Kahan p13 Example 3

Percentage Accurate: 100.0% → 100.0%

Time: 11.3s

Alternatives: 10

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	return 1.0 - (1.0 / (2.0 + (t_1 * 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 + (1.0d0 / t)))
    code = 1.0d0 - (1.0d0 / (2.0d0 + (t_1 * t_1)))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(t)
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	tmp = 1.0 - (1.0 / (2.0 + (t_1 * t_1)));
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]}, N[(1.0 - N[(1.0 / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	return 1.0 - (1.0 / (2.0 + (t_1 * 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 + (1.0d0 / t)))
    code = 1.0d0 - (1.0d0 / (2.0d0 + (t_1 * t_1)))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(t)
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	tmp = 1.0 - (1.0 / (2.0 + (t_1 * t_1)));
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]}, N[(1.0 - N[(1.0 / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{-1}{2 + \sqrt[3]{{\left(2 + \frac{2}{-1 - t}\right)}^{6}}} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (+ 1.0 (/ -1.0 (+ 2.0 (cbrt (pow (+ 2.0 (/ 2.0 (- -1.0 t))) 6.0))))))

C

double code(double t) {
	return 1.0 + (-1.0 / (2.0 + cbrt(pow((2.0 + (2.0 / (-1.0 - t))), 6.0))));
}

Java

public static double code(double t) {
	return 1.0 + (-1.0 / (2.0 + Math.cbrt(Math.pow((2.0 + (2.0 / (-1.0 - t))), 6.0))));
}

Julia

function code(t)
	return Float64(1.0 + Float64(-1.0 / Float64(2.0 + cbrt((Float64(2.0 + Float64(2.0 / Float64(-1.0 - t))) ^ 6.0)))))
end

Wolfram

code[t_] := N[(1.0 + N[(-1.0 / N[(2.0 + N[Power[N[Power[N[(2.0 + N[(2.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 6.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. sub-neg 99.7%

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

   2. distribute-neg-frac 99.7%

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

   3. metadata-eval 99.7%

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

3. Simplified 100.0%

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

   1. add-cbrt-cube 100.0%

      \[\leadsto 1 + \frac{-1}{2 + \color{blue}{\sqrt[3]{\left(\left(\left(2 - \frac{-2}{-1 - t}\right) \cdot \left(2 - \frac{-2}{-1 - t}\right)\right) \cdot \left(\left(2 - \frac{-2}{-1 - t}\right) \cdot \left(2 - \frac{-2}{-1 - t}\right)\right)\right) \cdot \left(\left(2 - \frac{-2}{-1 - t}\right) \cdot \left(2 - \frac{-2}{-1 - t}\right)\right)}}} \]

   2. pow1/3 100.0%

      \[\leadsto 1 + \frac{-1}{2 + \color{blue}{{\left(\left(\left(\left(2 - \frac{-2}{-1 - t}\right) \cdot \left(2 - \frac{-2}{-1 - t}\right)\right) \cdot \left(\left(2 - \frac{-2}{-1 - t}\right) \cdot \left(2 - \frac{-2}{-1 - t}\right)\right)\right) \cdot \left(\left(2 - \frac{-2}{-1 - t}\right) \cdot \left(2 - \frac{-2}{-1 - t}\right)\right)\right)}^{0.3333333333333333}}} \]

   3. pow3 100.0%

      \[\leadsto 1 + \frac{-1}{2 + {\color{blue}{\left({\left(\left(2 - \frac{-2}{-1 - t}\right) \cdot \left(2 - \frac{-2}{-1 - t}\right)\right)}^{3}\right)}}^{0.3333333333333333}} \]

   4. pow2 100.0%

      \[\leadsto 1 + \frac{-1}{2 + {\left({\color{blue}{\left({\left(2 - \frac{-2}{-1 - t}\right)}^{2}\right)}}^{3}\right)}^{0.3333333333333333}} \]

   5. pow-pow 100.0%

      \[\leadsto 1 + \frac{-1}{2 + {\color{blue}{\left({\left(2 - \frac{-2}{-1 - t}\right)}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333}} \]

   6. div-inv 100.0%

      \[\leadsto 1 + \frac{-1}{2 + {\left({\left(2 - \color{blue}{-2 \cdot \frac{1}{-1 - t}}\right)}^{\left(2 \cdot 3\right)}\right)}^{0.3333333333333333}} \]

   7. cancel-sign-sub-inv 100.0%

      \[\leadsto 1 + \frac{-1}{2 + {\left({\color{blue}{\left(2 + \left(--2\right) \cdot \frac{1}{-1 - t}\right)}}^{\left(2 \cdot 3\right)}\right)}^{0.3333333333333333}} \]

   8. metadata-eval 100.0%

      \[\leadsto 1 + \frac{-1}{2 + {\left({\left(2 + \color{blue}{2} \cdot \frac{1}{-1 - t}\right)}^{\left(2 \cdot 3\right)}\right)}^{0.3333333333333333}} \]

   9. div-inv 100.0%

      \[\leadsto 1 + \frac{-1}{2 + {\left({\left(2 + \color{blue}{\frac{2}{-1 - t}}\right)}^{\left(2 \cdot 3\right)}\right)}^{0.3333333333333333}} \]

   10. metadata-eval 100.0%

       \[\leadsto 1 + \frac{-1}{2 + {\left({\left(2 + \frac{2}{-1 - t}\right)}^{\color{blue}{6}}\right)}^{0.3333333333333333}} \]

6. Applied egg-rr 100.0%

   \[\leadsto 1 + \frac{-1}{2 + \color{blue}{{\left({\left(2 + \frac{2}{-1 - t}\right)}^{6}\right)}^{0.3333333333333333}}} \]
7. Step-by-step derivation

   1. unpow1/3 100.0%

      \[\leadsto 1 + \frac{-1}{2 + \color{blue}{\sqrt[3]{{\left(2 + \frac{2}{-1 - t}\right)}^{6}}}} \]

8. Simplified 100.0%

   \[\leadsto 1 + \frac{-1}{2 + \color{blue}{\sqrt[3]{{\left(2 + \frac{2}{-1 - t}\right)}^{6}}}} \]

9. Final simplification 100.0%

   \[\leadsto 1 + \frac{-1}{2 + \sqrt[3]{{\left(2 + \frac{2}{-1 - t}\right)}^{6}}} \]
10. Add Preprocessing

Alternative 2: 99.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.65 \lor \neg \left(t \leq 1\right):\\ \;\;\;\;1 + \frac{-1}{6 + \frac{-8 + \frac{4}{t}}{1 + t}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{2 + t \cdot \left(4 + \frac{4}{-1 - t}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.65) (not (<= t 1.0)))
   (+ 1.0 (/ -1.0 (+ 6.0 (/ (+ -8.0 (/ 4.0 t)) (+ 1.0 t)))))
   (+ 1.0 (/ -1.0 (+ 2.0 (* t (+ 4.0 (/ 4.0 (- -1.0 t)))))))))

C

double code(double t) {
	double tmp;
	if ((t <= -0.65) || !(t <= 1.0)) {
		tmp = 1.0 + (-1.0 / (6.0 + ((-8.0 + (4.0 / t)) / (1.0 + t))));
	} else {
		tmp = 1.0 + (-1.0 / (2.0 + (t * (4.0 + (4.0 / (-1.0 - t))))));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.65d0)) .or. (.not. (t <= 1.0d0))) then
        tmp = 1.0d0 + ((-1.0d0) / (6.0d0 + (((-8.0d0) + (4.0d0 / t)) / (1.0d0 + t))))
    else
        tmp = 1.0d0 + ((-1.0d0) / (2.0d0 + (t * (4.0d0 + (4.0d0 / ((-1.0d0) - t))))))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.65) || !(t <= 1.0)) {
		tmp = 1.0 + (-1.0 / (6.0 + ((-8.0 + (4.0 / t)) / (1.0 + t))));
	} else {
		tmp = 1.0 + (-1.0 / (2.0 + (t * (4.0 + (4.0 / (-1.0 - t))))));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.65) or not (t <= 1.0):
		tmp = 1.0 + (-1.0 / (6.0 + ((-8.0 + (4.0 / t)) / (1.0 + t))))
	else:
		tmp = 1.0 + (-1.0 / (2.0 + (t * (4.0 + (4.0 / (-1.0 - t))))))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.65) || !(t <= 1.0))
		tmp = Float64(1.0 + Float64(-1.0 / Float64(6.0 + Float64(Float64(-8.0 + Float64(4.0 / t)) / Float64(1.0 + t)))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(2.0 + Float64(t * Float64(4.0 + Float64(4.0 / Float64(-1.0 - t)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.65) || ~((t <= 1.0)))
		tmp = 1.0 + (-1.0 / (6.0 + ((-8.0 + (4.0 / t)) / (1.0 + t))));
	else
		tmp = 1.0 + (-1.0 / (2.0 + (t * (4.0 + (4.0 / (-1.0 - t))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.65], N[Not[LessEqual[t, 1.0]], $MachinePrecision]], N[(1.0 + N[(-1.0 / N[(6.0 + N[(N[(-8.0 + N[(4.0 / t), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(2.0 + N[(t * N[(4.0 + N[(4.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.650000000000000022 or 1 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative 100.0%

         \[\leadsto 1 + \frac{-1}{6 + \color{blue}{\left(\frac{2}{-1 - t} - -4\right) \cdot \frac{2}{-1 - t}}} \]

      2. frac-2neg 100.0%

         \[\leadsto 1 + \frac{-1}{6 + \left(\frac{2}{-1 - t} - -4\right) \cdot \color{blue}{\frac{-2}{-\left(-1 - t\right)}}} \]

      3. metadata-eval 100.0%

         \[\leadsto 1 + \frac{-1}{6 + \left(\frac{2}{-1 - t} - -4\right) \cdot \frac{\color{blue}{-2}}{-\left(-1 - t\right)}} \]

      4. associate-*r/ 100.0%

         \[\leadsto 1 + \frac{-1}{6 + \color{blue}{\frac{\left(\frac{2}{-1 - t} - -4\right) \cdot -2}{-\left(-1 - t\right)}}} \]

      5. sub-neg 100.0%

         \[\leadsto 1 + \frac{-1}{6 + \frac{\color{blue}{\left(\frac{2}{-1 - t} + \left(--4\right)\right)} \cdot -2}{-\left(-1 - t\right)}} \]

      6. metadata-eval 100.0%

         \[\leadsto 1 + \frac{-1}{6 + \frac{\left(\frac{2}{-1 - t} + \color{blue}{4}\right) \cdot -2}{-\left(-1 - t\right)}} \]

   6. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

         \[\leadsto 1 + \frac{-1}{6 + \frac{\color{blue}{-2 \cdot \left(\frac{2}{-1 - t} + 4\right)}}{-\left(-1 - t\right)}} \]

      2. +-commutative 100.0%

         \[\leadsto 1 + \frac{-1}{6 + \frac{-2 \cdot \color{blue}{\left(4 + \frac{2}{-1 - t}\right)}}{-\left(-1 - t\right)}} \]

      3. distribute-rgt-in 100.0%

         \[\leadsto 1 + \frac{-1}{6 + \frac{\color{blue}{4 \cdot -2 + \frac{2}{-1 - t} \cdot -2}}{-\left(-1 - t\right)}} \]

      4. metadata-eval 100.0%

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

      5. associate-*l/ 100.0%

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

      6. metadata-eval 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. remove-double-neg 100.0%

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

      11. metadata-eval 100.0%

          \[\leadsto 1 + \frac{-1}{6 + \frac{-8 + \frac{-4}{-1 - t}}{t + \color{blue}{1}}} \]

   8. Simplified 100.0%

      \[\leadsto 1 + \frac{-1}{6 + \color{blue}{\frac{-8 + \frac{-4}{-1 - t}}{t + 1}}} \]

   9. Taylor expanded in t around inf 99.3%

      \[\leadsto 1 + \frac{-1}{6 + \frac{\color{blue}{4 \cdot \frac{1}{t} - 8}}{t + 1}} \]
   10. Step-by-step derivation

       1. sub-neg 99.3%

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

       2. associate-*r/ 99.3%

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

       3. metadata-eval 99.3%

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

       4. metadata-eval 99.3%

          \[\leadsto 1 + \frac{-1}{6 + \frac{\frac{4}{t} + \color{blue}{-8}}{t + 1}} \]

   11. Simplified 99.3%

       \[\leadsto 1 + \frac{-1}{6 + \frac{\color{blue}{\frac{4}{t} + -8}}{t + 1}} \]

   if -0.650000000000000022 < t < 1

   1. Initial program 99.3%

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

      1. sub-neg 99.3%

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

      2. distribute-neg-frac 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 99.8%

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

      1. *-commutative 99.8%

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

      2. div-inv 99.8%

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

      3. cancel-sign-sub-inv 99.8%

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

      4. metadata-eval 99.8%

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

      5. div-inv 99.8%

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

      6. distribute-lft-in 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. distribute-lft-out 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-*l* 99.8%

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

      4. distribute-lft-in 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-*r/ 99.8%

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

      7. metadata-eval 99.8%

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

   9. Simplified 99.8%

      \[\leadsto 1 + \frac{-1}{2 + \color{blue}{t \cdot \left(4 + \frac{4}{-1 - t}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.65 \lor \neg \left(t \leq 1\right):\\ \;\;\;\;1 + \frac{-1}{6 + \frac{-8 + \frac{4}{t}}{1 + t}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{2 + t \cdot \left(4 + \frac{4}{-1 - t}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.15:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.88:\\ \;\;\;\;1 + \frac{-1}{2 + t \cdot \left(4 + \frac{4}{-1 - t}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -1.15)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (if (<= t 0.88)
     (+ 1.0 (/ -1.0 (+ 2.0 (* t (+ 4.0 (/ 4.0 (- -1.0 t)))))))
     (+ 1.0 (+ -0.16666666666666666 (/ -0.2222222222222222 t))))))

C

double code(double t) {
	double tmp;
	if (t <= -1.15) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 0.88) {
		tmp = 1.0 + (-1.0 / (2.0 + (t * (4.0 + (4.0 / (-1.0 - t))))));
	} else {
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.15d0)) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else if (t <= 0.88d0) then
        tmp = 1.0d0 + ((-1.0d0) / (2.0d0 + (t * (4.0d0 + (4.0d0 / ((-1.0d0) - t))))))
    else
        tmp = 1.0d0 + ((-0.16666666666666666d0) + ((-0.2222222222222222d0) / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -1.15) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 0.88) {
		tmp = 1.0 + (-1.0 / (2.0 + (t * (4.0 + (4.0 / (-1.0 - t))))));
	} else {
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -1.15:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	elif t <= 0.88:
		tmp = 1.0 + (-1.0 / (2.0 + (t * (4.0 + (4.0 / (-1.0 - t))))))
	else:
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -1.15)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	elseif (t <= 0.88)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(2.0 + Float64(t * Float64(4.0 + Float64(4.0 / Float64(-1.0 - t)))))));
	else
		tmp = Float64(1.0 + Float64(-0.16666666666666666 + Float64(-0.2222222222222222 / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -1.15)
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	elseif (t <= 0.88)
		tmp = 1.0 + (-1.0 / (2.0 + (t * (4.0 + (4.0 / (-1.0 - t))))));
	else
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -1.15], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.88], N[(1.0 + N[(-1.0 / N[(2.0 + N[(t * N[(4.0 + N[(4.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.16666666666666666 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.1499999999999999

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 98.8%

      \[\leadsto 1 + \color{blue}{\left(-\left(0.16666666666666666 + 0.2222222222222222 \cdot \frac{1}{t}\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-neg-in 98.8%

         \[\leadsto 1 + \color{blue}{\left(\left(-0.16666666666666666\right) + \left(-0.2222222222222222 \cdot \frac{1}{t}\right)\right)} \]

      2. metadata-eval 98.8%

         \[\leadsto 1 + \left(\color{blue}{-0.16666666666666666} + \left(-0.2222222222222222 \cdot \frac{1}{t}\right)\right) \]

      3. associate-*r/ 98.8%

         \[\leadsto 1 + \left(-0.16666666666666666 + \left(-\color{blue}{\frac{0.2222222222222222 \cdot 1}{t}}\right)\right) \]

      4. metadata-eval 98.8%

         \[\leadsto 1 + \left(-0.16666666666666666 + \left(-\frac{\color{blue}{0.2222222222222222}}{t}\right)\right) \]

      5. distribute-neg-frac 98.8%

         \[\leadsto 1 + \left(-0.16666666666666666 + \color{blue}{\frac{-0.2222222222222222}{t}}\right) \]

      6. metadata-eval 98.8%

         \[\leadsto 1 + \left(-0.16666666666666666 + \frac{\color{blue}{-0.2222222222222222}}{t}\right) \]

   7. Simplified 98.8%

      \[\leadsto 1 + \color{blue}{\left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)} \]

   8. Taylor expanded in t around 0 98.8%

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

      1. associate-*r/ 98.8%

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

      2. metadata-eval 98.8%

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

   10. Simplified 98.8%

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

   if -1.1499999999999999 < t < 0.880000000000000004

   1. Initial program 99.3%

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

      1. sub-neg 99.3%

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

      2. distribute-neg-frac 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 99.8%

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

      1. *-commutative 99.8%

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

      2. div-inv 99.8%

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

      3. cancel-sign-sub-inv 99.8%

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

      4. metadata-eval 99.8%

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

      5. div-inv 99.8%

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

      6. distribute-lft-in 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. distribute-lft-out 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-*l* 99.8%

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

      4. distribute-lft-in 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-*r/ 99.8%

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

      7. metadata-eval 99.8%

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

   9. Simplified 99.8%

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

   if 0.880000000000000004 < t

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 99.1%

      \[\leadsto 1 + \color{blue}{\left(-\left(0.16666666666666666 + 0.2222222222222222 \cdot \frac{1}{t}\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-neg-in 99.1%

         \[\leadsto 1 + \color{blue}{\left(\left(-0.16666666666666666\right) + \left(-0.2222222222222222 \cdot \frac{1}{t}\right)\right)} \]

      2. metadata-eval 99.1%

         \[\leadsto 1 + \left(\color{blue}{-0.16666666666666666} + \left(-0.2222222222222222 \cdot \frac{1}{t}\right)\right) \]

      3. associate-*r/ 99.1%

         \[\leadsto 1 + \left(-0.16666666666666666 + \left(-\color{blue}{\frac{0.2222222222222222 \cdot 1}{t}}\right)\right) \]

      4. metadata-eval 99.1%

         \[\leadsto 1 + \left(-0.16666666666666666 + \left(-\frac{\color{blue}{0.2222222222222222}}{t}\right)\right) \]

      5. distribute-neg-frac 99.1%

         \[\leadsto 1 + \left(-0.16666666666666666 + \color{blue}{\frac{-0.2222222222222222}{t}}\right) \]

      6. metadata-eval 99.1%

         \[\leadsto 1 + \left(-0.16666666666666666 + \frac{\color{blue}{-0.2222222222222222}}{t}\right) \]

   7. Simplified 99.1%

      \[\leadsto 1 + \color{blue}{\left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.15:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.88:\\ \;\;\;\;1 + \frac{-1}{2 + t \cdot \left(4 + \frac{4}{-1 - t}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.6:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.68:\\ \;\;\;\;1 + \frac{-1}{2 + t \cdot \left(t \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.6)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (if (<= t 0.68)
     (+ 1.0 (/ -1.0 (+ 2.0 (* t (* t 4.0)))))
     (+ 1.0 (+ -0.16666666666666666 (/ -0.2222222222222222 t))))))

C

double code(double t) {
	double tmp;
	if (t <= -0.6) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 0.68) {
		tmp = 1.0 + (-1.0 / (2.0 + (t * (t * 4.0))));
	} else {
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.6d0)) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else if (t <= 0.68d0) then
        tmp = 1.0d0 + ((-1.0d0) / (2.0d0 + (t * (t * 4.0d0))))
    else
        tmp = 1.0d0 + ((-0.16666666666666666d0) + ((-0.2222222222222222d0) / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.6) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 0.68) {
		tmp = 1.0 + (-1.0 / (2.0 + (t * (t * 4.0))));
	} else {
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.6:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	elif t <= 0.68:
		tmp = 1.0 + (-1.0 / (2.0 + (t * (t * 4.0))))
	else:
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.6)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	elseif (t <= 0.68)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(2.0 + Float64(t * Float64(t * 4.0)))));
	else
		tmp = Float64(1.0 + Float64(-0.16666666666666666 + Float64(-0.2222222222222222 / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.6)
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	elseif (t <= 0.68)
		tmp = 1.0 + (-1.0 / (2.0 + (t * (t * 4.0))));
	else
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.6], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.68], N[(1.0 + N[(-1.0 / N[(2.0 + N[(t * N[(t * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.16666666666666666 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.599999999999999978

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 98.8%

      \[\leadsto 1 + \color{blue}{\left(-\left(0.16666666666666666 + 0.2222222222222222 \cdot \frac{1}{t}\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-neg-in 98.8%

         \[\leadsto 1 + \color{blue}{\left(\left(-0.16666666666666666\right) + \left(-0.2222222222222222 \cdot \frac{1}{t}\right)\right)} \]

      2. metadata-eval 98.8%

         \[\leadsto 1 + \left(\color{blue}{-0.16666666666666666} + \left(-0.2222222222222222 \cdot \frac{1}{t}\right)\right) \]

      3. associate-*r/ 98.8%

         \[\leadsto 1 + \left(-0.16666666666666666 + \left(-\color{blue}{\frac{0.2222222222222222 \cdot 1}{t}}\right)\right) \]

      4. metadata-eval 98.8%

         \[\leadsto 1 + \left(-0.16666666666666666 + \left(-\frac{\color{blue}{0.2222222222222222}}{t}\right)\right) \]

      5. distribute-neg-frac 98.8%

         \[\leadsto 1 + \left(-0.16666666666666666 + \color{blue}{\frac{-0.2222222222222222}{t}}\right) \]

      6. metadata-eval 98.8%

         \[\leadsto 1 + \left(-0.16666666666666666 + \frac{\color{blue}{-0.2222222222222222}}{t}\right) \]

   7. Simplified 98.8%

      \[\leadsto 1 + \color{blue}{\left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)} \]

   8. Taylor expanded in t around 0 98.8%

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

      1. associate-*r/ 98.8%

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

      2. metadata-eval 98.8%

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

   10. Simplified 98.8%

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

   if -0.599999999999999978 < t < 0.680000000000000049

   1. Initial program 99.3%

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

      1. sub-neg 99.3%

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

      2. distribute-neg-frac 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 99.8%

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

      1. *-commutative 99.8%

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

      2. div-inv 99.8%

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

      3. cancel-sign-sub-inv 99.8%

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

      4. metadata-eval 99.8%

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

      5. div-inv 99.8%

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

      6. distribute-lft-in 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. distribute-lft-out 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-*l* 99.8%

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

      4. distribute-lft-in 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-*r/ 99.8%

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

      7. metadata-eval 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in t around 0 99.8%

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

       1. *-commutative 99.8%

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

   12. Simplified 99.8%

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

   if 0.680000000000000049 < t

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 99.1%

      \[\leadsto 1 + \color{blue}{\left(-\left(0.16666666666666666 + 0.2222222222222222 \cdot \frac{1}{t}\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-neg-in 99.1%

         \[\leadsto 1 + \color{blue}{\left(\left(-0.16666666666666666\right) + \left(-0.2222222222222222 \cdot \frac{1}{t}\right)\right)} \]

      2. metadata-eval 99.1%

         \[\leadsto 1 + \left(\color{blue}{-0.16666666666666666} + \left(-0.2222222222222222 \cdot \frac{1}{t}\right)\right) \]

      3. associate-*r/ 99.1%

         \[\leadsto 1 + \left(-0.16666666666666666 + \left(-\color{blue}{\frac{0.2222222222222222 \cdot 1}{t}}\right)\right) \]

      4. metadata-eval 99.1%

         \[\leadsto 1 + \left(-0.16666666666666666 + \left(-\frac{\color{blue}{0.2222222222222222}}{t}\right)\right) \]

      5. distribute-neg-frac 99.1%

         \[\leadsto 1 + \left(-0.16666666666666666 + \color{blue}{\frac{-0.2222222222222222}{t}}\right) \]

      6. metadata-eval 99.1%

         \[\leadsto 1 + \left(-0.16666666666666666 + \frac{\color{blue}{-0.2222222222222222}}{t}\right) \]

   7. Simplified 99.1%

      \[\leadsto 1 + \color{blue}{\left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.6:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.68:\\ \;\;\;\;1 + \frac{-1}{2 + t \cdot \left(t \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double t) {
	double t_1 = 2.0 + (-2.0 / (t - -1.0));
	return 1.0 + (-1.0 / (2.0 + (t_1 * 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)))
    code = 1.0d0 + ((-1.0d0) / (2.0d0 + (t_1 * t_1)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sub-neg 99.7%

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

   2. distribute-neg-frac 99.7%

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

   3. metadata-eval 99.7%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 6: 99.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.48:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.65:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.48)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (if (<= t 0.65)
     0.5
     (+ 1.0 (+ -0.16666666666666666 (/ -0.2222222222222222 t))))))

C

double code(double t) {
	double tmp;
	if (t <= -0.48) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 0.65) {
		tmp = 0.5;
	} else {
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.48d0)) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else if (t <= 0.65d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 + ((-0.16666666666666666d0) + ((-0.2222222222222222d0) / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.48) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 0.65) {
		tmp = 0.5;
	} else {
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.48:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	elif t <= 0.65:
		tmp = 0.5
	else:
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.48)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	elseif (t <= 0.65)
		tmp = 0.5;
	else
		tmp = Float64(1.0 + Float64(-0.16666666666666666 + Float64(-0.2222222222222222 / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.48)
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	elseif (t <= 0.65)
		tmp = 0.5;
	else
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.48], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.65], 0.5, N[(1.0 + N[(-0.16666666666666666 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.47999999999999998

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 98.8%

      \[\leadsto 1 + \color{blue}{\left(-\left(0.16666666666666666 + 0.2222222222222222 \cdot \frac{1}{t}\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-neg-in 98.8%

         \[\leadsto 1 + \color{blue}{\left(\left(-0.16666666666666666\right) + \left(-0.2222222222222222 \cdot \frac{1}{t}\right)\right)} \]

      2. metadata-eval 98.8%

         \[\leadsto 1 + \left(\color{blue}{-0.16666666666666666} + \left(-0.2222222222222222 \cdot \frac{1}{t}\right)\right) \]

      3. associate-*r/ 98.8%

         \[\leadsto 1 + \left(-0.16666666666666666 + \left(-\color{blue}{\frac{0.2222222222222222 \cdot 1}{t}}\right)\right) \]

      4. metadata-eval 98.8%

         \[\leadsto 1 + \left(-0.16666666666666666 + \left(-\frac{\color{blue}{0.2222222222222222}}{t}\right)\right) \]

      5. distribute-neg-frac 98.8%

         \[\leadsto 1 + \left(-0.16666666666666666 + \color{blue}{\frac{-0.2222222222222222}{t}}\right) \]

      6. metadata-eval 98.8%

         \[\leadsto 1 + \left(-0.16666666666666666 + \frac{\color{blue}{-0.2222222222222222}}{t}\right) \]

   7. Simplified 98.8%

      \[\leadsto 1 + \color{blue}{\left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)} \]

   8. Taylor expanded in t around 0 98.8%

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

      1. associate-*r/ 98.8%

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

      2. metadata-eval 98.8%

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

   10. Simplified 98.8%

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

   if -0.47999999999999998 < t < 0.650000000000000022

   1. Initial program 99.3%

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

      1. sub-neg 99.3%

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

      2. distribute-neg-frac 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 99.8%

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

   6. Taylor expanded in t around 0 99.8%

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

   7. Taylor expanded in t around 0 99.1%

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

   if 0.650000000000000022 < t

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 99.1%

      \[\leadsto 1 + \color{blue}{\left(-\left(0.16666666666666666 + 0.2222222222222222 \cdot \frac{1}{t}\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-neg-in 99.1%

         \[\leadsto 1 + \color{blue}{\left(\left(-0.16666666666666666\right) + \left(-0.2222222222222222 \cdot \frac{1}{t}\right)\right)} \]

      2. metadata-eval 99.1%

         \[\leadsto 1 + \left(\color{blue}{-0.16666666666666666} + \left(-0.2222222222222222 \cdot \frac{1}{t}\right)\right) \]

      3. associate-*r/ 99.1%

         \[\leadsto 1 + \left(-0.16666666666666666 + \left(-\color{blue}{\frac{0.2222222222222222 \cdot 1}{t}}\right)\right) \]

      4. metadata-eval 99.1%

         \[\leadsto 1 + \left(-0.16666666666666666 + \left(-\frac{\color{blue}{0.2222222222222222}}{t}\right)\right) \]

      5. distribute-neg-frac 99.1%

         \[\leadsto 1 + \left(-0.16666666666666666 + \color{blue}{\frac{-0.2222222222222222}{t}}\right) \]

      6. metadata-eval 99.1%

         \[\leadsto 1 + \left(-0.16666666666666666 + \frac{\color{blue}{-0.2222222222222222}}{t}\right) \]

   7. Simplified 99.1%

      \[\leadsto 1 + \color{blue}{\left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.48:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.65:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 100.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{-1}{6 + \frac{-8 + \frac{-4}{-1 - t}}{1 + t}} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (+ 1.0 (/ -1.0 (+ 6.0 (/ (+ -8.0 (/ -4.0 (- -1.0 t))) (+ 1.0 t))))))

C

double code(double t) {
	return 1.0 + (-1.0 / (6.0 + ((-8.0 + (-4.0 / (-1.0 - t))) / (1.0 + t))));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = 1.0d0 + ((-1.0d0) / (6.0d0 + (((-8.0d0) + ((-4.0d0) / ((-1.0d0) - t))) / (1.0d0 + t))))
end function

Java

public static double code(double t) {
	return 1.0 + (-1.0 / (6.0 + ((-8.0 + (-4.0 / (-1.0 - t))) / (1.0 + t))));
}

Python

def code(t):
	return 1.0 + (-1.0 / (6.0 + ((-8.0 + (-4.0 / (-1.0 - t))) / (1.0 + t))))

Julia

function code(t)
	return Float64(1.0 + Float64(-1.0 / Float64(6.0 + Float64(Float64(-8.0 + Float64(-4.0 / Float64(-1.0 - t))) / Float64(1.0 + t)))))
end

MATLAB

function tmp = code(t)
	tmp = 1.0 + (-1.0 / (6.0 + ((-8.0 + (-4.0 / (-1.0 - t))) / (1.0 + t))));
end

Wolfram

code[t_] := N[(1.0 + N[(-1.0 / N[(6.0 + N[(N[(-8.0 + N[(-4.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

3. Simplified 100.0%

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

   1. *-commutative 100.0%

      \[\leadsto 1 + \frac{-1}{6 + \color{blue}{\left(\frac{2}{-1 - t} - -4\right) \cdot \frac{2}{-1 - t}}} \]

   2. frac-2neg 100.0%

      \[\leadsto 1 + \frac{-1}{6 + \left(\frac{2}{-1 - t} - -4\right) \cdot \color{blue}{\frac{-2}{-\left(-1 - t\right)}}} \]

   3. metadata-eval 100.0%

      \[\leadsto 1 + \frac{-1}{6 + \left(\frac{2}{-1 - t} - -4\right) \cdot \frac{\color{blue}{-2}}{-\left(-1 - t\right)}} \]

   4. associate-*r/ 100.0%

      \[\leadsto 1 + \frac{-1}{6 + \color{blue}{\frac{\left(\frac{2}{-1 - t} - -4\right) \cdot -2}{-\left(-1 - t\right)}}} \]

   5. sub-neg 100.0%

      \[\leadsto 1 + \frac{-1}{6 + \frac{\color{blue}{\left(\frac{2}{-1 - t} + \left(--4\right)\right)} \cdot -2}{-\left(-1 - t\right)}} \]

   6. metadata-eval 100.0%

      \[\leadsto 1 + \frac{-1}{6 + \frac{\left(\frac{2}{-1 - t} + \color{blue}{4}\right) \cdot -2}{-\left(-1 - t\right)}} \]

6. Applied egg-rr 100.0%

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

   1. *-commutative 100.0%

      \[\leadsto 1 + \frac{-1}{6 + \frac{\color{blue}{-2 \cdot \left(\frac{2}{-1 - t} + 4\right)}}{-\left(-1 - t\right)}} \]

   2. +-commutative 100.0%

      \[\leadsto 1 + \frac{-1}{6 + \frac{-2 \cdot \color{blue}{\left(4 + \frac{2}{-1 - t}\right)}}{-\left(-1 - t\right)}} \]

   3. distribute-rgt-in 100.0%

      \[\leadsto 1 + \frac{-1}{6 + \frac{\color{blue}{4 \cdot -2 + \frac{2}{-1 - t} \cdot -2}}{-\left(-1 - t\right)}} \]

   4. metadata-eval 100.0%

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

   5. associate-*l/ 100.0%

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

   6. metadata-eval 100.0%

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

   7. sub-neg 100.0%

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

   8. +-commutative 100.0%

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

   9. distribute-neg-in 100.0%

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

   10. remove-double-neg 100.0%

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

   11. metadata-eval 100.0%

       \[\leadsto 1 + \frac{-1}{6 + \frac{-8 + \frac{-4}{-1 - t}}{t + \color{blue}{1}}} \]

8. Simplified 100.0%

   \[\leadsto 1 + \frac{-1}{6 + \color{blue}{\frac{-8 + \frac{-4}{-1 - t}}{t + 1}}} \]

9. Final simplification 100.0%

   \[\leadsto 1 + \frac{-1}{6 + \frac{-8 + \frac{-4}{-1 - t}}{1 + t}} \]
10. Add Preprocessing

Alternative 8: 99.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.48 \lor \neg \left(t \leq 0.65\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.48) (not (<= t 0.65)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.48) || !(t <= 0.65)) {
		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 <= (-0.48d0)) .or. (.not. (t <= 0.65d0))) 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 <= -0.48) || !(t <= 0.65)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.48) or not (t <= 0.65):
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.48) || !(t <= 0.65))
		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 <= -0.48) || ~((t <= 0.65)))
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.48], N[Not[LessEqual[t, 0.65]], $MachinePrecision]], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if t < -0.47999999999999998 or 0.650000000000000022 < t

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 98.9%

      \[\leadsto 1 + \color{blue}{\left(-\left(0.16666666666666666 + 0.2222222222222222 \cdot \frac{1}{t}\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-neg-in 98.9%

         \[\leadsto 1 + \color{blue}{\left(\left(-0.16666666666666666\right) + \left(-0.2222222222222222 \cdot \frac{1}{t}\right)\right)} \]

      2. metadata-eval 98.9%

         \[\leadsto 1 + \left(\color{blue}{-0.16666666666666666} + \left(-0.2222222222222222 \cdot \frac{1}{t}\right)\right) \]

      3. associate-*r/ 98.9%

         \[\leadsto 1 + \left(-0.16666666666666666 + \left(-\color{blue}{\frac{0.2222222222222222 \cdot 1}{t}}\right)\right) \]

      4. metadata-eval 98.9%

         \[\leadsto 1 + \left(-0.16666666666666666 + \left(-\frac{\color{blue}{0.2222222222222222}}{t}\right)\right) \]

      5. distribute-neg-frac 98.9%

         \[\leadsto 1 + \left(-0.16666666666666666 + \color{blue}{\frac{-0.2222222222222222}{t}}\right) \]

      6. metadata-eval 98.9%

         \[\leadsto 1 + \left(-0.16666666666666666 + \frac{\color{blue}{-0.2222222222222222}}{t}\right) \]

   7. Simplified 98.9%

      \[\leadsto 1 + \color{blue}{\left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)} \]

   8. Taylor expanded in t around 0 98.9%

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

      1. associate-*r/ 98.9%

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

      2. metadata-eval 98.9%

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

   10. Simplified 98.9%

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

   if -0.47999999999999998 < t < 0.650000000000000022

   1. Initial program 99.3%

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

      1. sub-neg 99.3%

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

      2. distribute-neg-frac 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 99.8%

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

   6. Taylor expanded in t around 0 99.8%

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

   7. Taylor expanded in t around 0 99.1%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.48 \lor \neg \left(t \leq 0.65\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.7% accurate, 2.6× 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%

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

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 98.9%

      \[\leadsto 1 + \color{blue}{\left(-\left(0.16666666666666666 + 0.2222222222222222 \cdot \frac{1}{t}\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-neg-in 98.9%

         \[\leadsto 1 + \color{blue}{\left(\left(-0.16666666666666666\right) + \left(-0.2222222222222222 \cdot \frac{1}{t}\right)\right)} \]

      2. metadata-eval 98.9%

         \[\leadsto 1 + \left(\color{blue}{-0.16666666666666666} + \left(-0.2222222222222222 \cdot \frac{1}{t}\right)\right) \]

      3. associate-*r/ 98.9%

         \[\leadsto 1 + \left(-0.16666666666666666 + \left(-\color{blue}{\frac{0.2222222222222222 \cdot 1}{t}}\right)\right) \]

      4. metadata-eval 98.9%

         \[\leadsto 1 + \left(-0.16666666666666666 + \left(-\frac{\color{blue}{0.2222222222222222}}{t}\right)\right) \]

      5. distribute-neg-frac 98.9%

         \[\leadsto 1 + \left(-0.16666666666666666 + \color{blue}{\frac{-0.2222222222222222}{t}}\right) \]

      6. metadata-eval 98.9%

         \[\leadsto 1 + \left(-0.16666666666666666 + \frac{\color{blue}{-0.2222222222222222}}{t}\right) \]

   7. Simplified 98.9%

      \[\leadsto 1 + \color{blue}{\left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)} \]

   8. Taylor expanded in t around inf 97.2%

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

   if -0.330000000000000016 < t < 1

   1. Initial program 99.3%

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

      1. sub-neg 99.3%

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

      2. distribute-neg-frac 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 99.8%

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

   6. Taylor expanded in t around 0 99.8%

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

   7. Taylor expanded in t around 0 99.1%

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

4. Final simplification 98.0%

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

Alternative 10: 59.2% accurate, 29.0× 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 99.7%

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

   1. sub-neg 99.7%

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

   2. distribute-neg-frac 99.7%

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

   3. metadata-eval 99.7%

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

3. Simplified 100.0%

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

5. Taylor expanded in t around 0 56.2%

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

6. Taylor expanded in t around 0 56.2%

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

7. Taylor expanded in t around 0 55.1%

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

8. Final simplification 55.1%

   \[\leadsto 0.5 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024048 
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  (- 1.0 (/ 1.0 (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))))))