Kahan p13 Example 3

Percentage Accurate: 99.9% → 100.0%

Time: 19.3s

Alternatives: 9

Speedup: 0.2×

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: 99.9% 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, 0.2× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{1}{\left(2 + \frac{-2}{1 + t}\right) \cdot \frac{8 + \frac{-8}{{\left(1 + t\right)}^{3}}}{\frac{\frac{4}{-1 - t} - 4}{1 + t} - 4} - 2} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (+
  1.0
  (/
   1.0
   (-
    (*
     (+ 2.0 (/ -2.0 (+ 1.0 t)))
     (/
      (+ 8.0 (/ -8.0 (pow (+ 1.0 t) 3.0)))
      (- (/ (- (/ 4.0 (- -1.0 t)) 4.0) (+ 1.0 t)) 4.0)))
    2.0))))

C

double code(double t) {
	return 1.0 + (1.0 / (((2.0 + (-2.0 / (1.0 + t))) * ((8.0 + (-8.0 / pow((1.0 + t), 3.0))) / ((((4.0 / (-1.0 - t)) - 4.0) / (1.0 + t)) - 4.0))) - 2.0));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = 1.0d0 + (1.0d0 / (((2.0d0 + ((-2.0d0) / (1.0d0 + t))) * ((8.0d0 + ((-8.0d0) / ((1.0d0 + t) ** 3.0d0))) / ((((4.0d0 / ((-1.0d0) - t)) - 4.0d0) / (1.0d0 + t)) - 4.0d0))) - 2.0d0))
end function

Java

public static double code(double t) {
	return 1.0 + (1.0 / (((2.0 + (-2.0 / (1.0 + t))) * ((8.0 + (-8.0 / Math.pow((1.0 + t), 3.0))) / ((((4.0 / (-1.0 - t)) - 4.0) / (1.0 + t)) - 4.0))) - 2.0));
}

Python

def code(t):
	return 1.0 + (1.0 / (((2.0 + (-2.0 / (1.0 + t))) * ((8.0 + (-8.0 / math.pow((1.0 + t), 3.0))) / ((((4.0 / (-1.0 - t)) - 4.0) / (1.0 + t)) - 4.0))) - 2.0))

Julia

function code(t)
	return Float64(1.0 + Float64(1.0 / Float64(Float64(Float64(2.0 + Float64(-2.0 / Float64(1.0 + t))) * Float64(Float64(8.0 + Float64(-8.0 / (Float64(1.0 + t) ^ 3.0))) / Float64(Float64(Float64(Float64(4.0 / Float64(-1.0 - t)) - 4.0) / Float64(1.0 + t)) - 4.0))) - 2.0)))
end

MATLAB

function tmp = code(t)
	tmp = 1.0 + (1.0 / (((2.0 + (-2.0 / (1.0 + t))) * ((8.0 + (-8.0 / ((1.0 + t) ^ 3.0))) / ((((4.0 / (-1.0 - t)) - 4.0) / (1.0 + t)) - 4.0))) - 2.0));
end

Wolfram

code[t_] := N[(1.0 + N[(1.0 / N[(N[(N[(2.0 + N[(-2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(8.0 + N[(-8.0 / N[Power[N[(1.0 + t), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(4.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] - 4.0), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. Add Preprocessing
3. Step-by-step derivation

   1. flip3-- 100.0%

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

   2. associate-*r/ 100.0%

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

4. Applied egg-rr 100.0%

   \[\leadsto 1 - \frac{1}{2 + \color{blue}{\frac{\left(2 + \frac{\frac{-2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(8 - {\left(\frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}\right)}^{3}\right)}{\mathsf{fma}\left(\frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}, 2 + \frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}, 4\right)}}} \]

6. Simplified 100.0%

   \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(2 + \frac{-2}{t + 1}\right) \cdot \frac{8 + \frac{-8}{{\left(t + 1\right)}^{3}}}{4 + \frac{4 + \frac{4}{t + 1}}{t + 1}}}} \]

7. Final simplification 100.0%

   \[\leadsto 1 + \frac{1}{\left(2 + \frac{-2}{1 + t}\right) \cdot \frac{8 + \frac{-8}{{\left(1 + t\right)}^{3}}}{\frac{\frac{4}{-1 - t} - 4}{1 + t} - 4} - 2} \]
8. Add Preprocessing

Alternative 2: 99.3% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -1.15) (not (<= t 0.88)))
   (- 1.0 (+ 0.16666666666666666 (/ 0.2222222222222222 t)))
   (+ 1.0 (/ 1.0 (- (* (* 2.0 t) (- (/ -2.0 (- -1.0 t)) 2.0)) 2.0)))))

C

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

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-1.15d0)) .or. (.not. (t <= 0.88d0))) then
        tmp = 1.0d0 - (0.16666666666666666d0 + (0.2222222222222222d0 / t))
    else
        tmp = 1.0d0 + (1.0d0 / (((2.0d0 * t) * (((-2.0d0) / ((-1.0d0) - t)) - 2.0d0)) - 2.0d0))
    end if
    code = tmp
end function

Java

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

Python

def code(t):
	tmp = 0
	if (t <= -1.15) or not (t <= 0.88):
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t))
	else:
		tmp = 1.0 + (1.0 / (((2.0 * t) * ((-2.0 / (-1.0 - t)) - 2.0)) - 2.0))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[t_] := If[Or[LessEqual[t, -1.15], N[Not[LessEqual[t, 0.88]], $MachinePrecision]], N[(1.0 - N[(0.16666666666666666 + N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(1.0 / N[(N[(N[(2.0 * t), $MachinePrecision] * N[(N[(-2.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.1499999999999999 or 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. Add Preprocessing

   3. Taylor expanded in t around inf 98.1%

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

      1. associate-*r/ 98.1%

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

      2. metadata-eval 98.1%

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

   5. Simplified 98.1%

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

   6. Taylor expanded in t around inf 98.1%

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

      1. associate-*r/ 98.1%

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

      2. metadata-eval 98.1%

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

   8. Simplified 98.1%

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

   9. Taylor expanded in t around inf 98.2%

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

       1. associate-*r/ 98.2%

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

       2. metadata-eval 98.2%

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

   11. Simplified 98.2%

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

   if -1.1499999999999999 < t < 0.880000000000000004

   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. Add Preprocessing
   3. Step-by-step derivation

      1. add-cube-cbrt 100.0%

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

      2. associate-/l* 100.0%

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

      3. pow2 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{{\left(\sqrt[3]{\frac{2}{t}}\right)}^{2} \cdot \frac{\sqrt[3]{\frac{2}{t}}}{1 + \frac{1}{t}}}\right)} \]
   5. Step-by-step derivation

      1. cancel-sign-sub-inv 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \color{blue}{\left(2 + \left(-{\left(\sqrt[3]{\frac{2}{t}}\right)}^{2}\right) \cdot \frac{\sqrt[3]{\frac{2}{t}}}{1 + \frac{1}{t}}\right)}} \]
   7. Step-by-step derivation

      1. distribute-lft-neg-out 100.0%

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

      2. unsub-neg 100.0%

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

      3. associate-*r/ 100.0%

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

      4. unpow2 100.0%

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

      5. unpow3 100.0%

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

   8. Simplified 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \color{blue}{\left(2 - \frac{{\left(\sqrt[3]{\frac{2}{t}}\right)}^{3}}{1 + \frac{1}{t}}\right)}} \]
   9. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. distribute-neg-frac 100.0%

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

      4. metadata-eval 100.0%

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

   10. Applied egg-rr 100.0%

       \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(2 + \frac{\frac{-2}{t}}{1 + \frac{1}{t}}\right)} \cdot \left(2 - \frac{{\left(\sqrt[3]{\frac{2}{t}}\right)}^{3}}{1 + \frac{1}{t}}\right)} \]
   11. Step-by-step derivation

       1. associate-/r* 100.0%

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

       2. distribute-lft-in 100.0%

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

       3. *-rgt-identity 100.0%

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

       4. rgt-mult-inverse 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in t around 0 99.8%

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

4. Final simplification 99.1%

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

Alternative 3: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_] := N[(1.0 + N[(1.0 / N[(N[(N[(2.0 + N[(N[(2.0 / t), $MachinePrecision] / N[(-1.0 + N[(-1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. Add Preprocessing
3. Step-by-step derivation

   1. add-log-exp 97.9%

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

   2. *-un-lft-identity 97.9%

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

   3. log-prod 97.9%

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

   4. metadata-eval 97.9%

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

   5. add-log-exp 97.9%

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

   6. associate-/l/ 94.4%

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

   7. *-commutative 94.4%

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

4. Applied egg-rr 100.0%

   \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\left(0 + \frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}\right)}\right)} \]
5. Step-by-step derivation

   1. +-lft-identity 94.4%

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

   2. distribute-lft-in 94.4%

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

   3. *-rgt-identity 94.4%

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

   4. rgt-mult-inverse 97.9%

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

6. Simplified 100.0%

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

7. Final simplification 100.0%

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

Alternative 4: 97.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_] := N[(1.0 + N[(1.0 / N[(N[(N[(2.0 + N[(2.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / t), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. Add Preprocessing

3. Taylor expanded in t around inf 97.9%

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

   1. associate-*r/ 97.9%

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

   2. metadata-eval 97.9%

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

5. Simplified 97.9%

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

   1. add-log-exp 97.9%

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

   2. *-un-lft-identity 97.9%

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

   3. log-prod 97.9%

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

   4. metadata-eval 97.9%

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

   5. add-log-exp 97.9%

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

   6. associate-/l/ 94.4%

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

   7. *-commutative 94.4%

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

7. Applied egg-rr 94.4%

   \[\leadsto 1 - \frac{1}{2 + \left(2 - \color{blue}{\left(0 + \frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}\right)}\right) \cdot \left(2 - \frac{2}{t}\right)} \]
8. Step-by-step derivation

   1. +-lft-identity 94.4%

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

   2. distribute-lft-in 94.4%

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

   3. *-rgt-identity 94.4%

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

   4. rgt-mult-inverse 97.9%

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

9. Simplified 97.9%

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

10. Final simplification 97.9%

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

Alternative 5: 98.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_] := N[(1.0 - N[(1.0 / N[(2.0 + N[(2.0 * N[(2.0 + N[(N[(2.0 / t), $MachinePrecision] / N[(-1.0 + N[(-1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. Add Preprocessing

3. Taylor expanded in t around inf 98.3%

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

4. Final simplification 98.3%

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

Alternative 6: 99.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.68\right):\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.49) (not (<= t 0.68)))
   (- 1.0 (+ 0.16666666666666666 (/ 0.2222222222222222 t)))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.49) || !(t <= 0.68)) {
		tmp = 1.0 - (0.16666666666666666 + (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 = 1.0d0 - (0.16666666666666666d0 + (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 = 1.0 - (0.16666666666666666 + (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 = 1.0 - (0.16666666666666666 + (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(1.0 - Float64(0.16666666666666666 + 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 = 1.0 - (0.16666666666666666 + (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[(1.0 - N[(0.16666666666666666 + N[(0.2222222222222222 / t), $MachinePrecision]), $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%

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

   3. Taylor expanded in t around inf 98.1%

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

      1. associate-*r/ 98.1%

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

      2. metadata-eval 98.1%

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

   5. Simplified 98.1%

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

   6. Taylor expanded in t around inf 98.1%

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

      1. associate-*r/ 98.1%

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

      2. metadata-eval 98.1%

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

   8. Simplified 98.1%

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

   9. Taylor expanded in t around inf 98.2%

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

       1. associate-*r/ 98.2%

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

       2. metadata-eval 98.2%

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

   11. Simplified 98.2%

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

   if -0.48999999999999999 < t < 0.680000000000000049

   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. Add Preprocessing

   3. Taylor expanded in t around inf 99.0%

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

   4. Taylor expanded in t around 0 99.6%

      \[\leadsto 1 - \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.68\right):\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.44 \lor \neg \left(t \leq 0.33\right):\\ \;\;\;\;0.8333333333333334 + \frac{-0.1111111111111111}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.44) (not (<= t 0.33)))
   (+ 0.8333333333333334 (/ -0.1111111111111111 t))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.44) || !(t <= 0.33)) {
		tmp = 0.8333333333333334 + (-0.1111111111111111 / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.44d0)) .or. (.not. (t <= 0.33d0))) then
        tmp = 0.8333333333333334d0 + ((-0.1111111111111111d0) / t)
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.44) || !(t <= 0.33)) {
		tmp = 0.8333333333333334 + (-0.1111111111111111 / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.44) or not (t <= 0.33):
		tmp = 0.8333333333333334 + (-0.1111111111111111 / t)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.44) || !(t <= 0.33))
		tmp = Float64(0.8333333333333334 + Float64(-0.1111111111111111 / t));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.44) || ~((t <= 0.33)))
		tmp = 0.8333333333333334 + (-0.1111111111111111 / t);
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.44], N[Not[LessEqual[t, 0.33]], $MachinePrecision]], N[(0.8333333333333334 + N[(-0.1111111111111111 / t), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if t < -0.440000000000000002 or 0.330000000000000016 < 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. Add Preprocessing

   3. Taylor expanded in t around inf 97.5%

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

   4. Taylor expanded in t around inf 97.5%

      \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + 0.1111111111111111 \cdot \frac{1}{t}\right)} \]
   5. Step-by-step derivation

      1. associate-*r/ 97.5%

         \[\leadsto 1 - \left(0.16666666666666666 + \color{blue}{\frac{0.1111111111111111 \cdot 1}{t}}\right) \]

      2. metadata-eval 97.5%

         \[\leadsto 1 - \left(0.16666666666666666 + \frac{\color{blue}{0.1111111111111111}}{t}\right) \]

   6. Simplified 97.5%

      \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + \frac{0.1111111111111111}{t}\right)} \]

   7. Taylor expanded in t around 0 97.5%

      \[\leadsto \color{blue}{0.8333333333333334 - 0.1111111111111111 \cdot \frac{1}{t}} \]
   8. Step-by-step derivation

      1. cancel-sign-sub-inv 97.5%

         \[\leadsto \color{blue}{0.8333333333333334 + \left(-0.1111111111111111\right) \cdot \frac{1}{t}} \]

      2. metadata-eval 97.5%

         \[\leadsto 0.8333333333333334 + \color{blue}{-0.1111111111111111} \cdot \frac{1}{t} \]

      3. associate-*r/ 97.5%

         \[\leadsto 0.8333333333333334 + \color{blue}{\frac{-0.1111111111111111 \cdot 1}{t}} \]

      4. metadata-eval 97.5%

         \[\leadsto 0.8333333333333334 + \frac{\color{blue}{-0.1111111111111111}}{t} \]

   9. Simplified 97.5%

      \[\leadsto \color{blue}{0.8333333333333334 + \frac{-0.1111111111111111}{t}} \]

   if -0.440000000000000002 < t < 0.330000000000000016

   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. Add Preprocessing

   3. Taylor expanded in t around inf 99.0%

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

   4. Taylor expanded in t around 0 99.6%

      \[\leadsto 1 - \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.44 \lor \neg \left(t \leq 0.33\right):\\ \;\;\;\;0.8333333333333334 + \frac{-0.1111111111111111}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.6% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < -0.340000000000000024 or 1 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 97.5%

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

   4. Taylor expanded in t around inf 97.5%

      \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + 0.1111111111111111 \cdot \frac{1}{t}\right)} \]
   5. Step-by-step derivation

      1. associate-*r/ 97.5%

         \[\leadsto 1 - \left(0.16666666666666666 + \color{blue}{\frac{0.1111111111111111 \cdot 1}{t}}\right) \]

      2. metadata-eval 97.5%

         \[\leadsto 1 - \left(0.16666666666666666 + \frac{\color{blue}{0.1111111111111111}}{t}\right) \]

   6. Simplified 97.5%

      \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + \frac{0.1111111111111111}{t}\right)} \]

   7. Taylor expanded in t around inf 97.5%

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

   if -0.340000000000000024 < t < 1

   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. Add Preprocessing

   3. Taylor expanded in t around inf 99.0%

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

   4. Taylor expanded in t around 0 99.6%

      \[\leadsto 1 - \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

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

Alternative 9: 58.6% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ 0.8333333333333334 \end{array} \]

FPCore

(FPCore (t) :precision binary64 0.8333333333333334)

C

double code(double t) {
	return 0.8333333333333334;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = 0.8333333333333334d0
end function

Java

public static double code(double t) {
	return 0.8333333333333334;
}

Python

def code(t):
	return 0.8333333333333334

Julia

function code(t)
	return 0.8333333333333334
end

MATLAB

function tmp = code(t)
	tmp = 0.8333333333333334;
end

Wolfram

code[t_] := 0.8333333333333334

Derivation

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. Add Preprocessing

3. Taylor expanded in t around inf 98.3%

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

4. Taylor expanded in t around inf 47.1%

   \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + 0.1111111111111111 \cdot \frac{1}{t}\right)} \]
5. Step-by-step derivation

   1. associate-*r/ 47.1%

      \[\leadsto 1 - \left(0.16666666666666666 + \color{blue}{\frac{0.1111111111111111 \cdot 1}{t}}\right) \]

   2. metadata-eval 47.1%

      \[\leadsto 1 - \left(0.16666666666666666 + \frac{\color{blue}{0.1111111111111111}}{t}\right) \]

6. Simplified 47.1%

   \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + \frac{0.1111111111111111}{t}\right)} \]

7. Taylor expanded in t around inf 55.8%

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

8. Final simplification 55.8%

   \[\leadsto 0.8333333333333334 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024053 
(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)))))))))