Kahan p13 Example 3

Percentage Accurate: 100.0% → 100.0%

Time: 12.8s

Alternatives: 6

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

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 / (2.0d0 + (((4.0d0 - ((-4.0d0) / ((1.0d0 + t) * ((-1.0d0) - t)))) / (2.0d0 - t_1)) * (2.0d0 + t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. sub-neg 100.0%

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

   2. flip-+ 100.0%

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

   3. metadata-eval 100.0%

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

   4. associate-/l/ 100.0%

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

   5. distribute-neg-frac 100.0%

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

   6. metadata-eval 100.0%

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

   7. *-commutative 100.0%

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

   8. associate-/l/ 100.0%

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

   9. distribute-neg-frac 100.0%

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

   10. metadata-eval 100.0%

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

   11. *-commutative 100.0%

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

   12. associate-/l/ 100.0%

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

   13. distribute-neg-frac 100.0%

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

4. Applied egg-rr 100.0%

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

   1. frac-2neg 100.0%

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

   2. metadata-eval 100.0%

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

   3. frac-times 100.0%

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

   4. metadata-eval 100.0%

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

   5. distribute-rgt-in 100.0%

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

   6. *-un-lft-identity 100.0%

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

   7. inv-pow 100.0%

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

   8. pow-plus 100.0%

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

   9. metadata-eval 100.0%

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

   10. metadata-eval 100.0%

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

   11. distribute-rgt-in 100.0%

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

   12. *-un-lft-identity 100.0%

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

   13. inv-pow 100.0%

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

   14. pow-plus 100.0%

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

   15. metadata-eval 100.0%

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

   16. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in t around 0 100.0%

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

   1. +-commutative 100.0%

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

9. Simplified 100.0%

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

    1. sub-neg 100.0%

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

    2. associate-/l/ 100.0%

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

    3. distribute-neg-frac 100.0%

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

    4. metadata-eval 100.0%

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

    5. *-commutative 100.0%

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

11. Applied egg-rr 100.0%

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

    1. distribute-lft-in 100.0%

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

    2. rgt-mult-inverse 100.0%

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

    3. *-rgt-identity 100.0%

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

13. Simplified 100.0%

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

14. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

Python

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

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(2.0 / 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 / (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[(2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 + N[(-1.0 / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 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. expm1-log1p-u 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)}\right)} \]

   2. expm1-udef 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} - 1\right)}\right)} \]

   3. associate-/l/ 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{2}{\left(1 + \frac{1}{t}\right) \cdot t}}\right)} - 1\right)\right)} \]

   4. *-commutative 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \left(e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{t \cdot \left(1 + \frac{1}{t}\right)}}\right)} - 1\right)\right)} \]

4. Applied egg-rr 100.0%

   \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}\right)} - 1\right)}\right)} \]
5. Step-by-step derivation

   1. expm1-def 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}\right)\right)}\right)} \]

   2. expm1-log1p 100.0%

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

   3. distribute-lft-in 100.0%

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

   4. rgt-mult-inverse 100.0%

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

   5. *-rgt-identity 100.0%

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

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. Step-by-step derivation

   1. expm1-log1p-u 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)}\right)} \]

   2. expm1-udef 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} - 1\right)}\right)} \]

   3. associate-/l/ 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{2}{\left(1 + \frac{1}{t}\right) \cdot t}}\right)} - 1\right)\right)} \]

   4. *-commutative 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \left(e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{t \cdot \left(1 + \frac{1}{t}\right)}}\right)} - 1\right)\right)} \]

8. Applied egg-rr 100.0%

   \[\leadsto 1 - \frac{1}{2 + \left(2 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}\right)} - 1\right)}\right) \cdot \left(2 - \frac{2}{t + 1}\right)} \]
9. Step-by-step derivation

   1. expm1-def 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}\right)\right)}\right)} \]

   2. expm1-log1p 100.0%

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

   3. distribute-lft-in 100.0%

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

   4. rgt-mult-inverse 100.0%

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

   5. *-rgt-identity 100.0%

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

10. Simplified 100.0%

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

11. Final simplification 100.0%

    \[\leadsto 1 + \frac{-1}{2 + \left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right)} \]
12. 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. associate-*l/ 100.0%

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

   2. frac-2neg 100.0%

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

   3. +-commutative 100.0%

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

   4. distribute-lft-in 100.0%

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

   5. metadata-eval 100.0%

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

   6. distribute-neg-in 100.0%

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

   7. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. distribute-neg-in 100.0%

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

   2. metadata-eval 100.0%

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

   3. associate-*r/ 100.0%

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

   4. metadata-eval 100.0%

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

   5. distribute-neg-frac 100.0%

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

   6. metadata-eval 100.0%

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

   7. +-commutative 100.0%

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

   8. unsub-neg 100.0%

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

8. Simplified 100.0%

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

9. Final simplification 100.0%

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

Alternative 4: 49.9% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ 0.8333333333333334 - \frac{0.2222222222222222}{t} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (- 0.8333333333333334 (/ 0.2222222222222222 t)))

C

double code(double t) {
	return 0.8333333333333334 - (0.2222222222222222 / t);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
end function

Java

public static double code(double t) {
	return 0.8333333333333334 - (0.2222222222222222 / t);
}

Python

def code(t):
	return 0.8333333333333334 - (0.2222222222222222 / t)

Julia

function code(t)
	return Float64(0.8333333333333334 - Float64(0.2222222222222222 / t))
end

MATLAB

function tmp = code(t)
	tmp = 0.8333333333333334 - (0.2222222222222222 / t);
end

Wolfram

code[t_] := N[(0.8333333333333334 - N[(0.2222222222222222 / t), $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. Taylor expanded in t around inf 51.4%

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

   1. associate-*r/ 51.4%

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

   2. metadata-eval 51.4%

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

7. Simplified 51.4%

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

8. Final simplification 51.4%

   \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222}{t} \]
9. Add Preprocessing

Alternative 5: 60.3% 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 59.2%

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

6. Final simplification 59.2%

   \[\leadsto 0.5 \]
7. Add Preprocessing

Alternative 6: 57.8% 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%

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

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 59.0%

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

6. Final simplification 59.0%

   \[\leadsto 0.8333333333333334 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024033 
(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)))))))))