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

Average Error: 16.5 → 7.0

Time: 20.4s

Precision: binary64

Cost: 4684

Math

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

↓

\[\begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ t_2 := \frac{t_1}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{if}\;t_2 \leq -4 \cdot 10^{-319}:\\ \;\;\;\;\frac{x + \frac{-1}{\frac{\frac{-t}{y}}{z}}}{1 + \left(a + \frac{b}{\frac{t}{y}}\right)}\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{elif}\;t_2 \leq 10^{+304}:\\ \;\;\;\;\frac{t_1}{\left(y \cdot b\right) \cdot \frac{1}{t} + \left(a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

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))) (t_2 (/ t_1 (+ (/ (* y b) t) (+ a 1.0)))))
   (if (<= t_2 -4e-319)
     (/ (+ x (/ -1.0 (/ (/ (- t) y) z))) (+ 1.0 (+ a (/ b (/ t y)))))
     (if (<= t_2 0.0)
       (+ (/ z b) (* (/ t y) (/ x b)))
       (if (<= t_2 1e+304)
         (/ t_1 (+ (* (* y b) (/ 1.0 t)) (+ a 1.0)))
         (/ 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);
	double t_2 = t_1 / (((y * b) / t) + (a + 1.0));
	double tmp;
	if (t_2 <= -4e-319) {
		tmp = (x + (-1.0 / ((-t / y) / z))) / (1.0 + (a + (b / (t / y))));
	} else if (t_2 <= 0.0) {
		tmp = (z / b) + ((t / y) * (x / b));
	} else if (t_2 <= 1e+304) {
		tmp = t_1 / (((y * b) * (1.0 / t)) + (a + 1.0));
	} 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)
    t_2 = t_1 / (((y * b) / t) + (a + 1.0d0))
    if (t_2 <= (-4d-319)) then
        tmp = (x + ((-1.0d0) / ((-t / y) / z))) / (1.0d0 + (a + (b / (t / y))))
    else if (t_2 <= 0.0d0) then
        tmp = (z / b) + ((t / y) * (x / b))
    else if (t_2 <= 1d+304) then
        tmp = t_1 / (((y * b) * (1.0d0 / t)) + (a + 1.0d0))
    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);
	double t_2 = t_1 / (((y * b) / t) + (a + 1.0));
	double tmp;
	if (t_2 <= -4e-319) {
		tmp = (x + (-1.0 / ((-t / y) / z))) / (1.0 + (a + (b / (t / y))));
	} else if (t_2 <= 0.0) {
		tmp = (z / b) + ((t / y) * (x / b));
	} else if (t_2 <= 1e+304) {
		tmp = t_1 / (((y * b) * (1.0 / t)) + (a + 1.0));
	} 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)
	t_2 = t_1 / (((y * b) / t) + (a + 1.0))
	tmp = 0
	if t_2 <= -4e-319:
		tmp = (x + (-1.0 / ((-t / y) / z))) / (1.0 + (a + (b / (t / y))))
	elif t_2 <= 0.0:
		tmp = (z / b) + ((t / y) * (x / b))
	elif t_2 <= 1e+304:
		tmp = t_1 / (((y * b) * (1.0 / t)) + (a + 1.0))
	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(x + Float64(Float64(y * z) / t))
	t_2 = Float64(t_1 / Float64(Float64(Float64(y * b) / t) + Float64(a + 1.0)))
	tmp = 0.0
	if (t_2 <= -4e-319)
		tmp = Float64(Float64(x + Float64(-1.0 / Float64(Float64(Float64(-t) / y) / z))) / Float64(1.0 + Float64(a + Float64(b / Float64(t / y)))));
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(z / b) + Float64(Float64(t / y) * Float64(x / b)));
	elseif (t_2 <= 1e+304)
		tmp = Float64(t_1 / Float64(Float64(Float64(y * b) * Float64(1.0 / t)) + Float64(a + 1.0)));
	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);
	t_2 = t_1 / (((y * b) / t) + (a + 1.0));
	tmp = 0.0;
	if (t_2 <= -4e-319)
		tmp = (x + (-1.0 / ((-t / y) / z))) / (1.0 + (a + (b / (t / y))));
	elseif (t_2 <= 0.0)
		tmp = (z / b) + ((t / y) * (x / b));
	elseif (t_2 <= 1e+304)
		tmp = t_1 / (((y * b) * (1.0 / t)) + (a + 1.0));
	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[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e-319], N[(N[(x + N[(-1.0 / N[(N[((-t) / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(a + N[(b / N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(z / b), $MachinePrecision] + N[(N[(t / y), $MachinePrecision] * N[(x / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+304], N[(t$95$1 / N[(N[(N[(y * b), $MachinePrecision] * N[(1.0 / t), $MachinePrecision]), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]]]]

Target

Original	16.5
Target	12.9
Herbie	7.0

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

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))) < -4.0000049e-319

   1. Initial program 7.6

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

   2. Simplified 6.4

      \[\leadsto \color{blue}{\frac{x + \frac{y}{t} \cdot z}{1 + \left(a + \frac{y}{t} \cdot b\right)}} \]
      Proof
      (/.f64 (+.f64 x (*.f64 (/.f64 y t) z)) (+.f64 1 (+.f64 a (*.f64 (/.f64 y t) b)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 x (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y z) t))) (+.f64 1 (+.f64 a (*.f64 (/.f64 y t) b)))): 25 points increase in error, 20 points decrease in error
      (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 1 (+.f64 a (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y b) t))))): 9 points increase in error, 7 points decrease in error
      (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 1 a) (/.f64 (*.f64 y b) t)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 a 1)) (/.f64 (*.f64 y b) t))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 6.4

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

   4. Applied egg-rr 8.7

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

   5. Applied egg-rr 6.2

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

   if -4.0000049e-319 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 0.0

   1. Initial program 30.1

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

   2. Simplified 19.4

      \[\leadsto \color{blue}{\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}} \]
      Proof
      (/.f64 (+.f64 x (/.f64 y (/.f64 t z))) (+.f64 (+.f64 a 1) (/.f64 y (/.f64 t b)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 x (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y z) t))) (+.f64 (+.f64 a 1) (/.f64 y (/.f64 t b)))): 21 points increase in error, 21 points decrease in error
      (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y b) t)))): 9 points increase in error, 14 points decrease in error

   3. Taylor expanded in t around 0 26.7

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

   4. Simplified 22.1

      \[\leadsto \color{blue}{\frac{z}{b} + t \cdot \left(\frac{\frac{x}{y}}{b} - \frac{1 + a}{y} \cdot \frac{z}{b \cdot b}\right)} \]
      Proof
      (+.f64 (/.f64 z b) (*.f64 t (-.f64 (/.f64 (/.f64 x y) b) (*.f64 (/.f64 (+.f64 1 a) y) (/.f64 z (*.f64 b b)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 z b) (*.f64 t (-.f64 (Rewrite<= associate-/r*_binary64 (/.f64 x (*.f64 y b))) (*.f64 (/.f64 (+.f64 1 a) y) (/.f64 z (*.f64 b b)))))): 4 points increase in error, 13 points decrease in error
      (+.f64 (/.f64 z b) (*.f64 t (-.f64 (/.f64 x (*.f64 y b)) (*.f64 (/.f64 (+.f64 1 a) y) (/.f64 z (Rewrite<= unpow2_binary64 (pow.f64 b 2))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 z b) (*.f64 t (-.f64 (/.f64 x (*.f64 y b)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (+.f64 1 a) z) (*.f64 y (pow.f64 b 2))))))): 10 points increase in error, 11 points decrease in error
      (+.f64 (/.f64 z b) (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 (/.f64 x (*.f64 y b)) (/.f64 (*.f64 (+.f64 1 a) z) (*.f64 y (pow.f64 b 2)))) t))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in x around inf 26.2

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

   6. Simplified 17.8

      \[\leadsto \frac{z}{b} + \color{blue}{\frac{x}{b} \cdot \frac{t}{y}} \]
      Proof
      (*.f64 (/.f64 x b) (/.f64 t y)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 t y) (/.f64 x b))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 t x) (*.f64 y b))): 54 points increase in error, 46 points decrease in error

   if 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 9.9999999999999994e303

   1. Initial program 0.4

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

   2. Applied egg-rr 0.4

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

   if 9.9999999999999994e303 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t)))

   1. Initial program 63.8

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

   2. Simplified 51.5

      \[\leadsto \color{blue}{\frac{x + \frac{y}{t} \cdot z}{1 + \left(a + \frac{y}{t} \cdot b\right)}} \]
      Proof
      (/.f64 (+.f64 x (*.f64 (/.f64 y t) z)) (+.f64 1 (+.f64 a (*.f64 (/.f64 y t) b)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 x (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y z) t))) (+.f64 1 (+.f64 a (*.f64 (/.f64 y t) b)))): 25 points increase in error, 20 points decrease in error
      (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 1 (+.f64 a (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y b) t))))): 9 points increase in error, 7 points decrease in error
      (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 1 a) (/.f64 (*.f64 y b) t)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 a 1)) (/.f64 (*.f64 y b) t))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in y around inf 12.5

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

4. Final simplification 7.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -4 \cdot 10^{-319}:\\ \;\;\;\;\frac{x + \frac{-1}{\frac{\frac{-t}{y}}{z}}}{1 + \left(a + \frac{b}{\frac{t}{y}}\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 10^{+304}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(y \cdot b\right) \cdot \frac{1}{t} + \left(a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.0
Cost	4684

\[\begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ t_2 := \frac{t_1}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{if}\;t_2 \leq -4 \cdot 10^{-319}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{1 + \left(a + b \cdot \frac{y}{t}\right)}\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{elif}\;t_2 \leq 10^{+304}:\\ \;\;\;\;\frac{t_1}{\left(y \cdot b\right) \cdot \frac{1}{t} + \left(a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 2
Error	7.0
Cost	4556

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{-319}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{1 + \left(a + b \cdot \frac{y}{t}\right)}\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{elif}\;t_1 \leq 10^{+304}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 3
Error	12.5
Cost	1616

\[\begin{array}{l} t_1 := \frac{x + z \cdot \frac{y}{t}}{1 + \left(a + b \cdot \frac{y}{t}\right)}\\ \mathbf{if}\;t \leq -2.9 \cdot 10^{-92}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-220}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 9.4 \cdot 10^{-67}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	12.5
Cost	1616

\[\begin{array}{l} t_1 := x + z \cdot \frac{y}{t}\\ t_2 := \frac{t_1}{1 + \left(a + b \cdot \frac{y}{t}\right)}\\ \mathbf{if}\;t \leq -5.6 \cdot 10^{-92}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-220}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{elif}\;t \leq 5.9 \cdot 10^{-111}:\\ \;\;\;\;\frac{t_1}{1 + \left(a + \frac{b}{\frac{t}{y}}\right)}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-67}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	12.5
Cost	1616

\[\begin{array}{l} t_1 := x + z \cdot \frac{y}{t}\\ \mathbf{if}\;t \leq -2.9 \cdot 10^{-92}:\\ \;\;\;\;\frac{t_1}{1 + \left(a + b \cdot \frac{y}{t}\right)}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-219}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-116}:\\ \;\;\;\;\frac{t_1}{1 + \left(a + \frac{b}{\frac{t}{y}}\right)}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-72}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\frac{y}{\frac{t}{b}} + \left(a + 1\right)}\\ \end{array} \]

Alternative 6
Error	29.9
Cost	1236

\[\begin{array}{l} t_1 := \frac{x}{a + 1}\\ \mathbf{if}\;y \leq -1.4 \cdot 10^{+59}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{1 + b \cdot \frac{y}{t}}\\ \mathbf{elif}\;y \leq -7 \cdot 10^{-50}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-49}:\\ \;\;\;\;\frac{y}{\frac{t}{z} \cdot \left(a + 1\right)}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 7
Error	30.0
Cost	1236

\[\begin{array}{l} t_1 := \frac{x}{a + 1}\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{+56}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -1.15 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{1 + b \cdot \frac{y}{t}}\\ \mathbf{elif}\;y \leq -9.5 \cdot 10^{-49}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 6.3 \cdot 10^{-83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-49}:\\ \;\;\;\;\frac{\frac{y \cdot z}{t}}{a + 1}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+120}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 8
Error	28.2
Cost	1232

\[\begin{array}{l} t_1 := \frac{x}{a + 1}\\ t_2 := \frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{if}\;y \leq -4.4 \cdot 10^{-49}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-49}:\\ \;\;\;\;\frac{\frac{y \cdot z}{t}}{a + 1}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+120}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	23.9
Cost	1232

\[\begin{array}{l} t_1 := \frac{x}{1 + \left(a + b \cdot \frac{y}{t}\right)}\\ \mathbf{if}\;t \leq -1.42 \cdot 10^{+51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.85 \cdot 10^{-34}:\\ \;\;\;\;\frac{y}{\frac{t}{z} \cdot \left(a + 1\right)}\\ \mathbf{elif}\;t \leq -4.8 \cdot 10^{-107}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.75 \cdot 10^{-65}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	23.9
Cost	1232

\[\begin{array}{l} t_1 := \frac{x}{1 + \left(a + b \cdot \frac{y}{t}\right)}\\ \mathbf{if}\;t \leq -1.42 \cdot 10^{+51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.4 \cdot 10^{-34}:\\ \;\;\;\;\frac{y}{\frac{t}{z} \cdot \left(a + 1\right)}\\ \mathbf{elif}\;t \leq -9.5 \cdot 10^{-113}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-65}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{\frac{t}{b}} + \left(a + 1\right)}\\ \end{array} \]

Alternative 11
Error	30.0
Cost	1104

\[\begin{array}{l} t_1 := \frac{z}{t} \cdot \frac{y}{a + 1}\\ t_2 := \frac{x}{a + 1}\\ \mathbf{if}\;t \leq -1.85 \cdot 10^{+77}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.3 \cdot 10^{-34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{-75}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -3 \cdot 10^{-86}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 12
Error	37.6
Cost	984

\[\begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{+121}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq -4.7 \cdot 10^{+33}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -12000000:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq 3.9 \cdot 10^{-195}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{-67}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 660000:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

Alternative 13
Error	19.8
Cost	968

\[\begin{array}{l} t_1 := \frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{if}\;b \leq -2.45 \cdot 10^{+51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	29.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-49}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+122}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 15
Error	36.7
Cost	456

\[\begin{array}{l} \mathbf{if}\;a \leq -0.000106:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

Alternative 16
Error	51.3
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022325 
(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))))