Kahan p13 Example 3

Percentage Accurate: 100.0% → 100.0%

Time: 15.4s

Alternatives: 8

Speedup: 1.4×

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, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-2}{-1 - t} - 2\\ 1 + \frac{2 - t\_1 \cdot t\_1}{{\left(2 + \frac{-2}{1 + t}\right)}^{4} - 4} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- (/ -2.0 (- -1.0 t)) 2.0)))
   (+
    1.0
    (/ (- 2.0 (* t_1 t_1)) (- (pow (+ 2.0 (/ -2.0 (+ 1.0 t))) 4.0) 4.0)))))

C

double code(double t) {
	double t_1 = (-2.0 / (-1.0 - t)) - 2.0;
	return 1.0 + ((2.0 - (t_1 * t_1)) / (pow((2.0 + (-2.0 / (1.0 + t))), 4.0) - 4.0));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = ((-2.0d0) / ((-1.0d0) - t)) - 2.0d0
    code = 1.0d0 + ((2.0d0 - (t_1 * t_1)) / (((2.0d0 + ((-2.0d0) / (1.0d0 + t))) ** 4.0d0) - 4.0d0))
end function

Java

public static double code(double t) {
	double t_1 = (-2.0 / (-1.0 - t)) - 2.0;
	return 1.0 + ((2.0 - (t_1 * t_1)) / (Math.pow((2.0 + (-2.0 / (1.0 + t))), 4.0) - 4.0));
}

Python

def code(t):
	t_1 = (-2.0 / (-1.0 - t)) - 2.0
	return 1.0 + ((2.0 - (t_1 * t_1)) / (math.pow((2.0 + (-2.0 / (1.0 + t))), 4.0) - 4.0))

Julia

function code(t)
	t_1 = Float64(Float64(-2.0 / Float64(-1.0 - t)) - 2.0)
	return Float64(1.0 + Float64(Float64(2.0 - Float64(t_1 * t_1)) / Float64((Float64(2.0 + Float64(-2.0 / Float64(1.0 + t))) ^ 4.0) - 4.0)))
end

MATLAB

function tmp = code(t)
	t_1 = (-2.0 / (-1.0 - t)) - 2.0;
	tmp = 1.0 + ((2.0 - (t_1 * t_1)) / (((2.0 + (-2.0 / (1.0 + t))) ^ 4.0) - 4.0));
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(-2.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]}, N[(1.0 + N[(N[(2.0 - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(2.0 + N[(-2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.0], $MachinePrecision] - 4.0), $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. flip-+ 100.0%

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

   2. associate-/r/ 100.0%

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

4. Applied egg-rr 100.0%

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

   1. associate-*l/ 100.0%

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

6. Simplified 100.0%

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

   1. unpow2 100.0%

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

8. Applied egg-rr 100.0%

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

9. Final simplification 100.0%

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

Alternative 2: 99.2% accurate, 1.2× speedup

Math

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

FPCore

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

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 / ((t * ((4.0 / (1.0 + t)) - 4.0)) - 2.0));
	} else {
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	}
	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 / ((t * ((4.0d0 / (1.0d0 + t)) - 4.0d0)) - 2.0d0))
    else
        tmp = 1.0d0 - ((0.2222222222222222d0 / t) + 0.16666666666666666d0)
    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 / ((t * ((4.0 / (1.0 + t)) - 4.0)) - 2.0));
	} else {
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	}
	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 / ((t * ((4.0 / (1.0 + t)) - 4.0)) - 2.0))
	else:
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666)
	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(Float64(t * Float64(Float64(4.0 / Float64(1.0 + t)) - 4.0)) - 2.0)));
	else
		tmp = Float64(1.0 - Float64(Float64(0.2222222222222222 / t) + 0.16666666666666666));
	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 / ((t * ((4.0 / (1.0 + t)) - 4.0)) - 2.0));
	else
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	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[(N[(t * N[(N[(4.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(0.2222222222222222 / t), $MachinePrecision] + 0.16666666666666666), $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. Add Preprocessing

   3. Taylor expanded in t around inf 98.6%

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

      1. associate-*r/ 98.6%

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

      2. metadata-eval 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in t around 0 98.6%

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

      1. associate-*r/ 98.6%

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

      2. metadata-eval 98.6%

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

   8. Simplified 98.6%

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

   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. Taylor expanded in t around 0 98.7%

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

      1. sub-neg 98.7%

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

      2. distribute-neg-frac 98.7%

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

      3. distribute-neg-frac 98.7%

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

      4. metadata-eval 98.7%

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

   5. Applied egg-rr 98.7%

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

      1. associate-/r* 98.7%

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

      2. distribute-lft-in 98.7%

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

      3. rgt-mult-inverse 98.7%

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

      4. *-rgt-identity 98.7%

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

   7. Simplified 98.7%

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

      1. add0 98.7%

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

      2. *-commutative 98.7%

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

      3. *-commutative 98.7%

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

      4. associate-*l* 98.7%

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

      5. distribute-lft-in 98.7%

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

      6. metadata-eval 98.7%

         \[\leadsto 1 - \frac{1}{2 + \left(t \cdot \left(\color{blue}{4} + 2 \cdot \frac{-2}{t + 1}\right) + 0\right)} \]

      7. associate-*r/ 98.7%

         \[\leadsto 1 - \frac{1}{2 + \left(t \cdot \left(4 + \color{blue}{\frac{2 \cdot -2}{t + 1}}\right) + 0\right)} \]

      8. metadata-eval 98.7%

         \[\leadsto 1 - \frac{1}{2 + \left(t \cdot \left(4 + \frac{\color{blue}{-4}}{t + 1}\right) + 0\right)} \]

      9. +-commutative 98.7%

         \[\leadsto 1 - \frac{1}{2 + \left(t \cdot \left(4 + \frac{-4}{\color{blue}{1 + t}}\right) + 0\right)} \]

   9. Applied egg-rr 98.7%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(t \cdot \left(4 + \frac{-4}{1 + t}\right) + 0\right)}} \]
   10. Step-by-step derivation

       1. add0 98.7%

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

       2. metadata-eval 98.7%

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

       3. associate-*l/ 98.7%

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

       4. metadata-eval 98.7%

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

       5. distribute-neg-frac 98.7%

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

       6. distribute-neg-frac2 98.7%

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

       7. distribute-neg-in 98.7%

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

       8. metadata-eval 98.7%

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

       9. associate-*l/ 98.7%

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

       10. metadata-eval 98.7%

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

       11. unsub-neg 98.7%

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

   11. Simplified 98.7%

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

   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. Add Preprocessing

   3. Taylor expanded in t around inf 98.8%

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

      1. associate-*r/ 98.8%

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

      2. metadata-eval 98.8%

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

   5. Simplified 98.8%

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

4. Final simplification 98.7%

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

Alternative 3: 99.2% accurate, 1.3× speedup

Math

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

FPCore

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

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 + ((2.0 * t) * (2.0 * t))));
	} else {
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	}
	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 + ((2.0d0 * t) * (2.0d0 * t))))
    else
        tmp = 1.0d0 - ((0.2222222222222222d0 / t) + 0.16666666666666666d0)
    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 + ((2.0 * t) * (2.0 * t))));
	} else {
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	}
	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 + ((2.0 * t) * (2.0 * t))))
	else:
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666)
	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(Float64(2.0 * t) * Float64(2.0 * t)))));
	else
		tmp = Float64(1.0 - Float64(Float64(0.2222222222222222 / t) + 0.16666666666666666));
	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 + ((2.0 * t) * (2.0 * t))));
	else
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	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[(N[(2.0 * t), $MachinePrecision] * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(0.2222222222222222 / t), $MachinePrecision] + 0.16666666666666666), $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. Add Preprocessing

   3. Taylor expanded in t around inf 98.6%

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

      1. associate-*r/ 98.6%

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

      2. metadata-eval 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in t around 0 98.6%

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

      1. associate-*r/ 98.6%

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

      2. metadata-eval 98.6%

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

   8. Simplified 98.6%

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

   if -0.599999999999999978 < 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 0 98.7%

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

   4. Taylor expanded in t around 0 98.7%

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

   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. Add Preprocessing

   3. Taylor expanded in t around inf 98.8%

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

      1. associate-*r/ 98.8%

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

      2. metadata-eval 98.8%

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

   5. Simplified 98.8%

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

4. Final simplification 98.7%

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

Alternative 4: 100.0% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (+
  1.0
  (/ 1.0 (- (* (+ 2.0 (/ -2.0 (+ 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 / (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) / (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 / (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 / (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(-2.0 / 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 / (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[(-2.0 / N[(1.0 + t), $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. sub-neg 59.1%

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

   2. distribute-neg-frac 59.1%

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

   3. distribute-neg-frac 59.1%

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

   4. metadata-eval 59.1%

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

4. Applied egg-rr 100.0%

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

   1. associate-/r* 59.1%

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

   2. distribute-lft-in 59.1%

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

   3. rgt-mult-inverse 59.1%

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

   4. *-rgt-identity 59.1%

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

6. Simplified 100.0%

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

   1. sub-neg 59.1%

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

   2. distribute-neg-frac 59.1%

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

   3. distribute-neg-frac 59.1%

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

   4. metadata-eval 59.1%

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

8. Applied egg-rr 100.0%

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

   1. associate-/r* 59.1%

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

   2. distribute-lft-in 59.1%

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

   3. rgt-mult-inverse 59.1%

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

   4. *-rgt-identity 59.1%

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

10. Simplified 100.0%

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

11. Final simplification 100.0%

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

Alternative 5: 99.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.66:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.49)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (if (<= t 0.66)
     0.5
     (- 1.0 (+ (/ 0.2222222222222222 t) 0.16666666666666666)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.49) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 0.66) {
		tmp = 0.5;
	} else {
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.49d0)) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else if (t <= 0.66d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 - ((0.2222222222222222d0 / t) + 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.49) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 0.66) {
		tmp = 0.5;
	} else {
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.49:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	elif t <= 0.66:
		tmp = 0.5
	else:
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.49)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	elseif (t <= 0.66)
		tmp = 0.5;
	else
		tmp = Float64(1.0 - Float64(Float64(0.2222222222222222 / t) + 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.49)
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	elseif (t <= 0.66)
		tmp = 0.5;
	else
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.49], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.66], 0.5, N[(1.0 - N[(N[(0.2222222222222222 / t), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.48999999999999999

   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 98.6%

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

      1. associate-*r/ 98.6%

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

      2. metadata-eval 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in t around 0 98.6%

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

      1. associate-*r/ 98.6%

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

      2. metadata-eval 98.6%

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

   8. Simplified 98.6%

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

   if -0.48999999999999999 < t < 0.660000000000000031

   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 0 98.4%

      \[\leadsto 1 - \color{blue}{0.5} \]

   if 0.660000000000000031 < 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 98.8%

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

      1. associate-*r/ 98.8%

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

      2. metadata-eval 98.8%

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

   5. Simplified 98.8%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.66:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < -0.48999999999999999 or 0.660000000000000031 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 98.7%

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

      1. associate-*r/ 98.7%

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

      2. metadata-eval 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in t around 0 98.7%

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

      1. associate-*r/ 98.7%

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

      2. metadata-eval 98.7%

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

   8. Simplified 98.7%

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

   if -0.48999999999999999 < t < 0.660000000000000031

   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 0 98.4%

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

4. Final simplification 98.5%

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

Alternative 7: 98.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.335:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.335) 0.8333333333333334 (if (<= t 1.0) 0.5 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.335) {
		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 <= (-0.335d0)) 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 <= -0.335) {
		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 <= -0.335:
		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 <= -0.335)
		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 <= -0.335)
		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, -0.335], 0.8333333333333334, If[LessEqual[t, 1.0], 0.5, 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.33500000000000002 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 98.7%

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

      1. associate-*r/ 98.7%

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

      2. metadata-eval 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in t around inf 97.1%

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

   if -0.33500000000000002 < 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 0 98.4%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.335:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.1% 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 52.0%

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

   1. associate-*r/ 52.0%

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

   2. metadata-eval 52.0%

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

5. Simplified 52.0%

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

6. Taylor expanded in t around inf 59.3%

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

7. Final simplification 59.3%

   \[\leadsto 0.8333333333333334 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024046 
(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)))))))))