Average Error: 0.0 → 0.0
Time: 10.7s
Precision: binary64
Cost: 2248
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
↓
\[\begin{array}{l} t_1 := \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}\\ \mathbf{if}\;t \leq -2.4 \cdot 10^{+15}:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 100000000:\\ \;\;\;\;\frac{1 + t_1}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{5 - \frac{8}{t}}{6 + \frac{-8}{t}}\\ \end{array} \]
(FPCore (t)
 :precision binary64
 (/
  (+ 1.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t))))
  (+ 2.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t))))))
↓
(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (/ (* t (* t 4.0)) (+ 1.0 t)) (+ 1.0 t))))
   (if (<= t -2.4e+15)
     0.8333333333333334
     (if (<= t 100000000.0)
       (/ (+ 1.0 t_1) (+ 2.0 t_1))
       (/ (- 5.0 (/ 8.0 t)) (+ 6.0 (/ -8.0 t)))))))
double code(double t) {
	return (1.0 + (((2.0 * t) / (1.0 + t)) * ((2.0 * t) / (1.0 + t)))) / (2.0 + (((2.0 * t) / (1.0 + t)) * ((2.0 * t) / (1.0 + t))));
}
↓
double code(double t) {
	double t_1 = ((t * (t * 4.0)) / (1.0 + t)) / (1.0 + t);
	double tmp;
	if (t <= -2.4e+15) {
		tmp = 0.8333333333333334;
	} else if (t <= 100000000.0) {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	} else {
		tmp = (5.0 - (8.0 / t)) / (6.0 + (-8.0 / t));
	}
	return tmp;
}

real(8) function code(t)
    real(8), intent (in) :: t
    code = (1.0d0 + (((2.0d0 * t) / (1.0d0 + t)) * ((2.0d0 * t) / (1.0d0 + t)))) / (2.0d0 + (((2.0d0 * t) / (1.0d0 + t)) * ((2.0d0 * t) / (1.0d0 + t))))
end function
↓
real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((t * (t * 4.0d0)) / (1.0d0 + t)) / (1.0d0 + t)
    if (t <= (-2.4d+15)) then
        tmp = 0.8333333333333334d0
    else if (t <= 100000000.0d0) then
        tmp = (1.0d0 + t_1) / (2.0d0 + t_1)
    else
        tmp = (5.0d0 - (8.0d0 / t)) / (6.0d0 + ((-8.0d0) / t))
    end if
    code = tmp
end function

public static double code(double t) {
	return (1.0 + (((2.0 * t) / (1.0 + t)) * ((2.0 * t) / (1.0 + t)))) / (2.0 + (((2.0 * t) / (1.0 + t)) * ((2.0 * t) / (1.0 + t))));
}
↓
public static double code(double t) {
	double t_1 = ((t * (t * 4.0)) / (1.0 + t)) / (1.0 + t);
	double tmp;
	if (t <= -2.4e+15) {
		tmp = 0.8333333333333334;
	} else if (t <= 100000000.0) {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	} else {
		tmp = (5.0 - (8.0 / t)) / (6.0 + (-8.0 / t));
	}
	return tmp;
}

def code(t):
	return (1.0 + (((2.0 * t) / (1.0 + t)) * ((2.0 * t) / (1.0 + t)))) / (2.0 + (((2.0 * t) / (1.0 + t)) * ((2.0 * t) / (1.0 + t))))
↓
def code(t):
	t_1 = ((t * (t * 4.0)) / (1.0 + t)) / (1.0 + t)
	tmp = 0
	if t <= -2.4e+15:
		tmp = 0.8333333333333334
	elif t <= 100000000.0:
		tmp = (1.0 + t_1) / (2.0 + t_1)
	else:
		tmp = (5.0 - (8.0 / t)) / (6.0 + (-8.0 / t))
	return tmp

function code(t)
	return Float64(Float64(1.0 + Float64(Float64(Float64(2.0 * t) / Float64(1.0 + t)) * Float64(Float64(2.0 * t) / Float64(1.0 + t)))) / Float64(2.0 + Float64(Float64(Float64(2.0 * t) / Float64(1.0 + t)) * Float64(Float64(2.0 * t) / Float64(1.0 + t)))))
end
↓
function code(t)
	t_1 = Float64(Float64(Float64(t * Float64(t * 4.0)) / Float64(1.0 + t)) / Float64(1.0 + t))
	tmp = 0.0
	if (t <= -2.4e+15)
		tmp = 0.8333333333333334;
	elseif (t <= 100000000.0)
		tmp = Float64(Float64(1.0 + t_1) / Float64(2.0 + t_1));
	else
		tmp = Float64(Float64(5.0 - Float64(8.0 / t)) / Float64(6.0 + Float64(-8.0 / t)));
	end
	return tmp
end

function tmp = code(t)
	tmp = (1.0 + (((2.0 * t) / (1.0 + t)) * ((2.0 * t) / (1.0 + t)))) / (2.0 + (((2.0 * t) / (1.0 + t)) * ((2.0 * t) / (1.0 + t))));
end
↓
function tmp_2 = code(t)
	t_1 = ((t * (t * 4.0)) / (1.0 + t)) / (1.0 + t);
	tmp = 0.0;
	if (t <= -2.4e+15)
		tmp = 0.8333333333333334;
	elseif (t <= 100000000.0)
		tmp = (1.0 + t_1) / (2.0 + t_1);
	else
		tmp = (5.0 - (8.0 / t)) / (6.0 + (-8.0 / t));
	end
	tmp_2 = tmp;
end

code[t_] := N[(N[(1.0 + N[(N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[t_] := Block[{t$95$1 = N[(N[(N[(t * N[(t * 4.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.4e+15], 0.8333333333333334, If[LessEqual[t, 100000000.0], N[(N[(1.0 + t$95$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(5.0 - N[(8.0 / t), $MachinePrecision]), $MachinePrecision] / N[(6.0 + N[(-8.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}
↓
\begin{array}{l}
t_1 := \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}\\
\mathbf{if}\;t \leq -2.4 \cdot 10^{+15}:\\
\;\;\;\;0.8333333333333334\\

\mathbf{elif}\;t \leq 100000000:\\
\;\;\;\;\frac{1 + t_1}{2 + t_1}\\

\mathbf{else}:\\
\;\;\;\;\frac{5 - \frac{8}{t}}{6 + \frac{-8}{t}}\\


\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if t < -2.4e15

    1. Initial program 0.1

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

    2. Taylor expanded in t around inf 0

      \[\leadsto \color{blue}{0.8333333333333334} \]

    if -2.4e15 < t < 1e8

    1. Initial program 0.0

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

    2. Simplified0.0

      \[\leadsto \color{blue}{\frac{1 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}{2 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}} \]
      Proof
      (/.f64 (+.f64 1 (/.f64 (/.f64 (*.f64 t (*.f64 t 4)) (+.f64 1 t)) (+.f64 1 t))) (+.f64 2 (/.f64 (/.f64 (*.f64 t (*.f64 t 4)) (+.f64 1 t)) (+.f64 1 t)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 1 (/.f64 (/.f64 (*.f64 t (*.f64 t (Rewrite<= metadata-eval (*.f64 2 2)))) (+.f64 1 t)) (+.f64 1 t))) (+.f64 2 (/.f64 (/.f64 (*.f64 t (*.f64 t 4)) (+.f64 1 t)) (+.f64 1 t)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 1 (/.f64 (/.f64 (*.f64 t (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 t 2) 2))) (+.f64 1 t)) (+.f64 1 t))) (+.f64 2 (/.f64 (/.f64 (*.f64 t (*.f64 t 4)) (+.f64 1 t)) (+.f64 1 t)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 1 (/.f64 (/.f64 (*.f64 t (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 t)) 2)) (+.f64 1 t)) (+.f64 1 t))) (+.f64 2 (/.f64 (/.f64 (*.f64 t (*.f64 t 4)) (+.f64 1 t)) (+.f64 1 t)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 1 (/.f64 (/.f64 (*.f64 t (Rewrite<= *-commutative_binary64 (*.f64 2 (*.f64 2 t)))) (+.f64 1 t)) (+.f64 1 t))) (+.f64 2 (/.f64 (/.f64 (*.f64 t (*.f64 t 4)) (+.f64 1 t)) (+.f64 1 t)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 1 (/.f64 (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 t 2) (*.f64 2 t))) (+.f64 1 t)) (+.f64 1 t))) (+.f64 2 (/.f64 (/.f64 (*.f64 t (*.f64 t 4)) (+.f64 1 t)) (+.f64 1 t)))): 15 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 1 (/.f64 (/.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 t)) (*.f64 2 t)) (+.f64 1 t)) (+.f64 1 t))) (+.f64 2 (/.f64 (/.f64 (*.f64 t (*.f64 t 4)) (+.f64 1 t)) (+.f64 1 t)))): 0 points increase in error, 15 points decrease in error
      (/.f64 (+.f64 1 (/.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (*.f64 2 t) (+.f64 1 t)) (*.f64 2 t))) (+.f64 1 t))) (+.f64 2 (/.f64 (/.f64 (*.f64 t (*.f64 t 4)) (+.f64 1 t)) (+.f64 1 t)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 1 (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 (*.f64 2 t) (+.f64 1 t)) (/.f64 (*.f64 2 t) (+.f64 1 t))))) (+.f64 2 (/.f64 (/.f64 (*.f64 t (*.f64 t 4)) (+.f64 1 t)) (+.f64 1 t)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 1 (*.f64 (/.f64 (*.f64 2 t) (+.f64 1 t)) (/.f64 (*.f64 2 t) (+.f64 1 t)))) (+.f64 2 (/.f64 (/.f64 (*.f64 t (*.f64 t (Rewrite<= metadata-eval (*.f64 2 2)))) (+.f64 1 t)) (+.f64 1 t)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 1 (*.f64 (/.f64 (*.f64 2 t) (+.f64 1 t)) (/.f64 (*.f64 2 t) (+.f64 1 t)))) (+.f64 2 (/.f64 (/.f64 (*.f64 t (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 t 2) 2))) (+.f64 1 t)) (+.f64 1 t)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 1 (*.f64 (/.f64 (*.f64 2 t) (+.f64 1 t)) (/.f64 (*.f64 2 t) (+.f64 1 t)))) (+.f64 2 (/.f64 (/.f64 (*.f64 t (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 t)) 2)) (+.f64 1 t)) (+.f64 1 t)))): 15 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 1 (*.f64 (/.f64 (*.f64 2 t) (+.f64 1 t)) (/.f64 (*.f64 2 t) (+.f64 1 t)))) (+.f64 2 (/.f64 (/.f64 (*.f64 t (Rewrite<= *-commutative_binary64 (*.f64 2 (*.f64 2 t)))) (+.f64 1 t)) (+.f64 1 t)))): 0 points increase in error, 15 points decrease in error
      (/.f64 (+.f64 1 (*.f64 (/.f64 (*.f64 2 t) (+.f64 1 t)) (/.f64 (*.f64 2 t) (+.f64 1 t)))) (+.f64 2 (/.f64 (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 t 2) (*.f64 2 t))) (+.f64 1 t)) (+.f64 1 t)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 1 (*.f64 (/.f64 (*.f64 2 t) (+.f64 1 t)) (/.f64 (*.f64 2 t) (+.f64 1 t)))) (+.f64 2 (/.f64 (/.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 t)) (*.f64 2 t)) (+.f64 1 t)) (+.f64 1 t)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 1 (*.f64 (/.f64 (*.f64 2 t) (+.f64 1 t)) (/.f64 (*.f64 2 t) (+.f64 1 t)))) (+.f64 2 (/.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (*.f64 2 t) (+.f64 1 t)) (*.f64 2 t))) (+.f64 1 t)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 1 (*.f64 (/.f64 (*.f64 2 t) (+.f64 1 t)) (/.f64 (*.f64 2 t) (+.f64 1 t)))) (+.f64 2 (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 (*.f64 2 t) (+.f64 1 t)) (/.f64 (*.f64 2 t) (+.f64 1 t)))))): 0 points increase in error, 0 points decrease in error

    if 1e8 < t

    1. Initial program 0.0

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

    2. Simplified32.6

      \[\leadsto \color{blue}{\frac{1 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}{2 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}} \]
      Proof
      (/.f64 (+.f64 1 (/.f64 (/.f64 (*.f64 t (*.f64 t 4)) (+.f64 1 t)) (+.f64 1 t))) (+.f64 2 (/.f64 (/.f64 (*.f64 t (*.f64 t 4)) (+.f64 1 t)) (+.f64 1 t)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 1 (/.f64 (/.f64 (*.f64 t (*.f64 t (Rewrite<= metadata-eval (*.f64 2 2)))) (+.f64 1 t)) (+.f64 1 t))) (+.f64 2 (/.f64 (/.f64 (*.f64 t (*.f64 t 4)) (+.f64 1 t)) (+.f64 1 t)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 1 (/.f64 (/.f64 (*.f64 t (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 t 2) 2))) (+.f64 1 t)) (+.f64 1 t))) (+.f64 2 (/.f64 (/.f64 (*.f64 t (*.f64 t 4)) (+.f64 1 t)) (+.f64 1 t)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 1 (/.f64 (/.f64 (*.f64 t (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 t)) 2)) (+.f64 1 t)) (+.f64 1 t))) (+.f64 2 (/.f64 (/.f64 (*.f64 t (*.f64 t 4)) (+.f64 1 t)) (+.f64 1 t)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 1 (/.f64 (/.f64 (*.f64 t (Rewrite<= *-commutative_binary64 (*.f64 2 (*.f64 2 t)))) (+.f64 1 t)) (+.f64 1 t))) (+.f64 2 (/.f64 (/.f64 (*.f64 t (*.f64 t 4)) (+.f64 1 t)) (+.f64 1 t)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 1 (/.f64 (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 t 2) (*.f64 2 t))) (+.f64 1 t)) (+.f64 1 t))) (+.f64 2 (/.f64 (/.f64 (*.f64 t (*.f64 t 4)) (+.f64 1 t)) (+.f64 1 t)))): 15 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 1 (/.f64 (/.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 t)) (*.f64 2 t)) (+.f64 1 t)) (+.f64 1 t))) (+.f64 2 (/.f64 (/.f64 (*.f64 t (*.f64 t 4)) (+.f64 1 t)) (+.f64 1 t)))): 0 points increase in error, 15 points decrease in error
      (/.f64 (+.f64 1 (/.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (*.f64 2 t) (+.f64 1 t)) (*.f64 2 t))) (+.f64 1 t))) (+.f64 2 (/.f64 (/.f64 (*.f64 t (*.f64 t 4)) (+.f64 1 t)) (+.f64 1 t)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 1 (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 (*.f64 2 t) (+.f64 1 t)) (/.f64 (*.f64 2 t) (+.f64 1 t))))) (+.f64 2 (/.f64 (/.f64 (*.f64 t (*.f64 t 4)) (+.f64 1 t)) (+.f64 1 t)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 1 (*.f64 (/.f64 (*.f64 2 t) (+.f64 1 t)) (/.f64 (*.f64 2 t) (+.f64 1 t)))) (+.f64 2 (/.f64 (/.f64 (*.f64 t (*.f64 t (Rewrite<= metadata-eval (*.f64 2 2)))) (+.f64 1 t)) (+.f64 1 t)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 1 (*.f64 (/.f64 (*.f64 2 t) (+.f64 1 t)) (/.f64 (*.f64 2 t) (+.f64 1 t)))) (+.f64 2 (/.f64 (/.f64 (*.f64 t (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 t 2) 2))) (+.f64 1 t)) (+.f64 1 t)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 1 (*.f64 (/.f64 (*.f64 2 t) (+.f64 1 t)) (/.f64 (*.f64 2 t) (+.f64 1 t)))) (+.f64 2 (/.f64 (/.f64 (*.f64 t (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 t)) 2)) (+.f64 1 t)) (+.f64 1 t)))): 15 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 1 (*.f64 (/.f64 (*.f64 2 t) (+.f64 1 t)) (/.f64 (*.f64 2 t) (+.f64 1 t)))) (+.f64 2 (/.f64 (/.f64 (*.f64 t (Rewrite<= *-commutative_binary64 (*.f64 2 (*.f64 2 t)))) (+.f64 1 t)) (+.f64 1 t)))): 0 points increase in error, 15 points decrease in error
      (/.f64 (+.f64 1 (*.f64 (/.f64 (*.f64 2 t) (+.f64 1 t)) (/.f64 (*.f64 2 t) (+.f64 1 t)))) (+.f64 2 (/.f64 (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 t 2) (*.f64 2 t))) (+.f64 1 t)) (+.f64 1 t)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 1 (*.f64 (/.f64 (*.f64 2 t) (+.f64 1 t)) (/.f64 (*.f64 2 t) (+.f64 1 t)))) (+.f64 2 (/.f64 (/.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 t)) (*.f64 2 t)) (+.f64 1 t)) (+.f64 1 t)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 1 (*.f64 (/.f64 (*.f64 2 t) (+.f64 1 t)) (/.f64 (*.f64 2 t) (+.f64 1 t)))) (+.f64 2 (/.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (*.f64 2 t) (+.f64 1 t)) (*.f64 2 t))) (+.f64 1 t)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 1 (*.f64 (/.f64 (*.f64 2 t) (+.f64 1 t)) (/.f64 (*.f64 2 t) (+.f64 1 t)))) (+.f64 2 (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 (*.f64 2 t) (+.f64 1 t)) (/.f64 (*.f64 2 t) (+.f64 1 t)))))): 0 points increase in error, 0 points decrease in error

    3. Taylor expanded in t around inf 32.6

      \[\leadsto \frac{1 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}{\color{blue}{6 - 8 \cdot \frac{1}{t}}} \]

    4. Simplified32.6

      \[\leadsto \frac{1 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}{\color{blue}{6 - \frac{8}{t}}} \]
      Proof
      (/.f64 (+.f64 1 (/.f64 (/.f64 (*.f64 t (*.f64 t 4)) (+.f64 1 t)) (+.f64 1 t))) (-.f64 6 (/.f64 8 t))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 1 (/.f64 (/.f64 (*.f64 t (*.f64 t 4)) (+.f64 1 t)) (+.f64 1 t))) (-.f64 6 (/.f64 (Rewrite<= metadata-eval (*.f64 8 1)) t))): 0 points increase in error, 3 points decrease in error
      (/.f64 (+.f64 1 (/.f64 (/.f64 (*.f64 t (*.f64 t 4)) (+.f64 1 t)) (+.f64 1 t))) (-.f64 6 (Rewrite<= associate-*r/_binary64 (*.f64 8 (/.f64 1 t))))): 3 points increase in error, 0 points decrease in error

    5. Taylor expanded in t around inf 0.0

      \[\leadsto \frac{\color{blue}{5 - 8 \cdot \frac{1}{t}}}{6 - \frac{8}{t}} \]

    6. Simplified0.0

      \[\leadsto \frac{\color{blue}{5 - \frac{8}{t}}}{6 - \frac{8}{t}} \]
      Proof
      (/.f64 (-.f64 5 (/.f64 8 t)) (-.f64 6 (/.f64 8 t))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 5 (/.f64 (Rewrite<= metadata-eval (*.f64 8 1)) t)) (-.f64 6 (/.f64 8 t))): 0 points increase in error, 3 points decrease in error
      (/.f64 (-.f64 5 (Rewrite<= associate-*r/_binary64 (*.f64 8 (/.f64 1 t)))) (-.f64 6 (/.f64 8 t))): 3 points increase in error, 0 points decrease in error
  3. Recombined 3 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.4 \cdot 10^{+15}:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 100000000:\\ \;\;\;\;\frac{1 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}{2 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{5 - \frac{8}{t}}{6 + \frac{-8}{t}}\\ \end{array} \]

Alternatives

Alternative 1
Error0.0
Cost2240
\[\begin{array}{l} t_1 := \frac{2 \cdot t}{1 + t}\\ t_2 := t_1 \cdot t_1\\ \frac{1 + t_2}{2 + t_2} \end{array} \]
Alternative 2
Error0.6
Cost968
\[\begin{array}{l} t_1 := 0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.8:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 0.34:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + t_1\\ \end{array} \]
Alternative 3
Error0.6
Cost968
\[\begin{array}{l} \mathbf{if}\;t \leq -0.55:\\ \;\;\;\;\frac{5 - \frac{8}{t}}{6 + \frac{-8}{t}}\\ \mathbf{elif}\;t \leq 0.34:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)\\ \end{array} \]
Alternative 4
Error0.6
Cost585
\[\begin{array}{l} \mathbf{if}\;t \leq -0.8 \lor \neg \left(t \leq 0.58\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]
Alternative 5
Error1.0
Cost584
\[\begin{array}{l} \mathbf{if}\;t \leq -0.42:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]
Alternative 6
Error1.1
Cost328
\[\begin{array}{l} \mathbf{if}\;t \leq -0.33:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]
Alternative 7
Error26.1
Cost64
\[0.5 \]

Error

Reproduce

herbie shell --seed 2022343 
(FPCore (t)
  :name "Kahan p13 Example 1"
  :precision binary64
  (/ (+ 1.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t)))) (+ 2.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t))))))