Kahan p13 Example 2

Percentage Accurate: 99.9% → 100.0%

Time: 31.3s

Alternatives: 7

Speedup: 1.5×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ t_2 := t\_1 \cdot t\_1\\ \frac{1 + t\_2}{2 + t\_2} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))) (t_2 (* t_1 t_1)))
   (/ (+ 1.0 t_2) (+ 2.0 t_2))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    t_1 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))
    t_2 = t_1 * t_1
    code = (1.0d0 + t_2) / (2.0d0 + t_2)
end function

Java

public static double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Python

def code(t):
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))
	t_2 = t_1 * t_1
	return (1.0 + t_2) / (2.0 + t_2)

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2))
end

MATLAB

function tmp = code(t)
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	t_2 = t_1 * t_1;
	tmp = (1.0 + t_2) / (2.0 + t_2);
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]]]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ t_2 := t\_1 \cdot t\_1\\ \frac{1 + t\_2}{2 + t\_2} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))) (t_2 (* t_1 t_1)))
   (/ (+ 1.0 t_2) (+ 2.0 t_2))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    t_1 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))
    t_2 = t_1 * t_1
    code = (1.0d0 + t_2) / (2.0d0 + t_2)
end function

Java

public static double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Python

def code(t):
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))
	t_2 = t_1 * t_1
	return (1.0 + t_2) / (2.0 + t_2)

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2))
end

MATLAB

function tmp = code(t)
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	t_2 = t_1 * t_1;
	tmp = (1.0 + t_2) / (2.0 + t_2);
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{-2}{-1 - t}\\ t_2 := t\_1 \cdot t\_1\\ \frac{1 + t\_2}{2 + t\_2} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ -2.0 (- -1.0 t)))) (t_2 (* t_1 t_1)))
   (/ (+ 1.0 t_2) (+ 2.0 t_2))))

C

double code(double t) {
	double t_1 = 2.0 - (-2.0 / (-1.0 - t));
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    t_1 = 2.0d0 - ((-2.0d0) / ((-1.0d0) - t))
    t_2 = t_1 * t_1
    code = (1.0d0 + t_2) / (2.0d0 + t_2)
end function

Java

public static double code(double t) {
	double t_1 = 2.0 - (-2.0 / (-1.0 - t));
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Python

def code(t):
	t_1 = 2.0 - (-2.0 / (-1.0 - t))
	t_2 = t_1 * t_1
	return (1.0 + t_2) / (2.0 + t_2)

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(-2.0 / Float64(-1.0 - t)))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2))
end

MATLAB

function tmp = code(t)
	t_1 = 2.0 - (-2.0 / (-1.0 - t));
	t_2 = t_1 * t_1;
	tmp = (1.0 + t_2) / (2.0 + t_2);
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 - N[(-2.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 99.6%

   \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. div-inv 99.6%

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

   2. associate-/l* 99.6%

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

4. Applied egg-rr 99.6%

   \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{2 \cdot \frac{\frac{1}{t}}{1 + \frac{1}{t}}}\right)} \]
5. Step-by-step derivation

   1. associate-/r* 99.6%

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

   2. associate-*r/ 99.6%

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

   3. metadata-eval 99.6%

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

   4. +-commutative 99.6%

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

   5. distribute-lft-in 99.6%

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

   6. rgt-mult-inverse 99.6%

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

   7. metadata-eval 99.6%

      \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{2}{\color{blue}{\left(0 - -1\right)} + t \cdot 1}\right)} \]

   8. *-rgt-identity 99.6%

      \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{2}{\left(0 - -1\right) + \color{blue}{t}}\right)} \]

   9. associate--r- 99.6%

      \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{2}{\color{blue}{0 - \left(-1 - t\right)}}\right)} \]

   10. neg-sub0 99.6%

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

   11. distribute-neg-frac2 99.6%

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

   12. distribute-neg-frac 99.6%

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

   13. metadata-eval 99.6%

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

6. Simplified 99.6%

   \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\frac{-2}{-1 - t}}\right)} \]
7. Step-by-step derivation

   1. div-inv 99.6%

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

   2. associate-/l* 99.6%

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

8. Applied egg-rr 99.6%

   \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \color{blue}{2 \cdot \frac{\frac{1}{t}}{1 + \frac{1}{t}}}\right) \cdot \left(2 - \frac{-2}{-1 - t}\right)} \]
9. Step-by-step derivation

   1. associate-/r* 99.6%

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

   2. associate-*r/ 99.6%

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

   3. metadata-eval 99.6%

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

   4. +-commutative 99.6%

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

   5. distribute-lft-in 99.6%

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

   6. rgt-mult-inverse 99.6%

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

   7. metadata-eval 99.6%

      \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{2}{\color{blue}{\left(0 - -1\right)} + t \cdot 1}\right)} \]

   8. *-rgt-identity 99.6%

      \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{2}{\left(0 - -1\right) + \color{blue}{t}}\right)} \]

   9. associate--r- 99.6%

      \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{2}{\color{blue}{0 - \left(-1 - t\right)}}\right)} \]

   10. neg-sub0 99.6%

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

   11. distribute-neg-frac2 99.6%

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

   12. distribute-neg-frac 99.6%

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

   13. metadata-eval 99.6%

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

10. Simplified 99.6%

    \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \color{blue}{\frac{-2}{-1 - t}}\right) \cdot \left(2 - \frac{-2}{-1 - t}\right)} \]
11. Step-by-step derivation

    1. div-inv 99.6%

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

    2. associate-/l* 99.6%

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

12. Applied egg-rr 99.6%

    \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{2 \cdot \frac{\frac{1}{t}}{1 + \frac{1}{t}}}\right)}{2 + \left(2 - \frac{-2}{-1 - t}\right) \cdot \left(2 - \frac{-2}{-1 - t}\right)} \]
13. Step-by-step derivation

    1. associate-/r* 99.6%

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

    2. associate-*r/ 99.6%

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

    3. metadata-eval 99.6%

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

    4. +-commutative 99.6%

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

    5. distribute-lft-in 99.6%

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

    6. rgt-mult-inverse 99.6%

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

    7. metadata-eval 99.6%

       \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{2}{\color{blue}{\left(0 - -1\right)} + t \cdot 1}\right)} \]

    8. *-rgt-identity 99.6%

       \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{2}{\left(0 - -1\right) + \color{blue}{t}}\right)} \]

    9. associate--r- 99.6%

       \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{2}{\color{blue}{0 - \left(-1 - t\right)}}\right)} \]

    10. neg-sub0 99.6%

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

    11. distribute-neg-frac2 99.6%

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

    12. distribute-neg-frac 99.6%

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

    13. metadata-eval 99.6%

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

14. Simplified 99.6%

    \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\frac{-2}{-1 - t}}\right)}{2 + \left(2 - \frac{-2}{-1 - t}\right) \cdot \left(2 - \frac{-2}{-1 - t}\right)} \]
15. Step-by-step derivation

    1. div-inv 99.6%

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

    2. associate-/l* 99.6%

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

16. Applied egg-rr 100.0%

    \[\leadsto \frac{1 + \left(2 - \color{blue}{2 \cdot \frac{\frac{1}{t}}{1 + \frac{1}{t}}}\right) \cdot \left(2 - \frac{-2}{-1 - t}\right)}{2 + \left(2 - \frac{-2}{-1 - t}\right) \cdot \left(2 - \frac{-2}{-1 - t}\right)} \]
17. Step-by-step derivation

    1. associate-/r* 99.6%

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

    2. associate-*r/ 99.6%

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

    3. metadata-eval 99.6%

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

    4. +-commutative 99.6%

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

    5. distribute-lft-in 99.6%

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

    6. rgt-mult-inverse 99.6%

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

    7. metadata-eval 99.6%

       \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{2}{\color{blue}{\left(0 - -1\right)} + t \cdot 1}\right)} \]

    8. *-rgt-identity 99.6%

       \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{2}{\left(0 - -1\right) + \color{blue}{t}}\right)} \]

    9. associate--r- 99.6%

       \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{2}{\color{blue}{0 - \left(-1 - t\right)}}\right)} \]

    10. neg-sub0 99.6%

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

    11. distribute-neg-frac2 99.6%

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

    12. distribute-neg-frac 99.6%

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

    13. metadata-eval 99.6%

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

18. Simplified 100.0%

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

19. Final simplification 100.0%

    \[\leadsto \frac{1 + \left(2 - \frac{-2}{-1 - t}\right) \cdot \left(2 - \frac{-2}{-1 - t}\right)}{2 + \left(2 - \frac{-2}{-1 - t}\right) \cdot \left(2 - \frac{-2}{-1 - t}\right)} \]
20. Add Preprocessing

Alternative 2: 99.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{-2}{-1 - t}\\ t_2 := \frac{\frac{12}{t} - 8}{t}\\ \mathbf{if}\;t \leq -0.37 \lor \neg \left(t \leq 0.6\right):\\ \;\;\;\;\frac{5 + t\_2}{6 + t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}{2 + t\_1 \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ -2.0 (- -1.0 t)))) (t_2 (/ (- (/ 12.0 t) 8.0) t)))
   (if (or (<= t -0.37) (not (<= t 0.6)))
     (/ (+ 5.0 t_2) (+ 6.0 t_2))
     (/ (+ 1.0 (* (* 2.0 t) (* 2.0 t))) (+ 2.0 (* t_1 t_1))))))

C

double code(double t) {
	double t_1 = 2.0 - (-2.0 / (-1.0 - t));
	double t_2 = ((12.0 / t) - 8.0) / t;
	double tmp;
	if ((t <= -0.37) || !(t <= 0.6)) {
		tmp = (5.0 + t_2) / (6.0 + t_2);
	} else {
		tmp = (1.0 + ((2.0 * t) * (2.0 * t))) / (2.0 + (t_1 * t_1));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 2.0d0 - ((-2.0d0) / ((-1.0d0) - t))
    t_2 = ((12.0d0 / t) - 8.0d0) / t
    if ((t <= (-0.37d0)) .or. (.not. (t <= 0.6d0))) then
        tmp = (5.0d0 + t_2) / (6.0d0 + t_2)
    else
        tmp = (1.0d0 + ((2.0d0 * t) * (2.0d0 * t))) / (2.0d0 + (t_1 * t_1))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 2.0 - (-2.0 / (-1.0 - t));
	double t_2 = ((12.0 / t) - 8.0) / t;
	double tmp;
	if ((t <= -0.37) || !(t <= 0.6)) {
		tmp = (5.0 + t_2) / (6.0 + t_2);
	} else {
		tmp = (1.0 + ((2.0 * t) * (2.0 * t))) / (2.0 + (t_1 * t_1));
	}
	return tmp;
}

Python

def code(t):
	t_1 = 2.0 - (-2.0 / (-1.0 - t))
	t_2 = ((12.0 / t) - 8.0) / t
	tmp = 0
	if (t <= -0.37) or not (t <= 0.6):
		tmp = (5.0 + t_2) / (6.0 + t_2)
	else:
		tmp = (1.0 + ((2.0 * t) * (2.0 * t))) / (2.0 + (t_1 * t_1))
	return tmp

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(-2.0 / Float64(-1.0 - t)))
	t_2 = Float64(Float64(Float64(12.0 / t) - 8.0) / t)
	tmp = 0.0
	if ((t <= -0.37) || !(t <= 0.6))
		tmp = Float64(Float64(5.0 + t_2) / Float64(6.0 + t_2));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(2.0 * t) * Float64(2.0 * t))) / Float64(2.0 + Float64(t_1 * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 2.0 - (-2.0 / (-1.0 - t));
	t_2 = ((12.0 / t) - 8.0) / t;
	tmp = 0.0;
	if ((t <= -0.37) || ~((t <= 0.6)))
		tmp = (5.0 + t_2) / (6.0 + t_2);
	else
		tmp = (1.0 + ((2.0 * t) * (2.0 * t))) / (2.0 + (t_1 * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 - N[(-2.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(12.0 / t), $MachinePrecision] - 8.0), $MachinePrecision] / t), $MachinePrecision]}, If[Or[LessEqual[t, -0.37], N[Not[LessEqual[t, 0.6]], $MachinePrecision]], N[(N[(5.0 + t$95$2), $MachinePrecision] / N[(6.0 + t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(2.0 * t), $MachinePrecision] * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.37 or 0.599999999999999978 < t

   1. Initial program 100.0%

      \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around -inf 100.0%

      \[\leadsto \frac{1 + \color{blue}{\left(4 + -1 \cdot \frac{8 - 12 \cdot \frac{1}{t}}{t}\right)}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto \frac{1 + \left(4 + \color{blue}{\left(-\frac{8 - 12 \cdot \frac{1}{t}}{t}\right)}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      2. unsub-neg 100.0%

         \[\leadsto \frac{1 + \color{blue}{\left(4 - \frac{8 - 12 \cdot \frac{1}{t}}{t}\right)}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      3. sub-neg 100.0%

         \[\leadsto \frac{1 + \left(4 - \frac{\color{blue}{8 + \left(-12 \cdot \frac{1}{t}\right)}}{t}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      4. associate-*r/ 100.0%

         \[\leadsto \frac{1 + \left(4 - \frac{8 + \left(-\color{blue}{\frac{12 \cdot 1}{t}}\right)}{t}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      5. metadata-eval 100.0%

         \[\leadsto \frac{1 + \left(4 - \frac{8 + \left(-\frac{\color{blue}{12}}{t}\right)}{t}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      6. distribute-neg-frac 100.0%

         \[\leadsto \frac{1 + \left(4 - \frac{8 + \color{blue}{\frac{-12}{t}}}{t}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      7. metadata-eval 100.0%

         \[\leadsto \frac{1 + \left(4 - \frac{8 + \frac{\color{blue}{-12}}{t}}{t}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   5. Simplified 100.0%

      \[\leadsto \frac{1 + \color{blue}{\left(4 - \frac{8 + \frac{-12}{t}}{t}\right)}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   6. Taylor expanded in t around -inf 100.0%

      \[\leadsto \frac{1 + \left(4 - \frac{8 + \frac{-12}{t}}{t}\right)}{2 + \color{blue}{\left(4 + -1 \cdot \frac{8 - 12 \cdot \frac{1}{t}}{t}\right)}} \]
   7. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto \frac{1 + \left(4 + \color{blue}{\left(-\frac{8 - 12 \cdot \frac{1}{t}}{t}\right)}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      2. unsub-neg 100.0%

         \[\leadsto \frac{1 + \color{blue}{\left(4 - \frac{8 - 12 \cdot \frac{1}{t}}{t}\right)}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      3. sub-neg 100.0%

         \[\leadsto \frac{1 + \left(4 - \frac{\color{blue}{8 + \left(-12 \cdot \frac{1}{t}\right)}}{t}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      4. associate-*r/ 100.0%

         \[\leadsto \frac{1 + \left(4 - \frac{8 + \left(-\color{blue}{\frac{12 \cdot 1}{t}}\right)}{t}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      5. metadata-eval 100.0%

         \[\leadsto \frac{1 + \left(4 - \frac{8 + \left(-\frac{\color{blue}{12}}{t}\right)}{t}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      6. distribute-neg-frac 100.0%

         \[\leadsto \frac{1 + \left(4 - \frac{8 + \color{blue}{\frac{-12}{t}}}{t}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      7. metadata-eval 100.0%

         \[\leadsto \frac{1 + \left(4 - \frac{8 + \frac{\color{blue}{-12}}{t}}{t}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   8. Simplified 100.0%

      \[\leadsto \frac{1 + \left(4 - \frac{8 + \frac{-12}{t}}{t}\right)}{2 + \color{blue}{\left(4 - \frac{8 + \frac{-12}{t}}{t}\right)}} \]
   9. Step-by-step derivation

      1. associate-+r- 100.0%

         \[\leadsto \frac{\color{blue}{\left(1 + 4\right) - \frac{8 + \frac{-12}{t}}{t}}}{2 + \left(4 - \frac{8 + \frac{-12}{t}}{t}\right)} \]

      2. metadata-eval 100.0%

         \[\leadsto \frac{\color{blue}{5} - \frac{8 + \frac{-12}{t}}{t}}{2 + \left(4 - \frac{8 + \frac{-12}{t}}{t}\right)} \]

      3. div-sub 100.0%

         \[\leadsto \color{blue}{\frac{5}{2 + \left(4 - \frac{8 + \frac{-12}{t}}{t}\right)} - \frac{\frac{8 + \frac{-12}{t}}{t}}{2 + \left(4 - \frac{8 + \frac{-12}{t}}{t}\right)}} \]

      4. associate-+r- 100.0%

         \[\leadsto \frac{5}{\color{blue}{\left(2 + 4\right) - \frac{8 + \frac{-12}{t}}{t}}} - \frac{\frac{8 + \frac{-12}{t}}{t}}{2 + \left(4 - \frac{8 + \frac{-12}{t}}{t}\right)} \]

      5. metadata-eval 100.0%

         \[\leadsto \frac{5}{\color{blue}{6} - \frac{8 + \frac{-12}{t}}{t}} - \frac{\frac{8 + \frac{-12}{t}}{t}}{2 + \left(4 - \frac{8 + \frac{-12}{t}}{t}\right)} \]

      6. associate-+r- 100.0%

         \[\leadsto \frac{5}{6 - \frac{8 + \frac{-12}{t}}{t}} - \frac{\frac{8 + \frac{-12}{t}}{t}}{\color{blue}{\left(2 + 4\right) - \frac{8 + \frac{-12}{t}}{t}}} \]

      7. metadata-eval 100.0%

         \[\leadsto \frac{5}{6 - \frac{8 + \frac{-12}{t}}{t}} - \frac{\frac{8 + \frac{-12}{t}}{t}}{\color{blue}{6} - \frac{8 + \frac{-12}{t}}{t}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{5}{6 - \frac{8 + \frac{-12}{t}}{t}} - \frac{\frac{8 + \frac{-12}{t}}{t}}{6 - \frac{8 + \frac{-12}{t}}{t}}} \]
   11. Step-by-step derivation

       1. div-sub 100.0%

          \[\leadsto \color{blue}{\frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 - \frac{8 + \frac{-12}{t}}{t}}} \]

       2. metadata-eval 100.0%

          \[\leadsto \frac{5 - \frac{8 + \frac{\color{blue}{-12}}{t}}{t}}{6 - \frac{8 + \frac{-12}{t}}{t}} \]

       3. distribute-neg-frac 100.0%

          \[\leadsto \frac{5 - \frac{8 + \color{blue}{\left(-\frac{12}{t}\right)}}{t}}{6 - \frac{8 + \frac{-12}{t}}{t}} \]

       4. metadata-eval 100.0%

          \[\leadsto \frac{5 - \frac{8 + \left(-\frac{\color{blue}{12 \cdot 1}}{t}\right)}{t}}{6 - \frac{8 + \frac{-12}{t}}{t}} \]

       5. associate-*r/ 100.0%

          \[\leadsto \frac{5 - \frac{8 + \left(-\color{blue}{12 \cdot \frac{1}{t}}\right)}{t}}{6 - \frac{8 + \frac{-12}{t}}{t}} \]

       6. sub-neg 100.0%

          \[\leadsto \frac{5 - \frac{\color{blue}{8 - 12 \cdot \frac{1}{t}}}{t}}{6 - \frac{8 + \frac{-12}{t}}{t}} \]

       7. associate-*r/ 100.0%

          \[\leadsto \frac{5 - \frac{8 - \color{blue}{\frac{12 \cdot 1}{t}}}{t}}{6 - \frac{8 + \frac{-12}{t}}{t}} \]

       8. metadata-eval 100.0%

          \[\leadsto \frac{5 - \frac{8 - \frac{\color{blue}{12}}{t}}{t}}{6 - \frac{8 + \frac{-12}{t}}{t}} \]

       9. metadata-eval 100.0%

          \[\leadsto \frac{5 - \frac{8 - \frac{12}{t}}{t}}{6 - \frac{8 + \frac{\color{blue}{-12}}{t}}{t}} \]

       10. distribute-neg-frac 100.0%

           \[\leadsto \frac{5 - \frac{8 - \frac{12}{t}}{t}}{6 - \frac{8 + \color{blue}{\left(-\frac{12}{t}\right)}}{t}} \]

       11. metadata-eval 100.0%

           \[\leadsto \frac{5 - \frac{8 - \frac{12}{t}}{t}}{6 - \frac{8 + \left(-\frac{\color{blue}{12 \cdot 1}}{t}\right)}{t}} \]

       12. associate-*r/ 100.0%

           \[\leadsto \frac{5 - \frac{8 - \frac{12}{t}}{t}}{6 - \frac{8 + \left(-\color{blue}{12 \cdot \frac{1}{t}}\right)}{t}} \]

       13. sub-neg 100.0%

           \[\leadsto \frac{5 - \frac{8 - \frac{12}{t}}{t}}{6 - \frac{\color{blue}{8 - 12 \cdot \frac{1}{t}}}{t}} \]

   12. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{5 - \frac{8 - \frac{12}{t}}{t}}{6 - \frac{8 - \frac{12}{t}}{t}}} \]

   if -0.37 < t < 0.599999999999999978

   1. Initial program 99.3%

      \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv 99.3%

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

      2. associate-/l* 99.3%

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

   4. Applied egg-rr 99.3%

      \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{2 \cdot \frac{\frac{1}{t}}{1 + \frac{1}{t}}}\right)} \]
   5. Step-by-step derivation

      1. associate-/r* 99.3%

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

      2. associate-*r/ 99.3%

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

      3. metadata-eval 99.3%

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

      4. +-commutative 99.3%

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

      5. distribute-lft-in 99.3%

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

      6. rgt-mult-inverse 99.3%

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

      7. metadata-eval 99.3%

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{2}{\color{blue}{\left(0 - -1\right)} + t \cdot 1}\right)} \]

      8. *-rgt-identity 99.3%

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{2}{\left(0 - -1\right) + \color{blue}{t}}\right)} \]

      9. associate--r- 99.3%

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{2}{\color{blue}{0 - \left(-1 - t\right)}}\right)} \]

      10. neg-sub0 99.3%

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

      11. distribute-neg-frac2 99.3%

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

      12. distribute-neg-frac 99.3%

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

      13. metadata-eval 99.3%

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

   6. Simplified 99.3%

      \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\frac{-2}{-1 - t}}\right)} \]
   7. Step-by-step derivation

      1. div-inv 99.3%

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

      2. associate-/l* 99.3%

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

   8. Applied egg-rr 99.3%

      \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \color{blue}{2 \cdot \frac{\frac{1}{t}}{1 + \frac{1}{t}}}\right) \cdot \left(2 - \frac{-2}{-1 - t}\right)} \]
   9. Step-by-step derivation

      1. associate-/r* 99.3%

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

      2. associate-*r/ 99.3%

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

      3. metadata-eval 99.3%

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

      4. +-commutative 99.3%

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

      5. distribute-lft-in 99.3%

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

      6. rgt-mult-inverse 99.3%

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

      7. metadata-eval 99.3%

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{2}{\color{blue}{\left(0 - -1\right)} + t \cdot 1}\right)} \]

      8. *-rgt-identity 99.3%

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{2}{\left(0 - -1\right) + \color{blue}{t}}\right)} \]

      9. associate--r- 99.3%

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{2}{\color{blue}{0 - \left(-1 - t\right)}}\right)} \]

      10. neg-sub0 99.3%

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

      11. distribute-neg-frac2 99.3%

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

      12. distribute-neg-frac 99.3%

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

      13. metadata-eval 99.3%

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

   10. Simplified 99.3%

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

   11. Taylor expanded in t around 0 98.9%

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

   12. Taylor expanded in t around 0 99.6%

       \[\leadsto \frac{1 + \color{blue}{\left(2 \cdot t\right)} \cdot \left(2 \cdot t\right)}{2 + \left(2 - \frac{-2}{-1 - t}\right) \cdot \left(2 - \frac{-2}{-1 - t}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.37 \lor \neg \left(t \leq 0.6\right):\\ \;\;\;\;\frac{5 + \frac{\frac{12}{t} - 8}{t}}{6 + \frac{\frac{12}{t} - 8}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}{2 + \left(2 - \frac{-2}{-1 - t}\right) \cdot \left(2 - \frac{-2}{-1 - t}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{12}{t} - 8}{t}\\ \mathbf{if}\;t \leq -0.41 \lor \neg \left(t \leq 1.2\right):\\ \;\;\;\;\frac{5 + t\_1}{6 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (- (/ 12.0 t) 8.0) t)))
   (if (or (<= t -0.41) (not (<= t 1.2))) (/ (+ 5.0 t_1) (+ 6.0 t_1)) 0.5)))

C

double code(double t) {
	double t_1 = ((12.0 / t) - 8.0) / t;
	double tmp;
	if ((t <= -0.41) || !(t <= 1.2)) {
		tmp = (5.0 + t_1) / (6.0 + t_1);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((12.0d0 / t) - 8.0d0) / t
    if ((t <= (-0.41d0)) .or. (.not. (t <= 1.2d0))) then
        tmp = (5.0d0 + t_1) / (6.0d0 + t_1)
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = ((12.0 / t) - 8.0) / t;
	double tmp;
	if ((t <= -0.41) || !(t <= 1.2)) {
		tmp = (5.0 + t_1) / (6.0 + t_1);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	t_1 = ((12.0 / t) - 8.0) / t
	tmp = 0
	if (t <= -0.41) or not (t <= 1.2):
		tmp = (5.0 + t_1) / (6.0 + t_1)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	t_1 = Float64(Float64(Float64(12.0 / t) - 8.0) / t)
	tmp = 0.0
	if ((t <= -0.41) || !(t <= 1.2))
		tmp = Float64(Float64(5.0 + t_1) / Float64(6.0 + t_1));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = ((12.0 / t) - 8.0) / t;
	tmp = 0.0;
	if ((t <= -0.41) || ~((t <= 1.2)))
		tmp = (5.0 + t_1) / (6.0 + t_1);
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(N[(12.0 / t), $MachinePrecision] - 8.0), $MachinePrecision] / t), $MachinePrecision]}, If[Or[LessEqual[t, -0.41], N[Not[LessEqual[t, 1.2]], $MachinePrecision]], N[(N[(5.0 + t$95$1), $MachinePrecision] / N[(6.0 + t$95$1), $MachinePrecision]), $MachinePrecision], 0.5]]

Derivation

1. Split input into 2 regimes

2. if t < -0.409999999999999976 or 1.19999999999999996 < t

   1. Initial program 100.0%

      \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around -inf 100.0%

      \[\leadsto \frac{1 + \color{blue}{\left(4 + -1 \cdot \frac{8 - 12 \cdot \frac{1}{t}}{t}\right)}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto \frac{1 + \left(4 + \color{blue}{\left(-\frac{8 - 12 \cdot \frac{1}{t}}{t}\right)}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      2. unsub-neg 100.0%

         \[\leadsto \frac{1 + \color{blue}{\left(4 - \frac{8 - 12 \cdot \frac{1}{t}}{t}\right)}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      3. sub-neg 100.0%

         \[\leadsto \frac{1 + \left(4 - \frac{\color{blue}{8 + \left(-12 \cdot \frac{1}{t}\right)}}{t}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      4. associate-*r/ 100.0%

         \[\leadsto \frac{1 + \left(4 - \frac{8 + \left(-\color{blue}{\frac{12 \cdot 1}{t}}\right)}{t}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      5. metadata-eval 100.0%

         \[\leadsto \frac{1 + \left(4 - \frac{8 + \left(-\frac{\color{blue}{12}}{t}\right)}{t}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      6. distribute-neg-frac 100.0%

         \[\leadsto \frac{1 + \left(4 - \frac{8 + \color{blue}{\frac{-12}{t}}}{t}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      7. metadata-eval 100.0%

         \[\leadsto \frac{1 + \left(4 - \frac{8 + \frac{\color{blue}{-12}}{t}}{t}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   5. Simplified 100.0%

      \[\leadsto \frac{1 + \color{blue}{\left(4 - \frac{8 + \frac{-12}{t}}{t}\right)}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   6. Taylor expanded in t around -inf 100.0%

      \[\leadsto \frac{1 + \left(4 - \frac{8 + \frac{-12}{t}}{t}\right)}{2 + \color{blue}{\left(4 + -1 \cdot \frac{8 - 12 \cdot \frac{1}{t}}{t}\right)}} \]
   7. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto \frac{1 + \left(4 + \color{blue}{\left(-\frac{8 - 12 \cdot \frac{1}{t}}{t}\right)}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      2. unsub-neg 100.0%

         \[\leadsto \frac{1 + \color{blue}{\left(4 - \frac{8 - 12 \cdot \frac{1}{t}}{t}\right)}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      3. sub-neg 100.0%

         \[\leadsto \frac{1 + \left(4 - \frac{\color{blue}{8 + \left(-12 \cdot \frac{1}{t}\right)}}{t}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      4. associate-*r/ 100.0%

         \[\leadsto \frac{1 + \left(4 - \frac{8 + \left(-\color{blue}{\frac{12 \cdot 1}{t}}\right)}{t}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      5. metadata-eval 100.0%

         \[\leadsto \frac{1 + \left(4 - \frac{8 + \left(-\frac{\color{blue}{12}}{t}\right)}{t}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      6. distribute-neg-frac 100.0%

         \[\leadsto \frac{1 + \left(4 - \frac{8 + \color{blue}{\frac{-12}{t}}}{t}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      7. metadata-eval 100.0%

         \[\leadsto \frac{1 + \left(4 - \frac{8 + \frac{\color{blue}{-12}}{t}}{t}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   8. Simplified 100.0%

      \[\leadsto \frac{1 + \left(4 - \frac{8 + \frac{-12}{t}}{t}\right)}{2 + \color{blue}{\left(4 - \frac{8 + \frac{-12}{t}}{t}\right)}} \]
   9. Step-by-step derivation

      1. associate-+r- 100.0%

         \[\leadsto \frac{\color{blue}{\left(1 + 4\right) - \frac{8 + \frac{-12}{t}}{t}}}{2 + \left(4 - \frac{8 + \frac{-12}{t}}{t}\right)} \]

      2. metadata-eval 100.0%

         \[\leadsto \frac{\color{blue}{5} - \frac{8 + \frac{-12}{t}}{t}}{2 + \left(4 - \frac{8 + \frac{-12}{t}}{t}\right)} \]

      3. div-sub 100.0%

         \[\leadsto \color{blue}{\frac{5}{2 + \left(4 - \frac{8 + \frac{-12}{t}}{t}\right)} - \frac{\frac{8 + \frac{-12}{t}}{t}}{2 + \left(4 - \frac{8 + \frac{-12}{t}}{t}\right)}} \]

      4. associate-+r- 100.0%

         \[\leadsto \frac{5}{\color{blue}{\left(2 + 4\right) - \frac{8 + \frac{-12}{t}}{t}}} - \frac{\frac{8 + \frac{-12}{t}}{t}}{2 + \left(4 - \frac{8 + \frac{-12}{t}}{t}\right)} \]

      5. metadata-eval 100.0%

         \[\leadsto \frac{5}{\color{blue}{6} - \frac{8 + \frac{-12}{t}}{t}} - \frac{\frac{8 + \frac{-12}{t}}{t}}{2 + \left(4 - \frac{8 + \frac{-12}{t}}{t}\right)} \]

      6. associate-+r- 100.0%

         \[\leadsto \frac{5}{6 - \frac{8 + \frac{-12}{t}}{t}} - \frac{\frac{8 + \frac{-12}{t}}{t}}{\color{blue}{\left(2 + 4\right) - \frac{8 + \frac{-12}{t}}{t}}} \]

      7. metadata-eval 100.0%

         \[\leadsto \frac{5}{6 - \frac{8 + \frac{-12}{t}}{t}} - \frac{\frac{8 + \frac{-12}{t}}{t}}{\color{blue}{6} - \frac{8 + \frac{-12}{t}}{t}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{5}{6 - \frac{8 + \frac{-12}{t}}{t}} - \frac{\frac{8 + \frac{-12}{t}}{t}}{6 - \frac{8 + \frac{-12}{t}}{t}}} \]
   11. Step-by-step derivation

       1. div-sub 100.0%

          \[\leadsto \color{blue}{\frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 - \frac{8 + \frac{-12}{t}}{t}}} \]

       2. metadata-eval 100.0%

          \[\leadsto \frac{5 - \frac{8 + \frac{\color{blue}{-12}}{t}}{t}}{6 - \frac{8 + \frac{-12}{t}}{t}} \]

       3. distribute-neg-frac 100.0%

          \[\leadsto \frac{5 - \frac{8 + \color{blue}{\left(-\frac{12}{t}\right)}}{t}}{6 - \frac{8 + \frac{-12}{t}}{t}} \]

       4. metadata-eval 100.0%

          \[\leadsto \frac{5 - \frac{8 + \left(-\frac{\color{blue}{12 \cdot 1}}{t}\right)}{t}}{6 - \frac{8 + \frac{-12}{t}}{t}} \]

       5. associate-*r/ 100.0%

          \[\leadsto \frac{5 - \frac{8 + \left(-\color{blue}{12 \cdot \frac{1}{t}}\right)}{t}}{6 - \frac{8 + \frac{-12}{t}}{t}} \]

       6. sub-neg 100.0%

          \[\leadsto \frac{5 - \frac{\color{blue}{8 - 12 \cdot \frac{1}{t}}}{t}}{6 - \frac{8 + \frac{-12}{t}}{t}} \]

       7. associate-*r/ 100.0%

          \[\leadsto \frac{5 - \frac{8 - \color{blue}{\frac{12 \cdot 1}{t}}}{t}}{6 - \frac{8 + \frac{-12}{t}}{t}} \]

       8. metadata-eval 100.0%

          \[\leadsto \frac{5 - \frac{8 - \frac{\color{blue}{12}}{t}}{t}}{6 - \frac{8 + \frac{-12}{t}}{t}} \]

       9. metadata-eval 100.0%

          \[\leadsto \frac{5 - \frac{8 - \frac{12}{t}}{t}}{6 - \frac{8 + \frac{\color{blue}{-12}}{t}}{t}} \]

       10. distribute-neg-frac 100.0%

           \[\leadsto \frac{5 - \frac{8 - \frac{12}{t}}{t}}{6 - \frac{8 + \color{blue}{\left(-\frac{12}{t}\right)}}{t}} \]

       11. metadata-eval 100.0%

           \[\leadsto \frac{5 - \frac{8 - \frac{12}{t}}{t}}{6 - \frac{8 + \left(-\frac{\color{blue}{12 \cdot 1}}{t}\right)}{t}} \]

       12. associate-*r/ 100.0%

           \[\leadsto \frac{5 - \frac{8 - \frac{12}{t}}{t}}{6 - \frac{8 + \left(-\color{blue}{12 \cdot \frac{1}{t}}\right)}{t}} \]

       13. sub-neg 100.0%

           \[\leadsto \frac{5 - \frac{8 - \frac{12}{t}}{t}}{6 - \frac{\color{blue}{8 - 12 \cdot \frac{1}{t}}}{t}} \]

   12. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{5 - \frac{8 - \frac{12}{t}}{t}}{6 - \frac{8 - \frac{12}{t}}{t}}} \]

   if -0.409999999999999976 < t < 1.19999999999999996

   1. Initial program 99.3%

      \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Step-by-step derivation

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(2 + \frac{2}{-1 - t}, 2 + \frac{2}{-1 - t}, 1\right)}{\mathsf{fma}\left(2 + \frac{2}{-1 - t}, 2 + \frac{2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 99.4%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.41 \lor \neg \left(t \leq 1.2\right):\\ \;\;\;\;\frac{5 + \frac{\frac{12}{t} - 8}{t}}{6 + \frac{\frac{12}{t} - 8}{t}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.52 \lor \neg \left(t \leq 0.23\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.52) (not (<= t 0.23)))
   (-
    0.8333333333333334
    (/ (+ 0.2222222222222222 (/ -0.037037037037037035 t)) t))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.52) || !(t <= 0.23)) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.52d0)) .or. (.not. (t <= 0.23d0))) then
        tmp = 0.8333333333333334d0 - ((0.2222222222222222d0 + ((-0.037037037037037035d0) / t)) / t)
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.52) || !(t <= 0.23)) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.52) or not (t <= 0.23):
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.52) || !(t <= 0.23))
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 + Float64(-0.037037037037037035 / t)) / t));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.52) || ~((t <= 0.23)))
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.52], N[Not[LessEqual[t, 0.23]], $MachinePrecision]], N[(0.8333333333333334 - N[(N[(0.2222222222222222 + N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if t < -0.52000000000000002 or 0.23000000000000001 < t

   1. Initial program 100.0%

      \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Step-by-step derivation

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(2 + \frac{2}{-1 - t}, 2 + \frac{2}{-1 - t}, 1\right)}{\mathsf{fma}\left(2 + \frac{2}{-1 - t}, 2 + \frac{2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around -inf 100.0%

      \[\leadsto \color{blue}{0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 - 0.037037037037037035 \cdot \frac{1}{t}}{t}} \]
   6. Step-by-step derivation

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

         \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 - 0.037037037037037035 \cdot \frac{1}{t}}{t}} \]

      3. sub-neg 100.0%

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

      4. associate-*r/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. distribute-neg-frac 100.0%

         \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222 + \color{blue}{\frac{-0.037037037037037035}{t}}}{t} \]

      7. metadata-eval 100.0%

         \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222 + \frac{\color{blue}{-0.037037037037037035}}{t}}{t} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}} \]

   if -0.52000000000000002 < t < 0.23000000000000001

   1. Initial program 99.3%

      \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Step-by-step derivation

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(2 + \frac{2}{-1 - t}, 2 + \frac{2}{-1 - t}, 1\right)}{\mathsf{fma}\left(2 + \frac{2}{-1 - t}, 2 + \frac{2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 99.4%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.52 \lor \neg \left(t \leq 0.23\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.66\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.49) (not (<= t 0.66)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.49) || !(t <= 0.66)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.49d0)) .or. (.not. (t <= 0.66d0))) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.49) || !(t <= 0.66)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.49) or not (t <= 0.66):
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.49) || !(t <= 0.66))
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.49) || ~((t <= 0.66)))
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.49], N[Not[LessEqual[t, 0.66]], $MachinePrecision]], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if t < -0.48999999999999999 or 0.660000000000000031 < t

   1. Initial program 100.0%

      \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Step-by-step derivation

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(2 + \frac{2}{-1 - t}, 2 + \frac{2}{-1 - t}, 1\right)}{\mathsf{fma}\left(2 + \frac{2}{-1 - t}, 2 + \frac{2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 99.8%

      \[\leadsto \color{blue}{0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t}} \]
   6. Step-by-step derivation

      1. associate-*r/ 99.8%

         \[\leadsto 0.8333333333333334 - \color{blue}{\frac{0.2222222222222222 \cdot 1}{t}} \]

      2. metadata-eval 99.8%

         \[\leadsto 0.8333333333333334 - \frac{\color{blue}{0.2222222222222222}}{t} \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222}{t}} \]

   if -0.48999999999999999 < t < 0.660000000000000031

   1. Initial program 99.3%

      \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Step-by-step derivation

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(2 + \frac{2}{-1 - t}, 2 + \frac{2}{-1 - t}, 1\right)}{\mathsf{fma}\left(2 + \frac{2}{-1 - t}, 2 + \frac{2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 99.4%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.66\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.34:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.34) 0.8333333333333334 (if (<= t 1.0) 0.5 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.34) {
		tmp = 0.8333333333333334;
	} else if (t <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.34d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 1.0d0) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.34) {
		tmp = 0.8333333333333334;
	} else if (t <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.34:
		tmp = 0.8333333333333334
	elif t <= 1.0:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.34)
		tmp = 0.8333333333333334;
	elseif (t <= 1.0)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.34)
		tmp = 0.8333333333333334;
	elseif (t <= 1.0)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.34], 0.8333333333333334, If[LessEqual[t, 1.0], 0.5, 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.340000000000000024 or 1 < t

   1. Initial program 100.0%

      \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Step-by-step derivation

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(2 + \frac{2}{-1 - t}, 2 + \frac{2}{-1 - t}, 1\right)}{\mathsf{fma}\left(2 + \frac{2}{-1 - t}, 2 + \frac{2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 98.7%

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

   if -0.340000000000000024 < t < 1

   1. Initial program 99.3%

      \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Step-by-step derivation

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(2 + \frac{2}{-1 - t}, 2 + \frac{2}{-1 - t}, 1\right)}{\mathsf{fma}\left(2 + \frac{2}{-1 - t}, 2 + \frac{2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 99.4%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.34:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.1% accurate, 51.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (t) :precision binary64 0.5)

C

double code(double t) {
	return 0.5;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = 0.5d0
end function

Java

public static double code(double t) {
	return 0.5;
}

Python

def code(t):
	return 0.5

Julia

function code(t)
	return 0.5
end

MATLAB

function tmp = code(t)
	tmp = 0.5;
end

Wolfram

code[t_] := 0.5

Derivation

1. Initial program 99.6%

   \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
2. Step-by-step derivation

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(2 + \frac{2}{-1 - t}, 2 + \frac{2}{-1 - t}, 1\right)}{\mathsf{fma}\left(2 + \frac{2}{-1 - t}, 2 + \frac{2}{-1 - t}, 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in t around 0 65.5%

   \[\leadsto \color{blue}{0.5} \]

6. Final simplification 65.5%

   \[\leadsto 0.5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024077 
(FPCore (t)
  :name "Kahan p13 Example 2"
  :precision binary64
  (/ (+ 1.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))) (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))))