Kahan p13 Example 3

Percentage Accurate: 100.0% → 100.0%

Time: 9.5s

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, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. 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. distribute-neg-frac 100.0%

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

   5. distribute-neg-frac 100.0%

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

   6. metadata-eval 100.0%

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

   7. distribute-neg-frac 100.0%

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

   8. distribute-neg-frac 100.0%

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

   9. metadata-eval 100.0%

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

   10. distribute-neg-frac 100.0%

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

   11. distribute-neg-frac 100.0%

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

   12. metadata-eval 100.0%

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

4. Applied egg-rr 100.0%

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

   1. sub-neg 100.0%

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

   2. distribute-neg-frac 100.0%

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

   3. distribute-neg-frac 100.0%

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

   4. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. associate-/r* 100.0%

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

   2. distribute-lft-in 100.0%

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

   3. *-rgt-identity 100.0%

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

   4. rgt-mult-inverse 100.0%

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

8. Simplified 100.0%

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

9. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. *-un-lft-identity 100.0%

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

4. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

   2. associate-/r* 100.0%

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

   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-identity 100.0%

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

   5. rgt-mult-inverse 100.0%

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

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. sub-neg 100.0%

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

   2. distribute-neg-frac 100.0%

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

   3. distribute-neg-frac 100.0%

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

   4. metadata-eval 100.0%

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

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* 100.0%

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

   2. distribute-lft-in 100.0%

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

   3. *-rgt-identity 100.0%

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

   4. rgt-mult-inverse 100.0%

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

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%

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

Alternative 3: 59.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.82 \lor \neg \left(t \leq 0.145\right):\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035}{t} + -0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;1 + t \cdot 0.125\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.82) (not (<= t 0.145)))
   (+
    0.8333333333333334
    (/ (+ (/ 0.037037037037037035 t) -0.2222222222222222) t))
   (+ 1.0 (* t 0.125))))

C

double code(double t) {
	double tmp;
	if ((t <= -0.82) || !(t <= 0.145)) {
		tmp = 0.8333333333333334 + (((0.037037037037037035 / t) + -0.2222222222222222) / t);
	} else {
		tmp = 1.0 + (t * 0.125);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.82d0)) .or. (.not. (t <= 0.145d0))) then
        tmp = 0.8333333333333334d0 + (((0.037037037037037035d0 / t) + (-0.2222222222222222d0)) / t)
    else
        tmp = 1.0d0 + (t * 0.125d0)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.82) || !(t <= 0.145)) {
		tmp = 0.8333333333333334 + (((0.037037037037037035 / t) + -0.2222222222222222) / t);
	} else {
		tmp = 1.0 + (t * 0.125);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.82) or not (t <= 0.145):
		tmp = 0.8333333333333334 + (((0.037037037037037035 / t) + -0.2222222222222222) / t)
	else:
		tmp = 1.0 + (t * 0.125)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.82) || !(t <= 0.145))
		tmp = Float64(0.8333333333333334 + Float64(Float64(Float64(0.037037037037037035 / t) + -0.2222222222222222) / t));
	else
		tmp = Float64(1.0 + Float64(t * 0.125));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.82) || ~((t <= 0.145)))
		tmp = 0.8333333333333334 + (((0.037037037037037035 / t) + -0.2222222222222222) / t);
	else
		tmp = 1.0 + (t * 0.125);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.82], N[Not[LessEqual[t, 0.145]], $MachinePrecision]], N[(0.8333333333333334 + N[(N[(N[(0.037037037037037035 / t), $MachinePrecision] + -0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(t * 0.125), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.819999999999999951 or 0.14499999999999999 < 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 98.4%

      \[\leadsto \color{blue}{\left(0.8333333333333334 + \frac{0.037037037037037035}{{t}^{2}}\right) - 0.2222222222222222 \cdot \frac{1}{t}} \]
   4. Step-by-step derivation

      1. associate--l+ 98.4%

         \[\leadsto \color{blue}{0.8333333333333334 + \left(\frac{0.037037037037037035}{{t}^{2}} - 0.2222222222222222 \cdot \frac{1}{t}\right)} \]

      2. unpow2 98.4%

         \[\leadsto 0.8333333333333334 + \left(\frac{0.037037037037037035}{\color{blue}{t \cdot t}} - 0.2222222222222222 \cdot \frac{1}{t}\right) \]

      3. associate-/r* 98.4%

         \[\leadsto 0.8333333333333334 + \left(\color{blue}{\frac{\frac{0.037037037037037035}{t}}{t}} - 0.2222222222222222 \cdot \frac{1}{t}\right) \]

      4. metadata-eval 98.4%

         \[\leadsto 0.8333333333333334 + \left(\frac{\frac{\color{blue}{0.037037037037037035 \cdot 1}}{t}}{t} - 0.2222222222222222 \cdot \frac{1}{t}\right) \]

      5. associate-*r/ 98.4%

         \[\leadsto 0.8333333333333334 + \left(\frac{\color{blue}{0.037037037037037035 \cdot \frac{1}{t}}}{t} - 0.2222222222222222 \cdot \frac{1}{t}\right) \]

      6. associate-*r/ 98.4%

         \[\leadsto 0.8333333333333334 + \left(\frac{0.037037037037037035 \cdot \frac{1}{t}}{t} - \color{blue}{\frac{0.2222222222222222 \cdot 1}{t}}\right) \]

      7. metadata-eval 98.4%

         \[\leadsto 0.8333333333333334 + \left(\frac{0.037037037037037035 \cdot \frac{1}{t}}{t} - \frac{\color{blue}{0.2222222222222222}}{t}\right) \]

      8. div-sub 98.4%

         \[\leadsto 0.8333333333333334 + \color{blue}{\frac{0.037037037037037035 \cdot \frac{1}{t} - 0.2222222222222222}{t}} \]

      9. remove-double-neg 98.4%

         \[\leadsto 0.8333333333333334 + \frac{\color{blue}{-\left(-\left(0.037037037037037035 \cdot \frac{1}{t} - 0.2222222222222222\right)\right)}}{t} \]

      10. mul-1-neg 98.4%

          \[\leadsto 0.8333333333333334 + \frac{-\color{blue}{-1 \cdot \left(0.037037037037037035 \cdot \frac{1}{t} - 0.2222222222222222\right)}}{t} \]

      11. sub-neg 98.4%

          \[\leadsto 0.8333333333333334 + \frac{--1 \cdot \color{blue}{\left(0.037037037037037035 \cdot \frac{1}{t} + \left(-0.2222222222222222\right)\right)}}{t} \]

      12. distribute-lft-in 98.4%

          \[\leadsto 0.8333333333333334 + \frac{-\color{blue}{\left(-1 \cdot \left(0.037037037037037035 \cdot \frac{1}{t}\right) + -1 \cdot \left(-0.2222222222222222\right)\right)}}{t} \]

      13. neg-mul-1 98.4%

          \[\leadsto 0.8333333333333334 + \frac{-\left(\color{blue}{\left(-0.037037037037037035 \cdot \frac{1}{t}\right)} + -1 \cdot \left(-0.2222222222222222\right)\right)}{t} \]

      14. metadata-eval 98.4%

          \[\leadsto 0.8333333333333334 + \frac{-\left(\left(-0.037037037037037035 \cdot \frac{1}{t}\right) + -1 \cdot \color{blue}{-0.2222222222222222}\right)}{t} \]

      15. metadata-eval 98.4%

          \[\leadsto 0.8333333333333334 + \frac{-\left(\left(-0.037037037037037035 \cdot \frac{1}{t}\right) + \color{blue}{0.2222222222222222}\right)}{t} \]

      16. +-commutative 98.4%

          \[\leadsto 0.8333333333333334 + \frac{-\color{blue}{\left(0.2222222222222222 + \left(-0.037037037037037035 \cdot \frac{1}{t}\right)\right)}}{t} \]

      17. sub-neg 98.4%

          \[\leadsto 0.8333333333333334 + \frac{-\color{blue}{\left(0.2222222222222222 - 0.037037037037037035 \cdot \frac{1}{t}\right)}}{t} \]

      18. distribute-neg-frac 98.4%

          \[\leadsto 0.8333333333333334 + \color{blue}{\left(-\frac{0.2222222222222222 - 0.037037037037037035 \cdot \frac{1}{t}}{t}\right)} \]

      19. mul-1-neg 98.4%

          \[\leadsto 0.8333333333333334 + \color{blue}{-1 \cdot \frac{0.2222222222222222 - 0.037037037037037035 \cdot \frac{1}{t}}{t}} \]

   5. Simplified 98.4%

      \[\leadsto \color{blue}{0.8333333333333334 + \frac{\frac{0.037037037037037035}{t} + -0.2222222222222222}{t}} \]

   if -0.819999999999999951 < t < 0.14499999999999999

   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 18.8%

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

      1. associate-*r/ 18.8%

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

      2. metadata-eval 18.8%

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

   5. Simplified 18.8%

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

   6. Taylor expanded in t around 0 18.8%

      \[\leadsto \color{blue}{1 + 0.125 \cdot t} \]
   7. Step-by-step derivation

      1. *-commutative 18.8%

         \[\leadsto 1 + \color{blue}{t \cdot 0.125} \]

   8. Simplified 18.8%

      \[\leadsto \color{blue}{1 + t \cdot 0.125} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.82 \lor \neg \left(t \leq 0.145\right):\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035}{t} + -0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;1 + t \cdot 0.125\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
2. Add Preprocessing
3. 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. distribute-neg-frac 100.0%

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

   5. distribute-neg-frac 100.0%

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

   6. metadata-eval 100.0%

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

   7. distribute-neg-frac 100.0%

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

   8. distribute-neg-frac 100.0%

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

   9. metadata-eval 100.0%

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

   10. distribute-neg-frac 100.0%

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

   11. distribute-neg-frac 100.0%

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

   12. metadata-eval 100.0%

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

4. Applied egg-rr 100.0%

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

   1. sub-neg 100.0%

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

   2. distribute-neg-frac 100.0%

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

   3. distribute-neg-frac 100.0%

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

   4. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. associate-/r* 100.0%

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

   2. distribute-lft-in 100.0%

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

   3. *-rgt-identity 100.0%

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

   4. rgt-mult-inverse 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in t around inf 97.5%

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

    1. associate-*r/ 97.5%

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

    2. metadata-eval 97.5%

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

11. Simplified 97.5%

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

12. Final simplification 97.5%

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

Alternative 5: 59.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.78 \lor \neg \left(t \leq 0.43\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;1 + t \cdot 0.125\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.78) (not (<= t 0.43)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (+ 1.0 (* t 0.125))))

C

double code(double t) {
	double tmp;
	if ((t <= -0.78) || !(t <= 0.43)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = 1.0 + (t * 0.125);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.78d0)) .or. (.not. (t <= 0.43d0))) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else
        tmp = 1.0d0 + (t * 0.125d0)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.78) || !(t <= 0.43)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = 1.0 + (t * 0.125);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.78) or not (t <= 0.43):
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = 1.0 + (t * 0.125)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.78) || !(t <= 0.43))
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = Float64(1.0 + Float64(t * 0.125));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.78) || ~((t <= 0.43)))
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	else
		tmp = 1.0 + (t * 0.125);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.78], N[Not[LessEqual[t, 0.43]], $MachinePrecision]], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(t * 0.125), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.78000000000000003 or 0.429999999999999993 < 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 98.0%

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

      1. associate-*r/ 98.0%

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

      2. metadata-eval 98.0%

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

   5. Simplified 98.0%

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

   if -0.78000000000000003 < t < 0.429999999999999993

   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 18.8%

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

      1. associate-*r/ 18.8%

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

      2. metadata-eval 18.8%

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

   5. Simplified 18.8%

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

   6. Taylor expanded in t around 0 18.8%

      \[\leadsto \color{blue}{1 + 0.125 \cdot t} \]
   7. Step-by-step derivation

      1. *-commutative 18.8%

         \[\leadsto 1 + \color{blue}{t \cdot 0.125} \]

   8. Simplified 18.8%

      \[\leadsto \color{blue}{1 + t \cdot 0.125} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.78 \lor \neg \left(t \leq 0.43\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;1 + t \cdot 0.125\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{1}{\frac{8 + \frac{-12}{t}}{t} - 6} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (+ 1.0 (/ 1.0 (- (/ (+ 8.0 (/ -12.0 t)) t) 6.0))))

C

double code(double t) {
	return 1.0 + (1.0 / (((8.0 + (-12.0 / t)) / t) - 6.0));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = 1.0d0 + (1.0d0 / (((8.0d0 + ((-12.0d0) / t)) / t) - 6.0d0))
end function

Java

public static double code(double t) {
	return 1.0 + (1.0 / (((8.0 + (-12.0 / t)) / t) - 6.0));
}

Python

def code(t):
	return 1.0 + (1.0 / (((8.0 + (-12.0 / t)) / t) - 6.0))

Julia

function code(t)
	return Float64(1.0 + Float64(1.0 / Float64(Float64(Float64(8.0 + Float64(-12.0 / t)) / t) - 6.0)))
end

MATLAB

function tmp = code(t)
	tmp = 1.0 + (1.0 / (((8.0 + (-12.0 / t)) / t) - 6.0));
end

Wolfram

code[t_] := N[(1.0 + N[(1.0 / N[(N[(N[(8.0 + N[(-12.0 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in t around -inf 52.6%

   \[\leadsto 1 - \frac{1}{\color{blue}{6 + -1 \cdot \frac{8 - 12 \cdot \frac{1}{t}}{t}}} \]
4. Step-by-step derivation

   1. mul-1-neg 52.6%

      \[\leadsto 1 - \frac{1}{6 + \color{blue}{\left(-\frac{8 - 12 \cdot \frac{1}{t}}{t}\right)}} \]

   2. unsub-neg 52.6%

      \[\leadsto 1 - \frac{1}{\color{blue}{6 - \frac{8 - 12 \cdot \frac{1}{t}}{t}}} \]

   3. sub-neg 52.6%

      \[\leadsto 1 - \frac{1}{6 - \frac{\color{blue}{8 + \left(-12 \cdot \frac{1}{t}\right)}}{t}} \]

   4. associate-*r/ 52.6%

      \[\leadsto 1 - \frac{1}{6 - \frac{8 + \left(-\color{blue}{\frac{12 \cdot 1}{t}}\right)}{t}} \]

   5. metadata-eval 52.6%

      \[\leadsto 1 - \frac{1}{6 - \frac{8 + \left(-\frac{\color{blue}{12}}{t}\right)}{t}} \]

   6. distribute-neg-frac 52.6%

      \[\leadsto 1 - \frac{1}{6 - \frac{8 + \color{blue}{\frac{-12}{t}}}{t}} \]

   7. metadata-eval 52.6%

      \[\leadsto 1 - \frac{1}{6 - \frac{8 + \frac{\color{blue}{-12}}{t}}{t}} \]

5. Simplified 52.6%

   \[\leadsto 1 - \frac{1}{\color{blue}{6 - \frac{8 + \frac{-12}{t}}{t}}} \]

6. Final simplification 52.6%

   \[\leadsto 1 + \frac{1}{\frac{8 + \frac{-12}{t}}{t} - 6} \]
7. Add Preprocessing

Alternative 7: 59.5% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in t around inf 52.4%

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

   1. associate-*r/ 52.4%

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

   2. metadata-eval 52.4%

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

5. Simplified 52.4%

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

6. Final simplification 52.4%

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

Alternative 8: 10.9% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ 1 + t \cdot 0.125 \end{array} \]

FPCore

(FPCore (t) :precision binary64 (+ 1.0 (* t 0.125)))

C

double code(double t) {
	return 1.0 + (t * 0.125);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = 1.0d0 + (t * 0.125d0)
end function

Java

public static double code(double t) {
	return 1.0 + (t * 0.125);
}

Python

def code(t):
	return 1.0 + (t * 0.125)

Julia

function code(t)
	return Float64(1.0 + Float64(t * 0.125))
end

MATLAB

function tmp = code(t)
	tmp = 1.0 + (t * 0.125);
end

Wolfram

code[t_] := N[(1.0 + N[(t * 0.125), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in t around inf 52.4%

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

   1. associate-*r/ 52.4%

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

   2. metadata-eval 52.4%

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

5. Simplified 52.4%

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

6. Taylor expanded in t around 0 12.4%

   \[\leadsto \color{blue}{1 + 0.125 \cdot t} \]
7. Step-by-step derivation

   1. *-commutative 12.4%

      \[\leadsto 1 + \color{blue}{t \cdot 0.125} \]

8. Simplified 12.4%

   \[\leadsto \color{blue}{1 + t \cdot 0.125} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024179 
(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)))))))))