Kahan p13 Example 1

Average Accuracy: 99.9% → 100.0%

Time: 16.5s

Precision: binary64

Cost: 2376

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -1600000:\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1 + \left(2 + \frac{t \cdot \frac{-4 \cdot t}{t + 1}}{-1 - t}\right)}{2 + \frac{\frac{\left(t \cdot t\right) \cdot 4}{t + 1}}{t + 1}}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

FPCore

(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
 (if (<= t -1600000.0)
   (+
    (/ 0.037037037037037035 (* t t))
    (+ 0.8333333333333334 (/ -0.2222222222222222 t)))
   (if (<= t 5e+15)
     (/
      (+ -1.0 (+ 2.0 (/ (* t (/ (* -4.0 t) (+ t 1.0))) (- -1.0 t))))
      (+ 2.0 (/ (/ (* (* t t) 4.0) (+ t 1.0)) (+ t 1.0))))
     0.8333333333333334)))

C

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 tmp;
	if (t <= -1600000.0) {
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 + (-0.2222222222222222 / t));
	} else if (t <= 5e+15) {
		tmp = (-1.0 + (2.0 + ((t * ((-4.0 * t) / (t + 1.0))) / (-1.0 - t)))) / (2.0 + ((((t * t) * 4.0) / (t + 1.0)) / (t + 1.0)));
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

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) :: tmp
    if (t <= (-1600000.0d0)) then
        tmp = (0.037037037037037035d0 / (t * t)) + (0.8333333333333334d0 + ((-0.2222222222222222d0) / t))
    else if (t <= 5d+15) then
        tmp = ((-1.0d0) + (2.0d0 + ((t * (((-4.0d0) * t) / (t + 1.0d0))) / ((-1.0d0) - t)))) / (2.0d0 + ((((t * t) * 4.0d0) / (t + 1.0d0)) / (t + 1.0d0)))
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

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 tmp;
	if (t <= -1600000.0) {
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 + (-0.2222222222222222 / t));
	} else if (t <= 5e+15) {
		tmp = (-1.0 + (2.0 + ((t * ((-4.0 * t) / (t + 1.0))) / (-1.0 - t)))) / (2.0 + ((((t * t) * 4.0) / (t + 1.0)) / (t + 1.0)));
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

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):
	tmp = 0
	if t <= -1600000.0:
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 + (-0.2222222222222222 / t))
	elif t <= 5e+15:
		tmp = (-1.0 + (2.0 + ((t * ((-4.0 * t) / (t + 1.0))) / (-1.0 - t)))) / (2.0 + ((((t * t) * 4.0) / (t + 1.0)) / (t + 1.0)))
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

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)
	tmp = 0.0
	if (t <= -1600000.0)
		tmp = Float64(Float64(0.037037037037037035 / Float64(t * t)) + Float64(0.8333333333333334 + Float64(-0.2222222222222222 / t)));
	elseif (t <= 5e+15)
		tmp = Float64(Float64(-1.0 + Float64(2.0 + Float64(Float64(t * Float64(Float64(-4.0 * t) / Float64(t + 1.0))) / Float64(-1.0 - t)))) / Float64(2.0 + Float64(Float64(Float64(Float64(t * t) * 4.0) / Float64(t + 1.0)) / Float64(t + 1.0))));
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

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)
	tmp = 0.0;
	if (t <= -1600000.0)
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 + (-0.2222222222222222 / t));
	elseif (t <= 5e+15)
		tmp = (-1.0 + (2.0 + ((t * ((-4.0 * t) / (t + 1.0))) / (-1.0 - t)))) / (2.0 + ((((t * t) * 4.0) / (t + 1.0)) / (t + 1.0)));
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

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_] := If[LessEqual[t, -1600000.0], N[(N[(0.037037037037037035 / N[(t * t), $MachinePrecision]), $MachinePrecision] + N[(0.8333333333333334 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5e+15], N[(N[(-1.0 + N[(2.0 + N[(N[(t * N[(N[(-4.0 * t), $MachinePrecision] / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(N[(N[(t * t), $MachinePrecision] * 4.0), $MachinePrecision] / N[(t + 1.0), $MachinePrecision]), $MachinePrecision] / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.8333333333333334]]

Derivation

1. Split input into 3 regimes

2. if t < -1.6e6

   1. Initial program 99.9%

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

   2. Taylor expanded in t around inf 100.0%

      \[\leadsto \color{blue}{\left(0.037037037037037035 \cdot \frac{1}{{t}^{2}} + 0.8333333333333334\right) - 0.2222222222222222 \cdot \frac{1}{t}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)} \]
      Proof

      [Start] 100.0	\[ \left(0.037037037037037035 \cdot \frac{1}{{t}^{2}} + 0.8333333333333334\right) - 0.2222222222222222 \cdot \frac{1}{t} \]
      associate--l+ [=>] 100.0	\[ \color{blue}{0.037037037037037035 \cdot \frac{1}{{t}^{2}} + \left(0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t}\right)} \]
      associate-*r/ [=>] 100.0	\[ \color{blue}{\frac{0.037037037037037035 \cdot 1}{{t}^{2}}} + \left(0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t}\right) \]
      metadata-eval [=>] 100.0	\[ \frac{\color{blue}{0.037037037037037035}}{{t}^{2}} + \left(0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t}\right) \]
      unpow2 [=>] 100.0	\[ \frac{0.037037037037037035}{\color{blue}{t \cdot t}} + \left(0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t}\right) \]
      associate-*r/ [=>] 100.0	\[ \frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 - \color{blue}{\frac{0.2222222222222222 \cdot 1}{t}}\right) \]
      metadata-eval [=>] 100.0	\[ \frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 - \frac{\color{blue}{0.2222222222222222}}{t}\right) \]

   if -1.6e6 < t < 5e15

   1. Initial program 100.0%

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

   2. Applied egg-rr 100.0%

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

      [Start] 100.0	\[ \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      expm1-log1p-u [=>] 100.0	\[ \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right)\right)}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      expm1-udef [=>] 100.0	\[ \frac{\color{blue}{e^{\mathsf{log1p}\left(1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right)} - 1}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \frac{\left(2 + \color{blue}{\frac{\left(2 \cdot \frac{t}{t + 1}\right) \cdot \left(2 \cdot t\right)}{t + 1}}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      Proof

      [Start] 100.0	\[ \frac{\left(2 + {\left(\frac{2}{t + 1} \cdot t\right)}^{2}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      unpow2 [=>] 100.0	\[ \frac{\left(2 + \color{blue}{\left(\frac{2}{t + 1} \cdot t\right) \cdot \left(\frac{2}{t + 1} \cdot t\right)}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-*l/ [=>] 100.0	\[ \frac{\left(2 + \left(\frac{2}{t + 1} \cdot t\right) \cdot \color{blue}{\frac{2 \cdot t}{t + 1}}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-*r/ [=>] 100.0	\[ \frac{\left(2 + \color{blue}{\frac{\left(\frac{2}{t + 1} \cdot t\right) \cdot \left(2 \cdot t\right)}{t + 1}}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-*l/ [=>] 100.0	\[ \frac{\left(2 + \frac{\color{blue}{\frac{2 \cdot t}{t + 1}} \cdot \left(2 \cdot t\right)}{t + 1}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      *-un-lft-identity [=>] 100.0	\[ \frac{\left(2 + \frac{\frac{2 \cdot t}{\color{blue}{1 \cdot \left(t + 1\right)}} \cdot \left(2 \cdot t\right)}{t + 1}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      times-frac [=>] 100.0	\[ \frac{\left(2 + \frac{\color{blue}{\left(\frac{2}{1} \cdot \frac{t}{t + 1}\right)} \cdot \left(2 \cdot t\right)}{t + 1}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      metadata-eval [=>] 100.0	\[ \frac{\left(2 + \frac{\left(\color{blue}{2} \cdot \frac{t}{t + 1}\right) \cdot \left(2 \cdot t\right)}{t + 1}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \frac{\left(2 + \color{blue}{\left(\frac{4}{\frac{t + 1}{t}} \cdot \left(-t\right)\right) \cdot \frac{1}{-1 - t}}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      Proof

      [Start] 100.0	\[ \frac{\left(2 + \frac{\left(2 \cdot \frac{t}{t + 1}\right) \cdot \left(2 \cdot t\right)}{t + 1}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      frac-2neg [=>] 100.0	\[ \frac{\left(2 + \color{blue}{\frac{-\left(2 \cdot \frac{t}{t + 1}\right) \cdot \left(2 \cdot t\right)}{-\left(t + 1\right)}}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      div-inv [=>] 100.0	\[ \frac{\left(2 + \color{blue}{\left(-\left(2 \cdot \frac{t}{t + 1}\right) \cdot \left(2 \cdot t\right)\right) \cdot \frac{1}{-\left(t + 1\right)}}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-*r* [=>] 100.0	\[ \frac{\left(2 + \left(-\color{blue}{\left(\left(2 \cdot \frac{t}{t + 1}\right) \cdot 2\right) \cdot t}\right) \cdot \frac{1}{-\left(t + 1\right)}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      distribute-rgt-neg-in [=>] 100.0	\[ \frac{\left(2 + \color{blue}{\left(\left(\left(2 \cdot \frac{t}{t + 1}\right) \cdot 2\right) \cdot \left(-t\right)\right)} \cdot \frac{1}{-\left(t + 1\right)}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-*r/ [=>] 100.0	\[ \frac{\left(2 + \left(\left(\color{blue}{\frac{2 \cdot t}{t + 1}} \cdot 2\right) \cdot \left(-t\right)\right) \cdot \frac{1}{-\left(t + 1\right)}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-/l* [=>] 100.0	\[ \frac{\left(2 + \left(\left(\color{blue}{\frac{2}{\frac{t + 1}{t}}} \cdot 2\right) \cdot \left(-t\right)\right) \cdot \frac{1}{-\left(t + 1\right)}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-*l/ [=>] 100.0	\[ \frac{\left(2 + \left(\color{blue}{\frac{2 \cdot 2}{\frac{t + 1}{t}}} \cdot \left(-t\right)\right) \cdot \frac{1}{-\left(t + 1\right)}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      metadata-eval [=>] 100.0	\[ \frac{\left(2 + \left(\frac{\color{blue}{4}}{\frac{t + 1}{t}} \cdot \left(-t\right)\right) \cdot \frac{1}{-\left(t + 1\right)}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      neg-sub0 [=>] 100.0	\[ \frac{\left(2 + \left(\frac{4}{\frac{t + 1}{t}} \cdot \left(-t\right)\right) \cdot \frac{1}{\color{blue}{0 - \left(t + 1\right)}}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      metadata-eval [<=] 100.0	\[ \frac{\left(2 + \left(\frac{4}{\frac{t + 1}{t}} \cdot \left(-t\right)\right) \cdot \frac{1}{\color{blue}{\log 1} - \left(t + 1\right)}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      +-commutative [=>] 100.0	\[ \frac{\left(2 + \left(\frac{4}{\frac{t + 1}{t}} \cdot \left(-t\right)\right) \cdot \frac{1}{\log 1 - \color{blue}{\left(1 + t\right)}}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate--r+ [=>] 100.0	\[ \frac{\left(2 + \left(\frac{4}{\frac{t + 1}{t}} \cdot \left(-t\right)\right) \cdot \frac{1}{\color{blue}{\left(\log 1 - 1\right) - t}}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      metadata-eval [=>] 100.0	\[ \frac{\left(2 + \left(\frac{4}{\frac{t + 1}{t}} \cdot \left(-t\right)\right) \cdot \frac{1}{\left(\color{blue}{0} - 1\right) - t}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      metadata-eval [=>] 100.0	\[ \frac{\left(2 + \left(\frac{4}{\frac{t + 1}{t}} \cdot \left(-t\right)\right) \cdot \frac{1}{\color{blue}{-1} - t}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   5. Simplified 100.0%

      \[\leadsto \frac{\left(2 + \color{blue}{\frac{\frac{-4 \cdot t}{t + 1} \cdot t}{-1 - t}}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      Proof

      [Start] 100.0	\[ \frac{\left(2 + \left(\frac{4}{\frac{t + 1}{t}} \cdot \left(-t\right)\right) \cdot \frac{1}{-1 - t}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-*r/ [=>] 100.0	\[ \frac{\left(2 + \color{blue}{\frac{\left(\frac{4}{\frac{t + 1}{t}} \cdot \left(-t\right)\right) \cdot 1}{-1 - t}}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      *-rgt-identity [=>] 100.0	\[ \frac{\left(2 + \frac{\color{blue}{\frac{4}{\frac{t + 1}{t}} \cdot \left(-t\right)}}{-1 - t}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-*l/ [=>] 100.0	\[ \frac{\left(2 + \frac{\color{blue}{\frac{4 \cdot \left(-t\right)}{\frac{t + 1}{t}}}}{-1 - t}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      *-lft-identity [<=] 100.0	\[ \frac{\left(2 + \frac{\frac{4 \cdot \left(-t\right)}{\color{blue}{1 \cdot \frac{t + 1}{t}}}}{-1 - t}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-*r/ [=>] 100.0	\[ \frac{\left(2 + \frac{\frac{4 \cdot \left(-t\right)}{\color{blue}{\frac{1 \cdot \left(t + 1\right)}{t}}}}{-1 - t}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-/r/ [=>] 100.0	\[ \frac{\left(2 + \frac{\color{blue}{\frac{4 \cdot \left(-t\right)}{1 \cdot \left(t + 1\right)} \cdot t}}{-1 - t}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      neg-mul-1 [=>] 100.0	\[ \frac{\left(2 + \frac{\frac{4 \cdot \color{blue}{\left(-1 \cdot t\right)}}{1 \cdot \left(t + 1\right)} \cdot t}{-1 - t}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-*r* [=>] 100.0	\[ \frac{\left(2 + \frac{\frac{\color{blue}{\left(4 \cdot -1\right) \cdot t}}{1 \cdot \left(t + 1\right)} \cdot t}{-1 - t}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      metadata-eval [=>] 100.0	\[ \frac{\left(2 + \frac{\frac{\color{blue}{-4} \cdot t}{1 \cdot \left(t + 1\right)} \cdot t}{-1 - t}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      distribute-rgt-in [=>] 100.0	\[ \frac{\left(2 + \frac{\frac{-4 \cdot t}{\color{blue}{t \cdot 1 + 1 \cdot 1}} \cdot t}{-1 - t}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      *-rgt-identity [=>] 100.0	\[ \frac{\left(2 + \frac{\frac{-4 \cdot t}{\color{blue}{t} + 1 \cdot 1} \cdot t}{-1 - t}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      metadata-eval [=>] 100.0	\[ \frac{\left(2 + \frac{\frac{-4 \cdot t}{t + \color{blue}{1}} \cdot t}{-1 - t}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{\left(2 + \frac{\frac{-4 \cdot t}{t + 1} \cdot t}{-1 - t}\right) - 1}{2 + \color{blue}{\frac{\frac{4 \cdot \left(t \cdot t\right)}{t + 1}}{t + 1}}} \]
      Proof

      [Start] 100.0	\[ \frac{\left(2 + \frac{\frac{-4 \cdot t}{t + 1} \cdot t}{-1 - t}\right) - 1}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-*l/ [=>] 100.0	\[ \frac{\left(2 + \frac{\frac{-4 \cdot t}{t + 1} \cdot t}{-1 - t}\right) - 1}{2 + \color{blue}{\frac{\left(2 \cdot t\right) \cdot \frac{2 \cdot t}{1 + t}}{1 + t}}} \]
      associate-*r/ [=>] 100.0	\[ \frac{\left(2 + \frac{\frac{-4 \cdot t}{t + 1} \cdot t}{-1 - t}\right) - 1}{2 + \frac{\color{blue}{\frac{\left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}{1 + t}}}{1 + t}} \]
      swap-sqr [=>] 100.0	\[ \frac{\left(2 + \frac{\frac{-4 \cdot t}{t + 1} \cdot t}{-1 - t}\right) - 1}{2 + \frac{\frac{\color{blue}{\left(2 \cdot 2\right) \cdot \left(t \cdot t\right)}}{1 + t}}{1 + t}} \]
      metadata-eval [=>] 100.0	\[ \frac{\left(2 + \frac{\frac{-4 \cdot t}{t + 1} \cdot t}{-1 - t}\right) - 1}{2 + \frac{\frac{\color{blue}{4} \cdot \left(t \cdot t\right)}{1 + t}}{1 + t}} \]
      +-commutative [=>] 100.0	\[ \frac{\left(2 + \frac{\frac{-4 \cdot t}{t + 1} \cdot t}{-1 - t}\right) - 1}{2 + \frac{\frac{4 \cdot \left(t \cdot t\right)}{\color{blue}{t + 1}}}{1 + t}} \]
      +-commutative [=>] 100.0	\[ \frac{\left(2 + \frac{\frac{-4 \cdot t}{t + 1} \cdot t}{-1 - t}\right) - 1}{2 + \frac{\frac{4 \cdot \left(t \cdot t\right)}{t + 1}}{\color{blue}{t + 1}}} \]

   if 5e15 < t

   1. Initial program 99.9%

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

   2. Taylor expanded in t around inf 100.0%

      \[\leadsto \color{blue}{0.8333333333333334} \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1600000:\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1 + \left(2 + \frac{t \cdot \frac{-4 \cdot t}{t + 1}}{-1 - t}\right)}{2 + \frac{\frac{\left(t \cdot t\right) \cdot 4}{t + 1}}{t + 1}}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.9%
Cost	2248

\[\begin{array}{l} t_1 := \frac{\frac{t \cdot \left(t \cdot 4\right)}{t + 1}}{t + 1}\\ \mathbf{if}\;t \leq -2 \cdot 10^{+154}:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + t_1}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 2
Accuracy	99.9%
Cost	2240

\[\begin{array}{l} t_1 := \frac{2 \cdot t}{t + 1}\\ \frac{-1 + \left(2 + \frac{t \cdot \frac{-4 \cdot t}{t + 1}}{-1 - t}\right)}{2 + t_1 \cdot t_1} \end{array} \]

Alternative 3
Accuracy	99.9%
Cost	2240

\[\begin{array}{l} t_1 := \frac{2 \cdot t}{t + 1}\\ t_2 := t_1 \cdot t_1\\ \frac{1 + t_2}{2 + t_2} \end{array} \]

Alternative 4
Accuracy	99.3%
Cost	1480

\[\begin{array}{l} t_1 := \frac{0.037037037037037035}{t \cdot t}\\ \mathbf{if}\;t \leq -0.82:\\ \;\;\;\;t_1 + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 0.37:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{1}{\left(\frac{0.08533333333333333}{t \cdot t} + 1.2\right) + \frac{0.32}{t}}\\ \end{array} \]

Alternative 5
Accuracy	99.4%
Cost	1480

\[\begin{array}{l} t_1 := \left(t \cdot t\right) \cdot 4\\ t_2 := \frac{0.037037037037037035}{t \cdot t}\\ \mathbf{if}\;t \leq -0.58:\\ \;\;\;\;t_2 + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 0.46:\\ \;\;\;\;\frac{1 + \frac{t_1}{t + 1}}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2 + \frac{1}{\left(\frac{0.08533333333333333}{t \cdot t} + 1.2\right) + \frac{0.32}{t}}\\ \end{array} \]

Alternative 6
Accuracy	99.3%
Cost	1224

\[\begin{array}{l} t_1 := \frac{0.037037037037037035}{t \cdot t}\\ t_2 := 0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.82:\\ \;\;\;\;t_1 + t_2\\ \mathbf{elif}\;t \leq 0.235:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{1}{\frac{1}{t_2}}\\ \end{array} \]

Alternative 7
Accuracy	99.3%
Cost	969

\[\begin{array}{l} \mathbf{if}\;t \leq -0.82 \lor \neg \left(t \leq 0.235\right):\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]

Alternative 8
Accuracy	99.2%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -0.85 \lor \neg \left(t \leq 0.39\right):\\ \;\;\;\;\frac{1}{1.2 + \frac{0.32}{t}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]

Alternative 9
Accuracy	99.2%
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -0.8 \lor \neg \left(t \leq 0.56\right):\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]

Alternative 10
Accuracy	98.7%
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -0.92:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 11
Accuracy	98.6%
Cost	328

\[\begin{array}{l} \mathbf{if}\;t \leq -0.33:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 12
Accuracy	59.1%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023135 
(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))))))