Kahan p13 Example 3

Percentage Accurate: 100.0% → 100.0%

Time: 9.1s

Alternatives: 6

Speedup: 0.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. sub-neg 100.0%

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

   2. distribute-neg-frac 100.0%

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

   3. metadata-eval 100.0%

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

   4. +-commutative 100.0%

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

   5. fma-define 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. clear-num 100.0%

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

   2. inv-pow 100.0%

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

   3. div-inv 100.0%

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

   4. clear-num 100.0%

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

   5. div-inv 100.0%

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

   6. metadata-eval 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - {\left(\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot \color{blue}{0.5}\right)\right)}^{-1}\right)} \]

4. Applied egg-rr 100.0%

   \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{{\left(\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot 0.5\right)\right)}^{-1}}\right)} \]
5. Step-by-step derivation

   1. unpow-1 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\frac{1}{\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot 0.5\right)}}\right)} \]

   2. *-commutative 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{\left(t \cdot 0.5\right) \cdot \left(1 + \frac{1}{t}\right)}}\right)} \]

   3. *-commutative 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{\left(0.5 \cdot t\right)} \cdot \left(1 + \frac{1}{t}\right)}\right)} \]

   4. associate-*l* 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{0.5 \cdot \left(t \cdot \left(1 + \frac{1}{t}\right)\right)}}\right)} \]

   5. distribute-lft-in 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \color{blue}{\left(t \cdot 1 + t \cdot \frac{1}{t}\right)}}\right)} \]

   6. *-rgt-identity 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \left(\color{blue}{t} + t \cdot \frac{1}{t}\right)}\right)} \]

   7. rgt-mult-inverse 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \left(t + \color{blue}{1}\right)}\right)} \]

6. Simplified 100.0%

   \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\frac{1}{0.5 \cdot \left(t + 1\right)}}\right)} \]
7. Step-by-step derivation

   1. sub-neg 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \color{blue}{\left(2 + \left(-\frac{1}{0.5 \cdot \left(t + 1\right)}\right)\right)}} \]

   2. associate-/r* 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 + \left(-\color{blue}{\frac{\frac{1}{0.5}}{t + 1}}\right)\right)} \]

   3. metadata-eval 100.0%

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

   4. distribute-neg-frac 100.0%

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

   5. metadata-eval 100.0%

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

   6. +-commutative 100.0%

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

8. Applied egg-rr 100.0%

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

   1. remove-double-neg 100.0%

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

   2. unsub-neg 100.0%

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

   3. distribute-neg-frac 100.0%

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

   4. metadata-eval 100.0%

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

   5. +-commutative 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%

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

11. Final simplification 100.0%

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

Alternative 3: 98.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < -0.48999999999999999 or 0.660000000000000031 < t

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 99.9%

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

      1. distribute-neg-in 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-*r/ 99.9%

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

      4. metadata-eval 99.9%

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

      5. distribute-neg-frac 99.9%

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

      6. metadata-eval 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in t around 0 99.9%

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

      1. associate-*r/ 99.9%

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

      2. metadata-eval 99.9%

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

   10. Simplified 99.9%

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

   if -0.48999999999999999 < t < 0.660000000000000031

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 100.0%

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

4. Final simplification 99.9%

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

Alternative 4: 98.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double t) {
	double tmp;
	if (t <= -0.49) {
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	} else if (t <= 0.66) {
		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 = 1.0d0 + ((-0.16666666666666666d0) + ((-0.2222222222222222d0) / t))
    else if (t <= 0.66d0) 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 = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	} else if (t <= 0.66) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.49:
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t))
	elif t <= 0.66:
		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(1.0 + Float64(-0.16666666666666666 + Float64(-0.2222222222222222 / t)));
	elseif (t <= 0.66)
		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 = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	elseif (t <= 0.66)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	end
	tmp_2 = tmp;
end

Wolfram

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

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

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 100.0%

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

      1. distribute-neg-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-*r/ 100.0%

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

      4. metadata-eval 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   if -0.48999999999999999 < t < 0.660000000000000031

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 100.0%

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

   if 0.660000000000000031 < t

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 99.8%

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

      1. distribute-neg-in 99.8%

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

      2. metadata-eval 99.8%

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

      3. associate-*r/ 99.8%

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

      4. metadata-eval 99.8%

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

      5. distribute-neg-frac 99.8%

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

      6. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in t around 0 99.8%

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

      1. associate-*r/ 99.8%

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

      2. metadata-eval 99.8%

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

   10. Simplified 99.8%

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

4. Final simplification 99.9%

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

Alternative 5: 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%

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

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 99.9%

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

      1. distribute-neg-in 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-*r/ 99.9%

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

      4. metadata-eval 99.9%

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

      5. distribute-neg-frac 99.9%

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

      6. metadata-eval 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in t around inf 98.9%

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

   if -0.330000000000000016 < t < 1

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 100.0%

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

4. Final simplification 99.5%

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

Alternative 6: 58.5% accurate, 29.0× speedup

Math

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

FPCore

(FPCore (t) :precision binary64 0.8333333333333334)

C

double code(double t) {
	return 0.8333333333333334;
}

Fortran

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

Java

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

Python

def code(t):
	return 0.8333333333333334

Julia

function code(t)
	return 0.8333333333333334
end

MATLAB

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

Wolfram

code[t_] := 0.8333333333333334

Derivation

1. Initial program 100.0%

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

   1. sub-neg 100.0%

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

   2. distribute-neg-frac 100.0%

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

   3. metadata-eval 100.0%

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

   4. +-commutative 100.0%

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

   5. fma-define 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in t around inf 49.0%

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

   1. distribute-neg-in 49.0%

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

   2. metadata-eval 49.0%

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

   3. associate-*r/ 49.0%

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

   4. metadata-eval 49.0%

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

   5. distribute-neg-frac 49.0%

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

   6. metadata-eval 49.0%

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

7. Simplified 49.0%

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

8. Taylor expanded in t around inf 57.1%

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

9. Final simplification 57.1%

   \[\leadsto 0.8333333333333334 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024044 
(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)))))))))