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

Average Error: 16.5 → 8.4

Time: 24.5s

Precision: binary64

Cost: 10184

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-318}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{z}{b} + \left(-\frac{\frac{\left(1 + a\right) \cdot \left(z \cdot t\right)}{{b}^{2}} - \frac{t \cdot x}{b}}{y}\right)\\ \mathbf{elif}\;t_1 \leq 10^{+307}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

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)) (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_1 -1e-318)
     t_1
     (if (<= t_1 0.0)
       (+
        (/ z b)
        (- (/ (- (/ (* (+ 1.0 a) (* z t)) (pow b 2.0)) (/ (* t x) b)) 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)) / ((a + 1.0) + ((y * b) / t));
	double tmp;
	if (t_1 <= -1e-318) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (z / b) + -(((((1.0 + a) * (z * t)) / pow(b, 2.0)) - ((t * x) / b)) / y);
	} else if (t_1 <= 1e+307) {
		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) :: tmp
    t_1 = (x + ((y * z) / t)) / ((a + 1.0d0) + ((y * b) / t))
    if (t_1 <= (-1d-318)) then
        tmp = t_1
    else if (t_1 <= 0.0d0) then
        tmp = (z / b) + -(((((1.0d0 + a) * (z * t)) / (b ** 2.0d0)) - ((t * x) / b)) / y)
    else if (t_1 <= 1d+307) 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)) / ((a + 1.0) + ((y * b) / t));
	double tmp;
	if (t_1 <= -1e-318) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (z / b) + -(((((1.0 + a) * (z * t)) / Math.pow(b, 2.0)) - ((t * x) / b)) / 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)) / ((a + 1.0) + ((y * b) / t))
	tmp = 0
	if t_1 <= -1e-318:
		tmp = t_1
	elif t_1 <= 0.0:
		tmp = (z / b) + -(((((1.0 + a) * (z * t)) / math.pow(b, 2.0)) - ((t * x) / b)) / 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(a + 1.0) + Float64(Float64(y * b) / t)))
	tmp = 0.0
	if (t_1 <= -1e-318)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(z / b) + Float64(-Float64(Float64(Float64(Float64(Float64(1.0 + a) * Float64(z * t)) / (b ^ 2.0)) - Float64(Float64(t * x) / b)) / 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)) / ((a + 1.0) + ((y * b) / t));
	tmp = 0.0;
	if (t_1 <= -1e-318)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = (z / b) + -(((((1.0 + a) * (z * t)) / (b ^ 2.0)) - ((t * x) / b)) / 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[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-318], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(z / b), $MachinePrecision] + (-N[(N[(N[(N[(N[(1.0 + a), $MachinePrecision] * N[(z * t), $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] - N[(N[(t * x), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$1, 1e+307], t$95$1, N[(z / b), $MachinePrecision]]]]]

Target

Original	16.5
Target	13.0
Herbie	8.4

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -9.9999875e-319 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 4.2

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

   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 29.6

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

   2. Applied egg-rr 50.8

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

   3. Taylor expanded in y around -inf 23.7

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

   4. Simplified 23.2

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

      [Start] 23.7	\[ -1 \cdot \frac{-1 \cdot \frac{t \cdot x}{b} - -1 \cdot \frac{t \cdot \left(\left(1 + a\right) \cdot z\right)}{{b}^{2}}}{y} + \frac{z}{b} \]
      rational_best.json-simplify-1 [=>] 23.7	\[ \color{blue}{\frac{z}{b} + -1 \cdot \frac{-1 \cdot \frac{t \cdot x}{b} - -1 \cdot \frac{t \cdot \left(\left(1 + a\right) \cdot z\right)}{{b}^{2}}}{y}} \]
      rational_best.json-simplify-2 [=>] 23.7	\[ \frac{z}{b} + \color{blue}{\frac{-1 \cdot \frac{t \cdot x}{b} - -1 \cdot \frac{t \cdot \left(\left(1 + a\right) \cdot z\right)}{{b}^{2}}}{y} \cdot -1} \]
      rational_best.json-simplify-12 [=>] 23.7	\[ \frac{z}{b} + \color{blue}{\left(-\frac{-1 \cdot \frac{t \cdot x}{b} - -1 \cdot \frac{t \cdot \left(\left(1 + a\right) \cdot z\right)}{{b}^{2}}}{y}\right)} \]

   if 9.99999999999999986e306 < (/.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. Taylor expanded in y around inf 11.9

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

4. Final simplification 8.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -1 \cdot 10^{-318}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 0:\\ \;\;\;\;\frac{z}{b} + \left(-\frac{\frac{\left(1 + a\right) \cdot \left(z \cdot t\right)}{{b}^{2}} - \frac{t \cdot x}{b}}{y}\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 10^{+307}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	8.9
Cost	4556

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

Alternative 2
Error	28.3
Cost	2024

\[\begin{array}{l} t_1 := \frac{x}{a - -1}\\ t_2 := \frac{y \cdot z}{t} + x\\ t_3 := \frac{t_2}{a}\\ t_4 := \frac{t \cdot x}{y \cdot b} + \frac{z}{b}\\ \mathbf{if}\;y \leq -13000000000:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y \leq -1.7 \cdot 10^{-28}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -3.5 \cdot 10^{-44}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y \leq -5.9 \cdot 10^{-61}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{-115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.12 \cdot 10^{-132}:\\ \;\;\;\;\frac{x}{\frac{y \cdot b}{t} + 1}\\ \mathbf{elif}\;y \leq -2.15 \cdot 10^{-135}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+33}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{+99}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 3
Error	23.9
Cost	1884

\[\begin{array}{l} t_1 := \frac{y \cdot z}{t} + x\\ t_2 := \frac{t \cdot x}{y \cdot b} + \frac{z}{b}\\ t_3 := \frac{y \cdot b}{t}\\ t_4 := \frac{x}{\left(a + 1\right) + t_3}\\ t_5 := \frac{t_1}{1 + t_3}\\ \mathbf{if}\;b \leq -3.2 \cdot 10^{+188}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -3.9 \cdot 10^{+106}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;b \leq -4.8 \cdot 10^{+66}:\\ \;\;\;\;\frac{y \cdot z}{t \cdot \left(1 + \left(t_3 + a\right)\right)}\\ \mathbf{elif}\;b \leq 5.9 \cdot 10^{-33}:\\ \;\;\;\;\frac{t_1}{1 + a}\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{+67}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{+176}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{+196}:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	22.8
Cost	1496

\[\begin{array}{l} t_1 := \frac{x}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_2 := \frac{t \cdot x}{y \cdot b} + \frac{z}{b}\\ \mathbf{if}\;b \leq -3.3 \cdot 10^{+188}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -8.5 \cdot 10^{+106}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -3.5 \cdot 10^{+66}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 4.9 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{y \cdot z}{t} + x}{1 + a}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+72}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{+199}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	23.4
Cost	1496

\[\begin{array}{l} t_1 := \frac{t \cdot x}{y \cdot b} + \frac{z}{b}\\ t_2 := \frac{y \cdot b}{t}\\ t_3 := \frac{x}{\left(a + 1\right) + t_2}\\ \mathbf{if}\;b \leq -3.5 \cdot 10^{+188}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -9.5 \cdot 10^{+106}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq -5.2 \cdot 10^{+66}:\\ \;\;\;\;\frac{y \cdot z}{t \cdot \left(1 + \left(t_2 + a\right)\right)}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{\frac{y \cdot z}{t} + x}{1 + a}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{+196}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	28.8
Cost	980

\[\begin{array}{l} t_1 := \frac{x}{a - -1}\\ \mathbf{if}\;y \leq -1.65 \cdot 10^{+63}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{-16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{-39}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{-57}:\\ \;\;\;\;\frac{y \cdot z}{t} + x\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{+99}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 7
Error	29.5
Cost	972

\[\begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{+65}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{\frac{y \cdot z}{t} + x}{a}\\ \mathbf{elif}\;y \leq -9.2 \cdot 10^{-133}:\\ \;\;\;\;\frac{x}{\frac{y \cdot b}{t} + 1}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+99}:\\ \;\;\;\;\frac{x}{a - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 8
Error	23.7
Cost	968

\[\begin{array}{l} t_1 := \frac{t \cdot x}{y \cdot b} + \frac{z}{b}\\ \mathbf{if}\;y \leq -6.6 \cdot 10^{+75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+99}:\\ \;\;\;\;\frac{x}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	39.1
Cost	720

\[\begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+61}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -6 \cdot 10^{-251}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-151}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+93}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 10
Error	39.2
Cost	720

\[\begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+62}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -4.9 \cdot 10^{-257}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-151}:\\ \;\;\;\;x - x \cdot a\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+93}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 11
Error	28.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+64}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 1.26 \cdot 10^{+99}:\\ \;\;\;\;\frac{x}{a - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 12
Error	36.5
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 13
Error	50.6
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))))