2) + ((y / t) * (z / t_2));
	} else if (t_1 <= -5e-309) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (t / y) * ((x + (y / (t / z))) / b);
	} else if (t_1 <= 1e+308) {
		tmp = t_1;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (x + ((y * z) / t)) / ((a + 1.0d0) + ((y * b) / t))
end function

↓

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x + ((y * z) / t)) / (((y * b) / t) + (a + 1.0d0))
    t_2 = (a + 1.0d0) + (y / (t / b))
    if (t_1 <= (-5d+151)) then
        tmp = (x / t_2) + ((y / t) * (z / t_2))
    else if (t_1 <= (-5d-309)) then
        tmp = t_1
    else if (t_1 <= 0.0d0) then
        tmp = (t / y) * ((x + (y / (t / z))) / b)
    else if (t_1 <= 1d+308) then
        tmp = t_1
    else
        tmp = z / b
    end if
    code = tmp
end function

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 <= -5e+151) {
		tmp = (x / t_2) + ((y / t) * (z / t_2));
	} else if (t_1 <= -5e-309) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (t / y) * ((x + (y / (t / z))) / b);
	} else if (t_1 <= 1e+308) {
		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 <= -5e+151:
		tmp = (x / t_2) + ((y / t) * (z / t_2))
	elif t_1 <= -5e-309:
		tmp = t_1
	elif t_1 <= 0.0:
		tmp = (t / y) * ((x + (y / (t / z))) / b)
	elif t_1 <= 1e+308:
		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 <= -5e+151)
		tmp = Float64(Float64(x / t_2) + Float64(Float64(y / t) * Float64(z / t_2)));
	elseif (t_1 <= -5e-309)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(t / y) * Float64(Float64(x + Float64(y / Float64(t / z))) / b));
	elseif (t_1 <= 1e+308)
		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 <= -5e+151)
		tmp = (x / t_2) + ((y / t) * (z / t_2));
	elseif (t_1 <= -5e-309)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = (t / y) * ((x + (y / (t / z))) / b);
	elseif (t_1 <= 1e+308)
		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, -5e+151], 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, -5e-309], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(t / y), $MachinePrecision] * N[(N[(x + N[(y / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+308], t$95$1, N[(z / b), $MachinePrecision]]]]]]]

Target

Original	74.4%
Target	80.0%
Herbie	90.3%

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

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))) < -5.0000000000000002e151

   1. Initial program 61.3%

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

   2. Simplified 76.1%

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

      [Start] 61.3	\[ \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      +-commutative [=>] 61.3	\[ \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-*l/ [<=] 76.1	\[ \frac{\color{blue}{\frac{y}{t} \cdot z} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      fma-def [=>] 76.1	\[ \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      +-commutative [=>] 76.1	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\color{blue}{\frac{y \cdot b}{t} + \left(a + 1\right)}} \]
      associate-+r+ [=>] 76.1	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\color{blue}{\left(\frac{y \cdot b}{t} + a\right) + 1}} \]
      +-commutative [=>] 76.1	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\color{blue}{1 + \left(\frac{y \cdot b}{t} + a\right)}} \]
      associate-*l/ [<=] 76.1	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + \left(\color{blue}{\frac{y}{t} \cdot b} + a\right)} \]
      fma-def [=>] 76.1	\[ \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + \color{blue}{\mathsf{fma}\left(\frac{y}{t}, b, a\right)}} \]

   3. Taylor expanded in z around 0 74.6%

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

   4. Simplified 91.8%

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

      [Start] 74.6	\[ \frac{y \cdot z}{t \cdot \left(\frac{y \cdot b}{t} + \left(1 + a\right)\right)} + \frac{x}{\frac{y \cdot b}{t} + \left(1 + a\right)} \]
      +-commutative [=>] 74.6	\[ \color{blue}{\frac{x}{\frac{y \cdot b}{t} + \left(1 + a\right)} + \frac{y \cdot z}{t \cdot \left(\frac{y \cdot b}{t} + \left(1 + a\right)\right)}} \]
      +-commutative [=>] 74.6	\[ \frac{x}{\color{blue}{\left(1 + a\right) + \frac{y \cdot b}{t}}} + \frac{y \cdot z}{t \cdot \left(\frac{y \cdot b}{t} + \left(1 + a\right)\right)} \]
      associate-/l* [=>] 74.4	\[ \frac{x}{\left(1 + a\right) + \color{blue}{\frac{y}{\frac{t}{b}}}} + \frac{y \cdot z}{t \cdot \left(\frac{y \cdot b}{t} + \left(1 + a\right)\right)} \]
      times-frac [=>] 92.7	\[ \frac{x}{\left(1 + a\right) + \frac{y}{\frac{t}{b}}} + \color{blue}{\frac{y}{t} \cdot \frac{z}{\frac{y \cdot b}{t} + \left(1 + a\right)}} \]
      +-commutative [=>] 92.7	\[ \frac{x}{\left(1 + a\right) + \frac{y}{\frac{t}{b}}} + \frac{y}{t} \cdot \frac{z}{\color{blue}{\left(1 + a\right) + \frac{y \cdot b}{t}}} \]
      associate-/l* [=>] 91.8	\[ \frac{x}{\left(1 + a\right) + \frac{y}{\frac{t}{b}}} + \frac{y}{t} \cdot \frac{z}{\left(1 + a\right) + \color{blue}{\frac{y}{\frac{t}{b}}}} \]

   if -5.0000000000000002e151 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -4.9999999999999995e-309 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 1e308

   1. Initial program 99.2%

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

   if -4.9999999999999995e-309 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 54.7%

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

   2. Simplified 69.0%

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

      [Start] 54.7	\[ \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      +-commutative [=>] 54.7	\[ \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-*r/ [<=] 55.4	\[ \frac{\color{blue}{y \cdot \frac{z}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      *-commutative [<=] 55.4	\[ \frac{\color{blue}{\frac{z}{t} \cdot y} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      fma-def [=>] 55.4	\[ \frac{\color{blue}{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-+l+ [=>] 55.4	\[ \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\color{blue}{a + \left(1 + \frac{y \cdot b}{t}\right)}} \]
      +-commutative [=>] 55.4	\[ \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{a + \color{blue}{\left(\frac{y \cdot b}{t} + 1\right)}} \]
      associate-*r/ [<=] 69.0	\[ \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{a + \left(\color{blue}{y \cdot \frac{b}{t}} + 1\right)} \]
      *-commutative [<=] 69.0	\[ \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{a + \left(\color{blue}{\frac{b}{t} \cdot y} + 1\right)} \]
      fma-def [=>] 69.0	\[ \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{a + \color{blue}{\mathsf{fma}\left(\frac{b}{t}, y, 1\right)}} \]

   3. Taylor expanded in b around inf 50.0%

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

   4. Simplified 64.7%

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

      [Start] 50.0	\[ \frac{t \cdot \left(\frac{y \cdot z}{t} + x\right)}{y \cdot b} \]
      times-frac [=>] 65.4	\[ \color{blue}{\frac{t}{y} \cdot \frac{\frac{y \cdot z}{t} + x}{b}} \]
      +-commutative [=>] 65.4	\[ \frac{t}{y} \cdot \frac{\color{blue}{x + \frac{y \cdot z}{t}}}{b} \]
      associate-/l* [=>] 64.7	\[ \frac{t}{y} \cdot \frac{x + \color{blue}{\frac{y}{\frac{t}{z}}}}{b} \]

   if 1e308 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t)))

   1. Initial program 0.0%

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

   2. Simplified 19.1%

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

      [Start] 0.0	\[ \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-/l* [=>] 12.6	\[ \frac{x + \color{blue}{\frac{y}{\frac{t}{z}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-+l+ [=>] 12.6	\[ \frac{x + \frac{y}{\frac{t}{z}}}{\color{blue}{a + \left(1 + \frac{y \cdot b}{t}\right)}} \]
      *-commutative [=>] 12.6	\[ \frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{\color{blue}{b \cdot y}}{t}\right)} \]
      associate-/l* [=>] 19.1	\[ \frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \color{blue}{\frac{b}{\frac{t}{y}}}\right)} \]

   3. Taylor expanded in y around inf 80.9%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -5 \cdot 10^{+151}:\\ \;\;\;\;\frac{x}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}} + \frac{y}{t} \cdot \frac{z}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 0:\\ \;\;\;\;\frac{t}{y} \cdot \frac{x + \frac{y}{\frac{t}{z}}}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 10^{+308}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	90.2%
Cost	5712

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{a + \left(1 + b \cdot \frac{y}{t}\right)}\\ \mathbf{elif}\;t_1 \leq -5 \cdot 10^{-309}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{t}{y} \cdot \frac{x + \frac{y}{\frac{t}{z}}}{b}\\ \mathbf{elif}\;t_1 \leq 10^{+308}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 2
Accuracy	59.5%
Cost	2280

\[\begin{array}{l} t_1 := \frac{y}{t} \cdot \frac{z}{a + \left(1 + b \cdot \frac{y}{t}\right)}\\ t_2 := \frac{x + \frac{y}{\frac{t}{z}}}{a + 1}\\ \mathbf{if}\;x \leq -0.08:\\ \;\;\;\;\frac{x}{a + \left(1 + \frac{b}{\frac{t}{y}}\right)}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-36}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;x \leq -1.12 \cdot 10^{-177}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.02 \cdot 10^{-189}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-289}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-303}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-192}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-141}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-14}:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+184}:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{a + 1} + \frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a + \left(1 + y \cdot \frac{b}{t}\right)}\\ \end{array} \]

Alternative 3
Accuracy	53.6%
Cost	1892

\[\begin{array}{l} t_1 := \frac{x}{a + \left(1 + b \cdot \frac{y}{t}\right)}\\ t_2 := \frac{y}{\left(a + 1\right) \cdot \frac{t}{z}}\\ \mathbf{if}\;x \leq -0.059:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-33}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-60}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-178}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.04 \cdot 10^{-190}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-243}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-191}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-138}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-14}:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	53.8%
Cost	1892

\[\begin{array}{l} t_1 := \frac{y}{\left(a + 1\right) \cdot \frac{t}{z}}\\ t_2 := \frac{x}{a + \left(1 + b \cdot \frac{y}{t}\right)}\\ \mathbf{if}\;x \leq -0.145:\\ \;\;\;\;\frac{x}{a + \left(1 + \frac{b}{\frac{t}{y}}\right)}\\ \mathbf{elif}\;x \leq -1.86 \cdot 10^{-32}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-61}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.95 \cdot 10^{-174}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-190}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-236}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-191}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-143}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-14}:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Accuracy	67.0%
Cost	1752

\[\begin{array}{l} t_1 := a + \left(1 + b \cdot \frac{y}{t}\right)\\ \mathbf{if}\;y \leq -4.7 \cdot 10^{+179}:\\ \;\;\;\;\frac{1}{b} \cdot \left(z + x \cdot \frac{t}{y}\right)\\ \mathbf{elif}\;y \leq -1.7 \cdot 10^{+64}:\\ \;\;\;\;\frac{x}{t_1}\\ \mathbf{elif}\;y \leq -420000:\\ \;\;\;\;\frac{y \cdot z}{y \cdot b + t \cdot \left(a + 1\right)}\\ \mathbf{elif}\;y \leq 250:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+128}:\\ \;\;\;\;\frac{x}{a + \left(1 + y \cdot \frac{b}{t}\right)}\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+161}:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \end{array} \]

Alternative 6
Accuracy	45.2%
Cost	1632

\[\begin{array}{l} t_1 := \frac{x}{a + 1}\\ t_2 := \frac{y}{\left(a + 1\right) \cdot \frac{t}{z}}\\ \mathbf{if}\;x \leq -0.7:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-37}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-60}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-178}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -8.4 \cdot 10^{-190}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;x \leq 1.08 \cdot 10^{-237}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-191}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-138}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-14}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Accuracy	54.6%
Cost	1632

\[\begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ t_2 := \frac{1}{b} \cdot \left(z + x \cdot \frac{t}{y}\right)\\ t_3 := \frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{if}\;a \leq -3.1 \cdot 10^{+106}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a}\\ \mathbf{elif}\;a \leq -7.8 \cdot 10^{-69}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -3.6 \cdot 10^{-242}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.55 \cdot 10^{-292}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{-262}:\\ \;\;\;\;\frac{x}{1 + y \cdot \frac{b}{t}}\\ \mathbf{elif}\;a \leq 3.1 \cdot 10^{-159}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 7.8 \cdot 10^{-57}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 800000000:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a}\\ \end{array} \]

Alternative 8
Accuracy	55.1%
Cost	1632

\[\begin{array}{l} t_1 := \frac{z + \frac{x \cdot t}{y}}{b}\\ t_2 := \frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ t_3 := x + \frac{y \cdot z}{t}\\ \mathbf{if}\;a \leq -1.8 \cdot 10^{+105}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a}\\ \mathbf{elif}\;a \leq -6 \cdot 10^{-66}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -2.2 \cdot 10^{-242}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -3.9 \cdot 10^{-287}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{-262}:\\ \;\;\;\;\frac{x}{1 + y \cdot \frac{b}{t}}\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{-126}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 1.46 \cdot 10^{-61}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 550000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a}\\ \end{array} \]

Alternative 9
Accuracy	55.3%
Cost	1500

\[\begin{array}{l} t_1 := \frac{z + \frac{x \cdot t}{y}}{b}\\ t_2 := x + z \cdot \frac{y}{t}\\ \mathbf{if}\;a \leq -1.18 \cdot 10^{+100}:\\ \;\;\;\;\frac{t_2}{a}\\ \mathbf{elif}\;a \leq -2.7 \cdot 10^{-68}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -5.5 \cdot 10^{-242}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \mathbf{elif}\;a \leq -1 \cdot 10^{-291}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{-301}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 4 \cdot 10^{-62}:\\ \;\;\;\;\frac{x}{1 + y \cdot \frac{b}{t}}\\ \mathbf{elif}\;a \leq 90000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a}\\ \end{array} \]

Alternative 10
Accuracy	65.8%
Cost	1496

\[\begin{array}{l} t_1 := \frac{x + \frac{y}{\frac{t}{z}}}{a + 1}\\ \mathbf{if}\;b \leq -4.6 \cdot 10^{+39}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{elif}\;b \leq 3.15 \cdot 10^{-28}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{+104}:\\ \;\;\;\;\frac{x}{a + \left(1 + y \cdot \frac{b}{t}\right)}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{+138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{+145}:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{+198}:\\ \;\;\;\;\frac{x}{a + \left(1 + b \cdot \frac{y}{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} \cdot \left(z + x \cdot \frac{t}{y}\right)\\ \end{array} \]

Alternative 11
Accuracy	41.2%
Cost	1380

\[\begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+63}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq -4.5 \cdot 10^{-69}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -6.2 \cdot 10^{-242}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{-295}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 7 \cdot 10^{-255}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{-126}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{-112}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{-35}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{+81}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

Alternative 12
Accuracy	52.4%
Cost	1368

\[\begin{array}{l} t_1 := \frac{x}{1 + y \cdot \frac{b}{t}}\\ \mathbf{if}\;y \leq -1.3 \cdot 10^{+144}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -1.3 \cdot 10^{+72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -3.7 \cdot 10^{+17}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-81}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+42}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+127}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 13
Accuracy	79.6%
Cost	1352

\[\begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+180}:\\ \;\;\;\;\frac{1}{b} \cdot \left(z + x \cdot \frac{t}{y}\right)\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+209}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{b}{\frac{t}{y}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \end{array} \]

Alternative 14
Accuracy	52.0%
Cost	1108

\[\begin{array}{l} t_1 := \frac{x}{a + 1}\\ \mathbf{if}\;y \leq -2.15 \cdot 10^{+17}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-81}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.85 \cdot 10^{+41}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+105}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+178}:\\ \;\;\;\;\frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 15
Accuracy	53.4%
Cost	1104

\[\begin{array}{l} t_1 := \frac{x + z \cdot \frac{y}{t}}{a}\\ \mathbf{if}\;a \leq -1.7 \cdot 10^{+105}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -3.2 \cdot 10^{-61}:\\ \;\;\;\;\frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{elif}\;a \leq -6.6 \cdot 10^{-193}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \mathbf{elif}\;a \leq 15200000:\\ \;\;\;\;\frac{x}{1 + y \cdot \frac{b}{t}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Accuracy	52.7%
Cost	1104

\[\begin{array}{l} \mathbf{if}\;a \leq -1.7 \cdot 10^{+105}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a}\\ \mathbf{elif}\;a \leq -2 \cdot 10^{-60}:\\ \;\;\;\;\frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{elif}\;a \leq -1.75 \cdot 10^{-194}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \mathbf{elif}\;a \leq 160000:\\ \;\;\;\;\frac{x}{1 + y \cdot \frac{b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a}\\ \end{array} \]

Alternative 17
Accuracy	42.4%
Cost	984

\[\begin{array}{l} \mathbf{if}\;a \leq -8.2 \cdot 10^{+68}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq -3 \cdot 10^{-68}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -2.8 \cdot 10^{-242}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.95 \cdot 10^{-295}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{-255}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.45 \cdot 10^{+81}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

Alternative 18
Accuracy	53.6%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1.26 \cdot 10^{+17}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-81}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 19
Accuracy	42.7%
Cost	456

\[\begin{array}{l} \mathbf{if}\;a \leq -1:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq 0.053:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

Alternative 20
Accuracy	21.0%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023135 
(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))))