Kahan p13 Example 3

Percentage Accurate: 99.9% → 100.0%

Time: 6.7s

Alternatives: 7

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: 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, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{1 + t}\\ 1 + \frac{-1}{6 + t\_1 \cdot \left(t\_1 + -4\right)} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ 2.0 (+ 1.0 t))))
   (+ 1.0 (/ -1.0 (+ 6.0 (* t_1 (+ t_1 -4.0)))))))

C

double code(double t) {
	double t_1 = 2.0 / (1.0 + t);
	return 1.0 + (-1.0 / (6.0 + (t_1 * (t_1 + -4.0))));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = 2.0d0 / (1.0d0 + t)
    code = 1.0d0 + ((-1.0d0) / (6.0d0 + (t_1 * (t_1 + (-4.0d0)))))
end function

Java

public static double code(double t) {
	double t_1 = 2.0 / (1.0 + t);
	return 1.0 + (-1.0 / (6.0 + (t_1 * (t_1 + -4.0))));
}

Python

def code(t):
	t_1 = 2.0 / (1.0 + t)
	return 1.0 + (-1.0 / (6.0 + (t_1 * (t_1 + -4.0))))

Julia

function code(t)
	t_1 = Float64(2.0 / Float64(1.0 + t))
	return Float64(1.0 + Float64(-1.0 / Float64(6.0 + Float64(t_1 * Float64(t_1 + -4.0)))))
end

MATLAB

function tmp = code(t)
	t_1 = 2.0 / (1.0 + t);
	tmp = 1.0 + (-1.0 / (6.0 + (t_1 * (t_1 + -4.0))));
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, N[(1.0 + N[(-1.0 / N[(6.0 + N[(t$95$1 * N[(t$95$1 + -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

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. Final simplification 100.0%

   \[\leadsto 1 + \frac{-1}{6 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)} \]
6. Add Preprocessing

Alternative 2: 99.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.42 \lor \neg \left(t \leq 0.78\right):\\ \;\;\;\;1 - \frac{1}{6 + \frac{-8 + \frac{4}{t}}{1 + t}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.42) (not (<= t 0.78)))
   (- 1.0 (/ 1.0 (+ 6.0 (/ (+ -8.0 (/ 4.0 t)) (+ 1.0 t)))))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.42) || !(t <= 0.78)) {
		tmp = 1.0 - (1.0 / (6.0 + ((-8.0 + (4.0 / t)) / (1.0 + t))));
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.42d0)) .or. (.not. (t <= 0.78d0))) then
        tmp = 1.0d0 - (1.0d0 / (6.0d0 + (((-8.0d0) + (4.0d0 / t)) / (1.0d0 + t))))
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.42) || !(t <= 0.78)) {
		tmp = 1.0 - (1.0 / (6.0 + ((-8.0 + (4.0 / t)) / (1.0 + t))));
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.42) or not (t <= 0.78):
		tmp = 1.0 - (1.0 / (6.0 + ((-8.0 + (4.0 / t)) / (1.0 + t))))
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.42) || !(t <= 0.78))
		tmp = Float64(1.0 - Float64(1.0 / Float64(6.0 + Float64(Float64(-8.0 + Float64(4.0 / t)) / Float64(1.0 + t)))));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.42) || ~((t <= 0.78)))
		tmp = 1.0 - (1.0 / (6.0 + ((-8.0 + (4.0 / t)) / (1.0 + t))));
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.42], N[Not[LessEqual[t, 0.78]], $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], 0.5]

Derivation

1. Split input into 2 regimes

2. if t < -0.419999999999999984 or 0.78000000000000003 < 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. *-un-lft-identity 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. times-frac 100.0%

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

      6. metadata-eval 100.0%

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

      7. fma-define 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. distribute-neg-frac 100.0%

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

      4. unsub-neg 100.0%

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

      5. fma-undefine 100.0%

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

      6. associate-*r/ 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-lft-in 100.0%

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

      9. metadata-eval 100.0%

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

      10. *-commutative 100.0%

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

      11. associate-*l/ 100.0%

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

      12. metadata-eval 100.0%

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

      13. +-commutative 100.0%

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

      14. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around inf 98.1%

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

       1. sub-neg 98.1%

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

       2. associate-*r/ 98.1%

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

       3. metadata-eval 98.1%

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

       4. metadata-eval 98.1%

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

   11. Simplified 98.1%

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

   if -0.419999999999999984 < t < 0.78000000000000003

   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. Taylor expanded in t around 0 99.0%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.42 \lor \neg \left(t \leq 0.78\right):\\ \;\;\;\;1 - \frac{1}{6 + \frac{-8 + \frac{4}{t}}{1 + t}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 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. *-un-lft-identity 100.0%

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

   2. +-commutative 100.0%

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

   3. associate-*l/ 100.0%

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

   4. *-un-lft-identity 100.0%

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

   5. times-frac 100.0%

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

   6. metadata-eval 100.0%

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

   7. fma-define 100.0%

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

6. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

   2. metadata-eval 100.0%

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

   3. distribute-neg-frac 100.0%

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

   4. unsub-neg 100.0%

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

   5. fma-undefine 100.0%

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

   6. associate-*r/ 100.0%

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

   7. +-commutative 100.0%

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

   8. distribute-lft-in 100.0%

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

   9. metadata-eval 100.0%

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

   10. *-commutative 100.0%

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

   11. associate-*l/ 100.0%

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

   12. metadata-eval 100.0%

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

   13. +-commutative 100.0%

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

   14. +-commutative 100.0%

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

8. Simplified 100.0%

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

9. Final simplification 100.0%

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

Alternative 4: 98.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.65\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.49) (not (<= t 0.65)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.49) || !(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 <= (-0.49d0)) .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 <= -0.49) || !(t <= 0.65)) {
		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 <= 0.65):
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.49) || !(t <= 0.65))
		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 <= -0.49) || ~((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, -0.49], 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 < -0.48999999999999999 or 0.650000000000000022 < t

   1. Initial program 100.0%

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

   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. Taylor expanded in t around inf 97.6%

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

      1. associate-*r/ 97.6%

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

      2. metadata-eval 97.6%

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

   7. Simplified 97.6%

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

   if -0.48999999999999999 < t < 0.650000000000000022

   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. Taylor expanded in t around 0 99.0%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.65\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;0.8333333333333334 - \left(1 + \left(\frac{0.2222222222222222}{t} + -1\right)\right)\\ \mathbf{elif}\;t \leq 0.65:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.49)
   (- 0.8333333333333334 (+ 1.0 (+ (/ 0.2222222222222222 t) -1.0)))
   (if (<= t 0.65) 0.5 (- 0.8333333333333334 (/ 0.2222222222222222 t)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.49) {
		tmp = 0.8333333333333334 - (1.0 + ((0.2222222222222222 / t) + -1.0));
	} else if (t <= 0.65) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.49d0)) then
        tmp = 0.8333333333333334d0 - (1.0d0 + ((0.2222222222222222d0 / t) + (-1.0d0)))
    else if (t <= 0.65d0) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.49) {
		tmp = 0.8333333333333334 - (1.0 + ((0.2222222222222222 / t) + -1.0));
	} else if (t <= 0.65) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.49:
		tmp = 0.8333333333333334 - (1.0 + ((0.2222222222222222 / t) + -1.0))
	elif t <= 0.65:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.49)
		tmp = Float64(0.8333333333333334 - Float64(1.0 + Float64(Float64(0.2222222222222222 / t) + -1.0)));
	elseif (t <= 0.65)
		tmp = 0.5;
	else
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.49)
		tmp = 0.8333333333333334 - (1.0 + ((0.2222222222222222 / t) + -1.0));
	elseif (t <= 0.65)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.49], N[(0.8333333333333334 - N[(1.0 + N[(N[(0.2222222222222222 / t), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.65], 0.5, N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.48999999999999999

   1. Initial program 100.0%

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

   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. Taylor expanded in t around inf 98.9%

      \[\leadsto \color{blue}{0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t}} \]
   6. 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%

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

   7. Simplified 98.9%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 96.3%

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

      3. frac-times 96.3%

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

      4. metadata-eval 96.3%

         \[\leadsto 0.8333333333333334 - \sqrt{\frac{\color{blue}{0.04938271604938271}}{t \cdot t}} \]

      5. metadata-eval 96.3%

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

      6. frac-times 96.3%

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

      7. sqrt-unprod 96.3%

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

      8. add-sqr-sqrt 96.3%

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

      9. expm1-log1p-u 96.3%

         \[\leadsto 0.8333333333333334 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.2222222222222222}{t}\right)\right)} \]

      10. expm1-undefine 96.3%

          \[\leadsto 0.8333333333333334 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{-0.2222222222222222}{t}\right)} - 1\right)} \]

      11. log1p-undefine 96.3%

          \[\leadsto 0.8333333333333334 - \left(e^{\color{blue}{\log \left(1 + \frac{-0.2222222222222222}{t}\right)}} - 1\right) \]

      12. add-exp-log 96.3%

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

      13. add-sqr-sqrt 96.3%

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

      14. sqrt-unprod 96.3%

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

      15. frac-times 96.3%

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

      16. metadata-eval 96.3%

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

      17. metadata-eval 96.3%

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

      18. frac-times 96.3%

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

      19. sqrt-unprod 0.0%

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

      20. add-sqr-sqrt 98.9%

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

   9. Applied egg-rr 98.9%

      \[\leadsto 0.8333333333333334 - \color{blue}{\left(\left(1 + \frac{0.2222222222222222}{t}\right) - 1\right)} \]
   10. Step-by-step derivation

       1. associate--l+ 98.9%

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

   11. Simplified 98.9%

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

   if -0.48999999999999999 < t < 0.650000000000000022

   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. Taylor expanded in t around 0 99.0%

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

   if 0.650000000000000022 < t

   1. Initial program 100.0%

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

   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. Taylor expanded in t around inf 96.6%

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

      1. associate-*r/ 96.6%

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

      2. metadata-eval 96.6%

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

   7. Simplified 96.6%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;0.8333333333333334 - \left(1 + \left(\frac{0.2222222222222222}{t} + -1\right)\right)\\ \mathbf{elif}\;t \leq 0.65:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.3% 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%

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

   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. Taylor expanded in t around inf 96.7%

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

   if -0.330000000000000016 < t < 1

   1. Initial program 100.0%

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

   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. Taylor expanded in t around 0 98.4%

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

4. Final simplification 97.6%

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

Alternative 7: 59.7% 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 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. Taylor expanded in t around 0 61.8%

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

6. Final simplification 61.8%

   \[\leadsto 0.5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024034 
(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)))))))))