Kahan p13 Example 2

Percentage Accurate: 99.9% → 100.0%

Time: 41.7s

Alternatives: 9

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

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 + \frac{-2}{t + 1}\right) \cdot \left(2 + \frac{2}{-1 - t}\right)\\ \frac{t\_1 + 1}{2 + t\_1} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. sub-neg 100.0%

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

   2. distribute-neg-frac 100.0%

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

   3. distribute-neg-frac 100.0%

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

   4. metadata-eval 100.0%

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

4. Applied egg-rr 100.0%

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

   1. associate-/r* 100.0%

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

   2. distribute-lft-in 100.0%

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

   3. *-rgt-identity 100.0%

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

   4. rgt-mult-inverse 100.0%

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

6. Simplified 100.0%

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

   1. div-inv 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. times-frac 100.0%

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

   4. metadata-eval 100.0%

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

8. Applied egg-rr 100.0%

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

   1. associate-/l/ 100.0%

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

   2. associate-*r/ 100.0%

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

   3. metadata-eval 100.0%

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

   4. *-commutative 100.0%

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

   5. distribute-rgt-in 100.0%

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

   6. *-lft-identity 100.0%

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

   7. lft-mult-inverse 100.0%

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

10. Simplified 100.0%

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

    1. div-inv 100.0%

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

    2. *-un-lft-identity 100.0%

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

    3. times-frac 100.0%

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

    4. metadata-eval 100.0%

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

12. Applied egg-rr 100.0%

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

    1. associate-/l/ 100.0%

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

    2. associate-*r/ 100.0%

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

    3. metadata-eval 100.0%

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

    4. *-commutative 100.0%

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

    5. distribute-rgt-in 100.0%

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

    6. *-lft-identity 100.0%

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

    7. lft-mult-inverse 100.0%

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

14. Simplified 100.0%

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

    1. sub-neg 100.0%

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

    2. distribute-neg-frac 100.0%

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

    3. distribute-neg-frac 100.0%

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

    4. metadata-eval 100.0%

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

16. Applied egg-rr 100.0%

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

    1. associate-/r* 100.0%

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

    2. distribute-lft-in 100.0%

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

    3. *-rgt-identity 100.0%

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

    4. rgt-mult-inverse 100.0%

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

18. Simplified 100.0%

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

19. Final simplification 100.0%

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

Alternative 2: 98.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{12 - \frac{16}{t}}{t} - 8}{t}\\ \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} + 1} \leq 10^{-8}:\\ \;\;\;\;\frac{5 + t\_1}{6 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(2 \cdot t\right) \cdot \left(\frac{2}{t + 1} - 2\right)}{2 + \left(2 + \frac{-2}{t + 1}\right) \cdot \left(2 + \frac{2}{-1 - t}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (- (/ (- 12.0 (/ 16.0 t)) t) 8.0) t)))
   (if (<= (/ (/ 2.0 t) (+ (/ 1.0 t) 1.0)) 1e-8)
     (/ (+ 5.0 t_1) (+ 6.0 t_1))
     (/
      (- 1.0 (* (* 2.0 t) (- (/ 2.0 (+ t 1.0)) 2.0)))
      (+ 2.0 (* (+ 2.0 (/ -2.0 (+ t 1.0))) (+ 2.0 (/ 2.0 (- -1.0 t)))))))))

C

double code(double t) {
	double t_1 = (((12.0 - (16.0 / t)) / t) - 8.0) / t;
	double tmp;
	if (((2.0 / t) / ((1.0 / t) + 1.0)) <= 1e-8) {
		tmp = (5.0 + t_1) / (6.0 + t_1);
	} else {
		tmp = (1.0 - ((2.0 * t) * ((2.0 / (t + 1.0)) - 2.0))) / (2.0 + ((2.0 + (-2.0 / (t + 1.0))) * (2.0 + (2.0 / (-1.0 - t)))));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (((12.0d0 - (16.0d0 / t)) / t) - 8.0d0) / t
    if (((2.0d0 / t) / ((1.0d0 / t) + 1.0d0)) <= 1d-8) then
        tmp = (5.0d0 + t_1) / (6.0d0 + t_1)
    else
        tmp = (1.0d0 - ((2.0d0 * t) * ((2.0d0 / (t + 1.0d0)) - 2.0d0))) / (2.0d0 + ((2.0d0 + ((-2.0d0) / (t + 1.0d0))) * (2.0d0 + (2.0d0 / ((-1.0d0) - t)))))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = (((12.0 - (16.0 / t)) / t) - 8.0) / t;
	double tmp;
	if (((2.0 / t) / ((1.0 / t) + 1.0)) <= 1e-8) {
		tmp = (5.0 + t_1) / (6.0 + t_1);
	} else {
		tmp = (1.0 - ((2.0 * t) * ((2.0 / (t + 1.0)) - 2.0))) / (2.0 + ((2.0 + (-2.0 / (t + 1.0))) * (2.0 + (2.0 / (-1.0 - t)))));
	}
	return tmp;
}

Python

def code(t):
	t_1 = (((12.0 - (16.0 / t)) / t) - 8.0) / t
	tmp = 0
	if ((2.0 / t) / ((1.0 / t) + 1.0)) <= 1e-8:
		tmp = (5.0 + t_1) / (6.0 + t_1)
	else:
		tmp = (1.0 - ((2.0 * t) * ((2.0 / (t + 1.0)) - 2.0))) / (2.0 + ((2.0 + (-2.0 / (t + 1.0))) * (2.0 + (2.0 / (-1.0 - t)))))
	return tmp

Julia

function code(t)
	t_1 = Float64(Float64(Float64(Float64(12.0 - Float64(16.0 / t)) / t) - 8.0) / t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(Float64(1.0 / t) + 1.0)) <= 1e-8)
		tmp = Float64(Float64(5.0 + t_1) / Float64(6.0 + t_1));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(2.0 * t) * Float64(Float64(2.0 / Float64(t + 1.0)) - 2.0))) / Float64(2.0 + Float64(Float64(2.0 + Float64(-2.0 / Float64(t + 1.0))) * Float64(2.0 + Float64(2.0 / Float64(-1.0 - t))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = (((12.0 - (16.0 / t)) / t) - 8.0) / t;
	tmp = 0.0;
	if (((2.0 / t) / ((1.0 / t) + 1.0)) <= 1e-8)
		tmp = (5.0 + t_1) / (6.0 + t_1);
	else
		tmp = (1.0 - ((2.0 * t) * ((2.0 / (t + 1.0)) - 2.0))) / (2.0 + ((2.0 + (-2.0 / (t + 1.0))) * (2.0 + (2.0 / (-1.0 - t)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(N[(N[(12.0 - N[(16.0 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - 8.0), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(N[(1.0 / t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 1e-8], N[(N[(5.0 + t$95$1), $MachinePrecision] / N[(6.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(2.0 * t), $MachinePrecision] * N[(N[(2.0 / N[(t + 1.0), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(2.0 + N[(-2.0 / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(2.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 1e-8

   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%

      \[\leadsto \frac{\color{blue}{5 + -1 \cdot \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto \frac{5 + \color{blue}{\left(-\frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}\right)}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      2. unsub-neg 100.0%

         \[\leadsto \frac{\color{blue}{5 - \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      3. mul-1-neg 100.0%

         \[\leadsto \frac{5 - \frac{8 + \color{blue}{\left(-\frac{12 - 16 \cdot \frac{1}{t}}{t}\right)}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      4. unsub-neg 100.0%

         \[\leadsto \frac{5 - \frac{\color{blue}{8 - \frac{12 - 16 \cdot \frac{1}{t}}{t}}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      5. associate-*r/ 100.0%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 - \color{blue}{\frac{16 \cdot 1}{t}}}{t}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      6. metadata-eval 100.0%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 - \frac{\color{blue}{16}}{t}}{t}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   5. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{5 - \frac{8 - \frac{12 - \frac{16}{t}}{t}}{t}}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   6. Taylor expanded in t around -inf 100.0%

      \[\leadsto \frac{5 - \frac{8 - \frac{12 - \frac{16}{t}}{t}}{t}}{\color{blue}{6 + -1 \cdot \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 - \frac{16}{t}}{t}}{t}}{6 + \color{blue}{\left(-\frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}\right)}} \]

      2. unsub-neg 100.0%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 - \frac{16}{t}}{t}}{t}}{\color{blue}{6 - \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}}} \]

      3. mul-1-neg 100.0%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 - \frac{16}{t}}{t}}{t}}{6 - \frac{8 + \color{blue}{\left(-\frac{12 - 16 \cdot \frac{1}{t}}{t}\right)}}{t}} \]

      4. unsub-neg 100.0%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 - \frac{16}{t}}{t}}{t}}{6 - \frac{\color{blue}{8 - \frac{12 - 16 \cdot \frac{1}{t}}{t}}}{t}} \]

      5. associate-*r/ 100.0%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 - \frac{16}{t}}{t}}{t}}{6 - \frac{8 - \frac{12 - \color{blue}{\frac{16 \cdot 1}{t}}}{t}}{t}} \]

      6. metadata-eval 100.0%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 - \frac{16}{t}}{t}}{t}}{6 - \frac{8 - \frac{12 - \frac{\color{blue}{16}}{t}}{t}}{t}} \]

   8. Simplified 100.0%

      \[\leadsto \frac{5 - \frac{8 - \frac{12 - \frac{16}{t}}{t}}{t}}{\color{blue}{6 - \frac{8 - \frac{12 - \frac{16}{t}}{t}}{t}}} \]

   if 1e-8 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) 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. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. distribute-neg-frac 100.0%

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

      4. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-/r* 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. rgt-mult-inverse 100.0%

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

   6. Simplified 100.0%

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

      1. div-inv 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. times-frac 100.0%

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

      4. metadata-eval 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. associate-/l/ 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. *-commutative 100.0%

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

      5. distribute-rgt-in 100.0%

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

      6. *-lft-identity 100.0%

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

      7. lft-mult-inverse 100.0%

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

   10. Simplified 100.0%

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

       1. div-inv 100.0%

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

       2. *-un-lft-identity 100.0%

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

       3. times-frac 100.0%

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

       4. metadata-eval 100.0%

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

   12. Applied egg-rr 100.0%

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

       1. associate-/l/ 100.0%

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

       2. associate-*r/ 100.0%

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

       3. metadata-eval 100.0%

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

       4. *-commutative 100.0%

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

       5. distribute-rgt-in 100.0%

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

       6. *-lft-identity 100.0%

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

       7. lft-mult-inverse 100.0%

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

   14. Simplified 100.0%

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

   15. Taylor expanded in t around 0 99.4%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} + 1} \leq 10^{-8}:\\ \;\;\;\;\frac{5 + \frac{\frac{12 - \frac{16}{t}}{t} - 8}{t}}{6 + \frac{\frac{12 - \frac{16}{t}}{t} - 8}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(2 \cdot t\right) \cdot \left(\frac{2}{t + 1} - 2\right)}{2 + \left(2 + \frac{-2}{t + 1}\right) \cdot \left(2 + \frac{2}{-1 - t}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{12 - \frac{16}{t}}{t} - 8}{t}\\ \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} + 1} \leq 10^{-8}:\\ \;\;\;\;\frac{5 + t\_1}{6 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (- (/ (- 12.0 (/ 16.0 t)) t) 8.0) t)))
   (if (<= (/ (/ 2.0 t) (+ (/ 1.0 t) 1.0)) 1e-8)
     (/ (+ 5.0 t_1) (+ 6.0 t_1))
     0.5)))

C

double code(double t) {
	double t_1 = (((12.0 - (16.0 / t)) / t) - 8.0) / t;
	double tmp;
	if (((2.0 / t) / ((1.0 / t) + 1.0)) <= 1e-8) {
		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 - (16.0d0 / t)) / t) - 8.0d0) / t
    if (((2.0d0 / t) / ((1.0d0 / t) + 1.0d0)) <= 1d-8) 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 - (16.0 / t)) / t) - 8.0) / t;
	double tmp;
	if (((2.0 / t) / ((1.0 / t) + 1.0)) <= 1e-8) {
		tmp = (5.0 + t_1) / (6.0 + t_1);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	t_1 = (((12.0 - (16.0 / t)) / t) - 8.0) / t
	tmp = 0
	if ((2.0 / t) / ((1.0 / t) + 1.0)) <= 1e-8:
		tmp = (5.0 + t_1) / (6.0 + t_1)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	t_1 = Float64(Float64(Float64(Float64(12.0 - Float64(16.0 / t)) / t) - 8.0) / t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(Float64(1.0 / t) + 1.0)) <= 1e-8)
		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 - (16.0 / t)) / t) - 8.0) / t;
	tmp = 0.0;
	if (((2.0 / t) / ((1.0 / t) + 1.0)) <= 1e-8)
		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[(N[(12.0 - N[(16.0 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - 8.0), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(N[(1.0 / t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 1e-8], 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 (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 1e-8

   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%

      \[\leadsto \frac{\color{blue}{5 + -1 \cdot \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto \frac{5 + \color{blue}{\left(-\frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}\right)}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      2. unsub-neg 100.0%

         \[\leadsto \frac{\color{blue}{5 - \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      3. mul-1-neg 100.0%

         \[\leadsto \frac{5 - \frac{8 + \color{blue}{\left(-\frac{12 - 16 \cdot \frac{1}{t}}{t}\right)}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      4. unsub-neg 100.0%

         \[\leadsto \frac{5 - \frac{\color{blue}{8 - \frac{12 - 16 \cdot \frac{1}{t}}{t}}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      5. associate-*r/ 100.0%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 - \color{blue}{\frac{16 \cdot 1}{t}}}{t}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      6. metadata-eval 100.0%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 - \frac{\color{blue}{16}}{t}}{t}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   5. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{5 - \frac{8 - \frac{12 - \frac{16}{t}}{t}}{t}}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   6. Taylor expanded in t around -inf 100.0%

      \[\leadsto \frac{5 - \frac{8 - \frac{12 - \frac{16}{t}}{t}}{t}}{\color{blue}{6 + -1 \cdot \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 - \frac{16}{t}}{t}}{t}}{6 + \color{blue}{\left(-\frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}\right)}} \]

      2. unsub-neg 100.0%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 - \frac{16}{t}}{t}}{t}}{\color{blue}{6 - \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}}} \]

      3. mul-1-neg 100.0%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 - \frac{16}{t}}{t}}{t}}{6 - \frac{8 + \color{blue}{\left(-\frac{12 - 16 \cdot \frac{1}{t}}{t}\right)}}{t}} \]

      4. unsub-neg 100.0%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 - \frac{16}{t}}{t}}{t}}{6 - \frac{\color{blue}{8 - \frac{12 - 16 \cdot \frac{1}{t}}{t}}}{t}} \]

      5. associate-*r/ 100.0%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 - \frac{16}{t}}{t}}{t}}{6 - \frac{8 - \frac{12 - \color{blue}{\frac{16 \cdot 1}{t}}}{t}}{t}} \]

      6. metadata-eval 100.0%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 - \frac{16}{t}}{t}}{t}}{6 - \frac{8 - \frac{12 - \frac{\color{blue}{16}}{t}}{t}}{t}} \]

   8. Simplified 100.0%

      \[\leadsto \frac{5 - \frac{8 - \frac{12 - \frac{16}{t}}{t}}{t}}{\color{blue}{6 - \frac{8 - \frac{12 - \frac{16}{t}}{t}}{t}}} \]

   if 1e-8 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) 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 0 99.4%

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

   4. Taylor expanded in t around 0 99.4%

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

      1. *-commutative 99.4%

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

   6. Simplified 99.4%

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

   7. 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} + 1} \leq 10^{-8}:\\ \;\;\;\;\frac{5 + \frac{\frac{12 - \frac{16}{t}}{t} - 8}{t}}{6 + \frac{\frac{12 - \frac{16}{t}}{t} - 8}{t}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} + 1} \leq 10^{-8}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ (/ 1.0 t) 1.0)) 1e-8)
   (+
    0.8333333333333334
    (/
     (-
      (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
      0.2222222222222222)
     t))
   0.5))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / ((1.0 / t) + 1.0)) <= 1e-8) {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((2.0d0 / t) / ((1.0d0 / t) + 1.0d0)) <= 1d-8) then
        tmp = 0.8333333333333334d0 + ((((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t) - 0.2222222222222222d0) / t)
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (((2.0 / t) / ((1.0 / t) + 1.0)) <= 1e-8) {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if ((2.0 / t) / ((1.0 / t) + 1.0)) <= 1e-8:
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(Float64(1.0 / t) + 1.0)) <= 1e-8)
		tmp = Float64(0.8333333333333334 + Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) - 0.2222222222222222) / t));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (((2.0 / t) / ((1.0 / t) + 1.0)) <= 1e-8)
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(N[(1.0 / t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 1e-8], N[(0.8333333333333334 + N[(N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 1e-8

   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%

      \[\leadsto \frac{\color{blue}{5 + -1 \cdot \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto \frac{5 + \color{blue}{\left(-\frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}\right)}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      2. unsub-neg 100.0%

         \[\leadsto \frac{\color{blue}{5 - \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      3. mul-1-neg 100.0%

         \[\leadsto \frac{5 - \frac{8 + \color{blue}{\left(-\frac{12 - 16 \cdot \frac{1}{t}}{t}\right)}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      4. unsub-neg 100.0%

         \[\leadsto \frac{5 - \frac{\color{blue}{8 - \frac{12 - 16 \cdot \frac{1}{t}}{t}}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      5. associate-*r/ 100.0%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 - \color{blue}{\frac{16 \cdot 1}{t}}}{t}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      6. metadata-eval 100.0%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 - \frac{\color{blue}{16}}{t}}{t}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   5. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{5 - \frac{8 - \frac{12 - \frac{16}{t}}{t}}{t}}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   6. Taylor expanded in t around -inf 100.0%

      \[\leadsto \frac{5 - \frac{8 - \frac{12 - \frac{16}{t}}{t}}{t}}{\color{blue}{6 + -1 \cdot \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 - \frac{16}{t}}{t}}{t}}{6 + \color{blue}{\left(-\frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}\right)}} \]

      2. unsub-neg 100.0%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 - \frac{16}{t}}{t}}{t}}{\color{blue}{6 - \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}}} \]

      3. mul-1-neg 100.0%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 - \frac{16}{t}}{t}}{t}}{6 - \frac{8 + \color{blue}{\left(-\frac{12 - 16 \cdot \frac{1}{t}}{t}\right)}}{t}} \]

      4. unsub-neg 100.0%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 - \frac{16}{t}}{t}}{t}}{6 - \frac{\color{blue}{8 - \frac{12 - 16 \cdot \frac{1}{t}}{t}}}{t}} \]

      5. associate-*r/ 100.0%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 - \frac{16}{t}}{t}}{t}}{6 - \frac{8 - \frac{12 - \color{blue}{\frac{16 \cdot 1}{t}}}{t}}{t}} \]

      6. metadata-eval 100.0%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 - \frac{16}{t}}{t}}{t}}{6 - \frac{8 - \frac{12 - \frac{\color{blue}{16}}{t}}{t}}{t}} \]

   8. Simplified 100.0%

      \[\leadsto \frac{5 - \frac{8 - \frac{12 - \frac{16}{t}}{t}}{t}}{\color{blue}{6 - \frac{8 - \frac{12 - \frac{16}{t}}{t}}{t}}} \]

   9. Taylor expanded in t around -inf 100.0%

      \[\leadsto \color{blue}{0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}} \]
   10. Step-by-step derivation

       1. mul-1-neg 100.0%

          \[\leadsto 0.8333333333333334 + \color{blue}{\left(-\frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}\right)} \]

       2. unsub-neg 100.0%

          \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}} \]

       3. mul-1-neg 100.0%

          \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222 + \color{blue}{\left(-\frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}\right)}}{t} \]

       4. unsub-neg 100.0%

          \[\leadsto 0.8333333333333334 - \frac{\color{blue}{0.2222222222222222 - \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}}{t} \]

       5. associate-*r/ 100.0%

          \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \color{blue}{\frac{0.04938271604938271 \cdot 1}{t}}}{t}}{t} \]

       6. metadata-eval 100.0%

          \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{\color{blue}{0.04938271604938271}}{t}}{t}}{t} \]

   11. Simplified 100.0%

       \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}} \]

   if 1e-8 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) 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 0 99.4%

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

   4. Taylor expanded in t around 0 99.4%

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

      1. *-commutative 99.4%

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

   6. Simplified 99.4%

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

   7. 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} + 1} \leq 10^{-8}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.5:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{elif}\;t \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{5 - \frac{8}{t}}{6 - \frac{8}{t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.5)
   (-
    0.8333333333333334
    (/ (+ 0.2222222222222222 (/ -0.037037037037037035 t)) t))
   (if (<= t 2.0) 0.5 (/ (- 5.0 (/ 8.0 t)) (- 6.0 (/ 8.0 t))))))

C

double code(double t) {
	double tmp;
	if (t <= -0.5) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else if (t <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = (5.0 - (8.0 / t)) / (6.0 - (8.0 / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.5d0)) then
        tmp = 0.8333333333333334d0 - ((0.2222222222222222d0 + ((-0.037037037037037035d0) / t)) / t)
    else if (t <= 2.0d0) then
        tmp = 0.5d0
    else
        tmp = (5.0d0 - (8.0d0 / t)) / (6.0d0 - (8.0d0 / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.5) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else if (t <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = (5.0 - (8.0 / t)) / (6.0 - (8.0 / t));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.5:
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t)
	elif t <= 2.0:
		tmp = 0.5
	else:
		tmp = (5.0 - (8.0 / t)) / (6.0 - (8.0 / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.5)
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 + Float64(-0.037037037037037035 / t)) / t));
	elseif (t <= 2.0)
		tmp = 0.5;
	else
		tmp = Float64(Float64(5.0 - Float64(8.0 / t)) / Float64(6.0 - Float64(8.0 / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.5)
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	elseif (t <= 2.0)
		tmp = 0.5;
	else
		tmp = (5.0 - (8.0 / t)) / (6.0 - (8.0 / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.5], N[(0.8333333333333334 - N[(N[(0.2222222222222222 + N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.0], 0.5, N[(N[(5.0 - N[(8.0 / t), $MachinePrecision]), $MachinePrecision] / N[(6.0 - N[(8.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.5

   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 99.6%

      \[\leadsto \frac{\color{blue}{5 + -1 \cdot \frac{8 - 12 \cdot \frac{1}{t}}{t}}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 99.6%

         \[\leadsto \frac{5 + \color{blue}{\left(-\frac{8 - 12 \cdot \frac{1}{t}}{t}\right)}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      2. unsub-neg 99.6%

         \[\leadsto \frac{\color{blue}{5 - \frac{8 - 12 \cdot \frac{1}{t}}{t}}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      3. sub-neg 99.6%

         \[\leadsto \frac{5 - \frac{\color{blue}{8 + \left(-12 \cdot \frac{1}{t}\right)}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      4. associate-*r/ 99.6%

         \[\leadsto \frac{5 - \frac{8 + \left(-\color{blue}{\frac{12 \cdot 1}{t}}\right)}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      5. metadata-eval 99.6%

         \[\leadsto \frac{5 - \frac{8 + \left(-\frac{\color{blue}{12}}{t}\right)}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      6. distribute-neg-frac 99.6%

         \[\leadsto \frac{5 - \frac{8 + \color{blue}{\frac{-12}{t}}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      7. metadata-eval 99.6%

         \[\leadsto \frac{5 - \frac{8 + \frac{\color{blue}{-12}}{t}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   5. Simplified 99.6%

      \[\leadsto \frac{\color{blue}{5 - \frac{8 + \frac{-12}{t}}{t}}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   6. Taylor expanded in t around -inf 99.7%

      \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{\color{blue}{6 + -1 \cdot \frac{8 - 12 \cdot \frac{1}{t}}{t}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 99.7%

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

      2. unsub-neg 99.7%

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

      3. sub-neg 99.7%

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

      4. associate-*r/ 99.7%

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

      5. metadata-eval 99.7%

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

      6. distribute-neg-frac 99.7%

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

      7. metadata-eval 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in t around -inf 99.7%

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

       1. mul-1-neg 99.7%

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

       2. unsub-neg 99.7%

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

       3. sub-neg 99.7%

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

       4. associate-*r/ 99.7%

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

       5. metadata-eval 99.7%

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

       6. distribute-neg-frac 99.7%

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

       7. metadata-eval 99.7%

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

   11. Simplified 99.7%

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

   if -0.5 < t < 2

   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 0 99.4%

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

   4. Taylor expanded in t around 0 99.4%

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

      1. *-commutative 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in t around 0 99.4%

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

   if 2 < 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%

      \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{6 - 8 \cdot \frac{1}{t}}} \]
   4. Step-by-step derivation

      1. associate-*r/ 100.0%

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 - \color{blue}{\frac{8 \cdot 1}{t}}} \]

      2. metadata-eval 100.0%

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 - \frac{\color{blue}{8}}{t}} \]

   5. Simplified 100.0%

      \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{6 - \frac{8}{t}}} \]

   6. Taylor expanded in t around inf 100.0%

      \[\leadsto \frac{\color{blue}{5 - 8 \cdot \frac{1}{t}}}{6 - \frac{8}{t}} \]
   7. Step-by-step derivation

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   8. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{5 - \frac{8}{t}}}{6 - \frac{8}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.5:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{elif}\;t \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{5 - \frac{8}{t}}{6 - \frac{8}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

double code(double t) {
	double tmp;
	if ((t <= -0.5) || !(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 <= (-0.5d0)) .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 <= -0.5) || !(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 <= -0.5) 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 <= -0.5) || !(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 <= -0.5) || ~((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, -0.5], 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 < -0.5 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. Add Preprocessing

   3. Taylor expanded in t around -inf 99.1%

      \[\leadsto \frac{\color{blue}{5 + -1 \cdot \frac{8 - 12 \cdot \frac{1}{t}}{t}}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 99.1%

         \[\leadsto \frac{5 + \color{blue}{\left(-\frac{8 - 12 \cdot \frac{1}{t}}{t}\right)}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      2. unsub-neg 99.1%

         \[\leadsto \frac{\color{blue}{5 - \frac{8 - 12 \cdot \frac{1}{t}}{t}}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      3. sub-neg 99.1%

         \[\leadsto \frac{5 - \frac{\color{blue}{8 + \left(-12 \cdot \frac{1}{t}\right)}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      4. associate-*r/ 99.1%

         \[\leadsto \frac{5 - \frac{8 + \left(-\color{blue}{\frac{12 \cdot 1}{t}}\right)}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      5. metadata-eval 99.1%

         \[\leadsto \frac{5 - \frac{8 + \left(-\frac{\color{blue}{12}}{t}\right)}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      6. distribute-neg-frac 99.1%

         \[\leadsto \frac{5 - \frac{8 + \color{blue}{\frac{-12}{t}}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

      7. metadata-eval 99.1%

         \[\leadsto \frac{5 - \frac{8 + \frac{\color{blue}{-12}}{t}}{t}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   5. Simplified 99.1%

      \[\leadsto \frac{\color{blue}{5 - \frac{8 + \frac{-12}{t}}{t}}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

   6. Taylor expanded in t around -inf 99.2%

      \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{\color{blue}{6 + -1 \cdot \frac{8 - 12 \cdot \frac{1}{t}}{t}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 99.2%

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

      2. unsub-neg 99.2%

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

      3. sub-neg 99.2%

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

      4. associate-*r/ 99.2%

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

      5. metadata-eval 99.2%

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

      6. distribute-neg-frac 99.2%

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

      7. metadata-eval 99.2%

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

   8. Simplified 99.2%

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

   9. Taylor expanded in t around -inf 99.3%

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

       1. mul-1-neg 99.3%

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

       2. unsub-neg 99.3%

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

       3. sub-neg 99.3%

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

       4. associate-*r/ 99.3%

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

       5. metadata-eval 99.3%

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

       6. distribute-neg-frac 99.3%

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

       7. metadata-eval 99.3%

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

   11. Simplified 99.3%

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

   if -0.5 < t < 0.23000000000000001

   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 0 100.0%

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

   4. Taylor expanded in t around 0 100.0%

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

      1. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t around 0 100.0%

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

4. Final simplification 99.6%

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

Alternative 7: 98.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.68\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.49) (not (<= t 0.68)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.49) || !(t <= 0.68)) {
		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 <= 0.68d0))) 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 <= 0.68)) {
		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 <= 0.68):
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.49) || !(t <= 0.68))
		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 <= 0.68)))
		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, 0.68]], $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 0.680000000000000049 < 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 99.4%

      \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{6 - 8 \cdot \frac{1}{t}}} \]
   4. Step-by-step derivation

      1. associate-*r/ 99.4%

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 - \color{blue}{\frac{8 \cdot 1}{t}}} \]

      2. metadata-eval 99.4%

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 - \frac{\color{blue}{8}}{t}} \]

   5. Simplified 99.4%

      \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{6 - \frac{8}{t}}} \]

   6. Taylor expanded in t around inf 99.6%

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

      1. associate-*r/ 99.6%

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

      2. metadata-eval 99.6%

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

   8. Simplified 99.6%

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

   if -0.48999999999999999 < t < 0.680000000000000049

   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 0 99.4%

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

   4. Taylor expanded in t around 0 99.4%

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

      1. *-commutative 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in t around 0 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.68\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.4% accurate, 4.6× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.33) 0.8333333333333334 (if (<= t 1.0) 0.5 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.33) {
		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 <= (-0.33d0)) 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 <= -0.33) {
		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 <= -0.33:
		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 <= -0.33)
		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 <= -0.33)
		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, -0.33], 0.8333333333333334, If[LessEqual[t, 1.0], 0.5, 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.330000000000000016 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. Add Preprocessing

   3. Taylor expanded in t around inf 99.4%

      \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{6 - 8 \cdot \frac{1}{t}}} \]
   4. Step-by-step derivation

      1. associate-*r/ 99.4%

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 - \color{blue}{\frac{8 \cdot 1}{t}}} \]

      2. metadata-eval 99.4%

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{6 - \frac{\color{blue}{8}}{t}} \]

   5. Simplified 99.4%

      \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{6 - \frac{8}{t}}} \]

   6. Taylor expanded in t around inf 98.6%

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

   if -0.330000000000000016 < t < 1

   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 0 99.4%

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

   4. Taylor expanded in t around 0 99.4%

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

      1. *-commutative 99.4%

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

   6. Simplified 99.4%

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

   7. 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%

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

Alternative 9: 59.5% 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 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 0 59.7%

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

4. Taylor expanded in t around 0 52.9%

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

   1. *-commutative 52.9%

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

6. Simplified 52.9%

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

7. Taylor expanded in t around 0 61.1%

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

8. Final simplification 61.1%

   \[\leadsto 0.5 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024072 
(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))))))))