Kahan p13 Example 1

Percentage Accurate: 99.9% → 100.0%

Time: 9.8s

Alternatives: 6

Speedup: 1.3×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 \cdot t}{1 + t}\\ t_2 := t_1 \cdot t_1\\ \frac{1 + t_2}{2 + t_2} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* 2.0 t) (+ 1.0 t))) (t_2 (* t_1 t_1)))
   (/ (+ 1.0 t_2) (+ 2.0 t_2))))

C

double code(double t) {
	double t_1 = (2.0 * t) / (1.0 + t);
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    t_1 = (2.0d0 * t) / (1.0d0 + t)
    t_2 = t_1 * t_1
    code = (1.0d0 + t_2) / (2.0d0 + t_2)
end function

Java

public static double code(double t) {
	double t_1 = (2.0 * t) / (1.0 + t);
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Python

def code(t):
	t_1 = (2.0 * t) / (1.0 + t)
	t_2 = t_1 * t_1
	return (1.0 + t_2) / (2.0 + t_2)

Julia

function code(t)
	t_1 = Float64(Float64(2.0 * t) / Float64(1.0 + t))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2))
end

MATLAB

function tmp = code(t)
	t_1 = (2.0 * t) / (1.0 + t);
	t_2 = t_1 * t_1;
	tmp = (1.0 + t_2) / (2.0 + t_2);
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]]]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 \cdot t}{1 + t}\\ t_2 := t_1 \cdot t_1\\ \frac{1 + t_2}{2 + t_2} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* 2.0 t) (+ 1.0 t))) (t_2 (* t_1 t_1)))
   (/ (+ 1.0 t_2) (+ 2.0 t_2))))

C

double code(double t) {
	double t_1 = (2.0 * t) / (1.0 + t);
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    t_1 = (2.0d0 * t) / (1.0d0 + t)
    t_2 = t_1 * t_1
    code = (1.0d0 + t_2) / (2.0d0 + t_2)
end function

Java

public static double code(double t) {
	double t_1 = (2.0 * t) / (1.0 + t);
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Python

def code(t):
	t_1 = (2.0 * t) / (1.0 + t)
	t_2 = t_1 * t_1
	return (1.0 + t_2) / (2.0 + t_2)

Julia

function code(t)
	t_1 = Float64(Float64(2.0 * t) / Float64(1.0 + t))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2))
end

MATLAB

function tmp = code(t)
	t_1 = (2.0 * t) / (1.0 + t);
	t_2 = t_1 * t_1;
	tmp = (1.0 + t_2) / (2.0 + t_2);
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 100.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{\left(0.25 + \frac{0.25}{t}\right) \cdot \left(1 + t\right)}\\ \frac{1 + t_1}{t_1 + 2} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ t (* (+ 0.25 (/ 0.25 t)) (+ 1.0 t)))))
   (/ (+ 1.0 t_1) (+ t_1 2.0))))

C

double code(double t) {
	double t_1 = t / ((0.25 + (0.25 / t)) * (1.0 + t));
	return (1.0 + t_1) / (t_1 + 2.0);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = t / ((0.25d0 + (0.25d0 / t)) * (1.0d0 + t))
    code = (1.0d0 + t_1) / (t_1 + 2.0d0)
end function

Java

public static double code(double t) {
	double t_1 = t / ((0.25 + (0.25 / t)) * (1.0 + t));
	return (1.0 + t_1) / (t_1 + 2.0);
}

Python

def code(t):
	t_1 = t / ((0.25 + (0.25 / t)) * (1.0 + t))
	return (1.0 + t_1) / (t_1 + 2.0)

Julia

function code(t)
	t_1 = Float64(t / Float64(Float64(0.25 + Float64(0.25 / t)) * Float64(1.0 + t)))
	return Float64(Float64(1.0 + t_1) / Float64(t_1 + 2.0))
end

MATLAB

function tmp = code(t)
	t_1 = t / ((0.25 + (0.25 / t)) * (1.0 + t));
	tmp = (1.0 + t_1) / (t_1 + 2.0);
end

Wolfram

code[t_] := Block[{t$95$1 = N[(t / N[(N[(0.25 + N[(0.25 / t), $MachinePrecision]), $MachinePrecision] * N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 + t$95$1), $MachinePrecision] / N[(t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

   1. div-inv 100.0%

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

   2. associate-*r* 99.6%

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

   3. associate-*l/ 73.8%

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

   4. swap-sqr 73.8%

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

   5. metadata-eval 73.8%

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

   6. *-commutative 73.8%

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

   7. associate-*r* 73.8%

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

   8. associate-/l* 99.6%

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

   9. frac-times 99.6%

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

   10. *-commutative 99.6%

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

   11. *-un-lft-identity 99.6%

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

   12. +-commutative 99.6%

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

   13. +-commutative 99.6%

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

3. Applied egg-rr 99.6%

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

   1. div-inv 100.0%

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

   2. associate-*r* 99.6%

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

   3. associate-*l/ 73.8%

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

   4. swap-sqr 73.8%

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

   5. metadata-eval 73.8%

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

   6. *-commutative 73.8%

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

   7. associate-*r* 73.8%

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

   8. associate-/l* 99.6%

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

   9. frac-times 99.6%

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

   10. *-commutative 99.6%

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

   11. *-un-lft-identity 99.6%

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

   12. +-commutative 99.6%

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

   13. +-commutative 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in t around 0 99.6%

   \[\leadsto \frac{1 + \frac{t}{\frac{t + 1}{t \cdot 4} \cdot \left(t + 1\right)}}{2 + \frac{t}{\color{blue}{\left(0.25 + 0.25 \cdot \frac{1}{t}\right)} \cdot \left(t + 1\right)}} \]
7. Step-by-step derivation

   1. associate-*r/ 99.6%

      \[\leadsto \frac{1 + \frac{t}{\frac{t + 1}{t \cdot 4} \cdot \left(t + 1\right)}}{2 + \frac{t}{\left(0.25 + \color{blue}{\frac{0.25 \cdot 1}{t}}\right) \cdot \left(t + 1\right)}} \]

   2. metadata-eval 99.6%

      \[\leadsto \frac{1 + \frac{t}{\frac{t + 1}{t \cdot 4} \cdot \left(t + 1\right)}}{2 + \frac{t}{\left(0.25 + \frac{\color{blue}{0.25}}{t}\right) \cdot \left(t + 1\right)}} \]

8. Simplified 99.6%

   \[\leadsto \frac{1 + \frac{t}{\frac{t + 1}{t \cdot 4} \cdot \left(t + 1\right)}}{2 + \frac{t}{\color{blue}{\left(0.25 + \frac{0.25}{t}\right)} \cdot \left(t + 1\right)}} \]

9. Taylor expanded in t around 0 100.0%

   \[\leadsto \frac{1 + \frac{t}{\color{blue}{\left(0.25 + 0.25 \cdot \frac{1}{t}\right)} \cdot \left(t + 1\right)}}{2 + \frac{t}{\left(0.25 + \frac{0.25}{t}\right) \cdot \left(t + 1\right)}} \]
10. Step-by-step derivation

    1. associate-*r/ 99.6%

       \[\leadsto \frac{1 + \frac{t}{\frac{t + 1}{t \cdot 4} \cdot \left(t + 1\right)}}{2 + \frac{t}{\left(0.25 + \color{blue}{\frac{0.25 \cdot 1}{t}}\right) \cdot \left(t + 1\right)}} \]

    2. metadata-eval 99.6%

       \[\leadsto \frac{1 + \frac{t}{\frac{t + 1}{t \cdot 4} \cdot \left(t + 1\right)}}{2 + \frac{t}{\left(0.25 + \frac{\color{blue}{0.25}}{t}\right) \cdot \left(t + 1\right)}} \]

11. Simplified 100.0%

    \[\leadsto \frac{1 + \frac{t}{\color{blue}{\left(0.25 + \frac{0.25}{t}\right)} \cdot \left(t + 1\right)}}{2 + \frac{t}{\left(0.25 + \frac{0.25}{t}\right) \cdot \left(t + 1\right)}} \]

12. Final simplification 100.0%

    \[\leadsto \frac{1 + \frac{t}{\left(0.25 + \frac{0.25}{t}\right) \cdot \left(1 + t\right)}}{\frac{t}{\left(0.25 + \frac{0.25}{t}\right) \cdot \left(1 + t\right)} + 2} \]

Alternative 2: 98.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{1 + t} \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\frac{1 + \frac{t}{\left(0.25 + \frac{0.25}{t}\right) \cdot \left(1 + t\right)}}{2 + \frac{t}{\frac{0.25}{t} + 0.5}}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* t 2.0) (+ 1.0 t)) 4e-11)
   (/
    (+ 1.0 (/ t (* (+ 0.25 (/ 0.25 t)) (+ 1.0 t))))
    (+ 2.0 (/ t (+ (/ 0.25 t) 0.5))))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))))

C

double code(double t) {
	double tmp;
	if (((t * 2.0) / (1.0 + t)) <= 4e-11) {
		tmp = (1.0 + (t / ((0.25 + (0.25 / t)) * (1.0 + t)))) / (2.0 + (t / ((0.25 / t) + 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 * 2.0d0) / (1.0d0 + t)) <= 4d-11) then
        tmp = (1.0d0 + (t / ((0.25d0 + (0.25d0 / t)) * (1.0d0 + t)))) / (2.0d0 + (t / ((0.25d0 / t) + 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 * 2.0) / (1.0 + t)) <= 4e-11) {
		tmp = (1.0 + (t / ((0.25 + (0.25 / t)) * (1.0 + t)))) / (2.0 + (t / ((0.25 / t) + 0.5)));
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if ((t * 2.0) / (1.0 + t)) <= 4e-11:
		tmp = (1.0 + (t / ((0.25 + (0.25 / t)) * (1.0 + t)))) / (2.0 + (t / ((0.25 / t) + 0.5)))
	else:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(t * 2.0) / Float64(1.0 + t)) <= 4e-11)
		tmp = Float64(Float64(1.0 + Float64(t / Float64(Float64(0.25 + Float64(0.25 / t)) * Float64(1.0 + t)))) / Float64(2.0 + Float64(t / Float64(Float64(0.25 / t) + 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 * 2.0) / (1.0 + t)) <= 4e-11)
		tmp = (1.0 + (t / ((0.25 + (0.25 / t)) * (1.0 + t)))) / (2.0 + (t / ((0.25 / t) + 0.5)));
	else
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[N[(N[(t * 2.0), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision], 4e-11], N[(N[(1.0 + N[(t / N[(N[(0.25 + N[(0.25 / t), $MachinePrecision]), $MachinePrecision] * N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(t / N[(N[(0.25 / t), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 2 t) (+.f64 1 t)) < 3.99999999999999976e-11

   1. Initial program 100.0%

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

      1. div-inv 100.0%

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

      2. associate-*r* 100.0%

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

      3. associate-*l/ 100.0%

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

      4. swap-sqr 100.0%

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

      5. metadata-eval 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. associate-/l* 100.0%

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

      9. frac-times 100.0%

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

      10. *-commutative 100.0%

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

      11. *-un-lft-identity 100.0%

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

      12. +-commutative 100.0%

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

      13. +-commutative 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. div-inv 100.0%

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

      2. associate-*r* 100.0%

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

      3. associate-*l/ 100.0%

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

      4. swap-sqr 100.0%

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

      5. metadata-eval 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. associate-/l* 100.0%

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

      9. frac-times 100.0%

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

      10. *-commutative 100.0%

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

      11. *-un-lft-identity 100.0%

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

      12. +-commutative 100.0%

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

      13. +-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in t around 0 99.8%

      \[\leadsto \frac{1 + \frac{t}{\frac{t + 1}{t \cdot 4} \cdot \left(t + 1\right)}}{2 + \frac{t}{\color{blue}{0.5 + 0.25 \cdot \frac{1}{t}}}} \]
   7. Step-by-step derivation

      1. +-commutative 99.8%

         \[\leadsto \frac{1 + \frac{t}{\frac{t + 1}{t \cdot 4} \cdot \left(t + 1\right)}}{2 + \frac{t}{\color{blue}{0.25 \cdot \frac{1}{t} + 0.5}}} \]

      2. associate-*r/ 99.8%

         \[\leadsto \frac{1 + \frac{t}{\frac{t + 1}{t \cdot 4} \cdot \left(t + 1\right)}}{2 + \frac{t}{\color{blue}{\frac{0.25 \cdot 1}{t}} + 0.5}} \]

      3. metadata-eval 99.8%

         \[\leadsto \frac{1 + \frac{t}{\frac{t + 1}{t \cdot 4} \cdot \left(t + 1\right)}}{2 + \frac{t}{\frac{\color{blue}{0.25}}{t} + 0.5}} \]

   8. Simplified 99.8%

      \[\leadsto \frac{1 + \frac{t}{\frac{t + 1}{t \cdot 4} \cdot \left(t + 1\right)}}{2 + \frac{t}{\color{blue}{\frac{0.25}{t} + 0.5}}} \]

   9. Taylor expanded in t around 0 99.8%

      \[\leadsto \frac{1 + \frac{t}{\color{blue}{\left(0.25 + 0.25 \cdot \frac{1}{t}\right)} \cdot \left(t + 1\right)}}{2 + \frac{t}{\frac{0.25}{t} + 0.5}} \]
   10. Step-by-step derivation

       1. associate-*r/ 100.0%

          \[\leadsto \frac{1 + \frac{t}{\frac{t + 1}{t \cdot 4} \cdot \left(t + 1\right)}}{2 + \frac{t}{\left(0.25 + \color{blue}{\frac{0.25 \cdot 1}{t}}\right) \cdot \left(t + 1\right)}} \]

       2. metadata-eval 100.0%

          \[\leadsto \frac{1 + \frac{t}{\frac{t + 1}{t \cdot 4} \cdot \left(t + 1\right)}}{2 + \frac{t}{\left(0.25 + \frac{\color{blue}{0.25}}{t}\right) \cdot \left(t + 1\right)}} \]

   11. Simplified 99.8%

       \[\leadsto \frac{1 + \frac{t}{\color{blue}{\left(0.25 + \frac{0.25}{t}\right)} \cdot \left(t + 1\right)}}{2 + \frac{t}{\frac{0.25}{t} + 0.5}} \]

   if 3.99999999999999976e-11 < (/.f64 (*.f64 2 t) (+.f64 1 t))

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   2. Taylor expanded in t around inf 97.9%

      \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{6 - 8 \cdot \frac{1}{t}}} \]
   3. Step-by-step derivation

      1. associate-*r/ 97.9%

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

      2. metadata-eval 97.9%

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

   4. Simplified 97.9%

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

   5. Taylor expanded in t around inf 98.4%

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

      1. associate-*r/ 98.4%

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

      2. metadata-eval 98.4%

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

   7. Simplified 98.4%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{1 + t} \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\frac{1 + \frac{t}{\left(0.25 + \frac{0.25}{t}\right) \cdot \left(1 + t\right)}}{2 + \frac{t}{\frac{0.25}{t} + 0.5}}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \]

Alternative 3: 98.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{1 + t} \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\frac{1 + \frac{t}{\frac{0.25}{t} \cdot \left(1 + t\right)}}{2 + \frac{t}{\frac{0.25}{t} + 0.5}}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* t 2.0) (+ 1.0 t)) 4e-11)
   (/ (+ 1.0 (/ t (* (/ 0.25 t) (+ 1.0 t)))) (+ 2.0 (/ t (+ (/ 0.25 t) 0.5))))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))))

C

double code(double t) {
	double tmp;
	if (((t * 2.0) / (1.0 + t)) <= 4e-11) {
		tmp = (1.0 + (t / ((0.25 / t) * (1.0 + t)))) / (2.0 + (t / ((0.25 / t) + 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 * 2.0d0) / (1.0d0 + t)) <= 4d-11) then
        tmp = (1.0d0 + (t / ((0.25d0 / t) * (1.0d0 + t)))) / (2.0d0 + (t / ((0.25d0 / t) + 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 * 2.0) / (1.0 + t)) <= 4e-11) {
		tmp = (1.0 + (t / ((0.25 / t) * (1.0 + t)))) / (2.0 + (t / ((0.25 / t) + 0.5)));
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if ((t * 2.0) / (1.0 + t)) <= 4e-11:
		tmp = (1.0 + (t / ((0.25 / t) * (1.0 + t)))) / (2.0 + (t / ((0.25 / t) + 0.5)))
	else:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(t * 2.0) / Float64(1.0 + t)) <= 4e-11)
		tmp = Float64(Float64(1.0 + Float64(t / Float64(Float64(0.25 / t) * Float64(1.0 + t)))) / Float64(2.0 + Float64(t / Float64(Float64(0.25 / t) + 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 * 2.0) / (1.0 + t)) <= 4e-11)
		tmp = (1.0 + (t / ((0.25 / t) * (1.0 + t)))) / (2.0 + (t / ((0.25 / t) + 0.5)));
	else
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[N[(N[(t * 2.0), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision], 4e-11], N[(N[(1.0 + N[(t / N[(N[(0.25 / t), $MachinePrecision] * N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(t / N[(N[(0.25 / t), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 2 t) (+.f64 1 t)) < 3.99999999999999976e-11

   1. Initial program 100.0%

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

      1. div-inv 100.0%

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

      2. associate-*r* 100.0%

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

      3. associate-*l/ 100.0%

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

      4. swap-sqr 100.0%

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

      5. metadata-eval 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. associate-/l* 100.0%

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

      9. frac-times 100.0%

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

      10. *-commutative 100.0%

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

      11. *-un-lft-identity 100.0%

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

      12. +-commutative 100.0%

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

      13. +-commutative 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. div-inv 100.0%

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

      2. associate-*r* 100.0%

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

      3. associate-*l/ 100.0%

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

      4. swap-sqr 100.0%

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

      5. metadata-eval 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. associate-/l* 100.0%

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

      9. frac-times 100.0%

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

      10. *-commutative 100.0%

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

      11. *-un-lft-identity 100.0%

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

      12. +-commutative 100.0%

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

      13. +-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in t around 0 99.8%

      \[\leadsto \frac{1 + \frac{t}{\frac{t + 1}{t \cdot 4} \cdot \left(t + 1\right)}}{2 + \frac{t}{\color{blue}{0.5 + 0.25 \cdot \frac{1}{t}}}} \]
   7. Step-by-step derivation

      1. +-commutative 99.8%

         \[\leadsto \frac{1 + \frac{t}{\frac{t + 1}{t \cdot 4} \cdot \left(t + 1\right)}}{2 + \frac{t}{\color{blue}{0.25 \cdot \frac{1}{t} + 0.5}}} \]

      2. associate-*r/ 99.8%

         \[\leadsto \frac{1 + \frac{t}{\frac{t + 1}{t \cdot 4} \cdot \left(t + 1\right)}}{2 + \frac{t}{\color{blue}{\frac{0.25 \cdot 1}{t}} + 0.5}} \]

      3. metadata-eval 99.8%

         \[\leadsto \frac{1 + \frac{t}{\frac{t + 1}{t \cdot 4} \cdot \left(t + 1\right)}}{2 + \frac{t}{\frac{\color{blue}{0.25}}{t} + 0.5}} \]

   8. Simplified 99.8%

      \[\leadsto \frac{1 + \frac{t}{\frac{t + 1}{t \cdot 4} \cdot \left(t + 1\right)}}{2 + \frac{t}{\color{blue}{\frac{0.25}{t} + 0.5}}} \]

   9. Taylor expanded in t around 0 99.7%

      \[\leadsto \frac{1 + \frac{t}{\color{blue}{\frac{0.25}{t}} \cdot \left(t + 1\right)}}{2 + \frac{t}{\frac{0.25}{t} + 0.5}} \]

   if 3.99999999999999976e-11 < (/.f64 (*.f64 2 t) (+.f64 1 t))

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   2. Taylor expanded in t around inf 97.9%

      \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{6 - 8 \cdot \frac{1}{t}}} \]
   3. Step-by-step derivation

      1. associate-*r/ 97.9%

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

      2. metadata-eval 97.9%

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

   4. Simplified 97.9%

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

   5. Taylor expanded in t around inf 98.4%

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

      1. associate-*r/ 98.4%

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

      2. metadata-eval 98.4%

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

   7. Simplified 98.4%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{1 + t} \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\frac{1 + \frac{t}{\frac{0.25}{t} \cdot \left(1 + t\right)}}{2 + \frac{t}{\frac{0.25}{t} + 0.5}}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \]

Alternative 4: 98.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{1 + t} \leq 4 \cdot 10^{-11}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (* t 2.0) (+ 1.0 t)) 4e-11)
   0.5
   (- 0.8333333333333334 (/ 0.2222222222222222 t))))

C

double code(double t) {
	double tmp;
	if (((t * 2.0) / (1.0 + t)) <= 4e-11) {
		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 * 2.0d0) / (1.0d0 + t)) <= 4d-11) 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 * 2.0) / (1.0 + t)) <= 4e-11) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if ((t * 2.0) / (1.0 + t)) <= 4e-11:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(t * 2.0) / Float64(1.0 + t)) <= 4e-11)
		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 * 2.0) / (1.0 + t)) <= 4e-11)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[N[(N[(t * 2.0), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision], 4e-11], 0.5, N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 2 t) (+.f64 1 t)) < 3.99999999999999976e-11

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. associate-*r/ 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. swap-sqr 100.0%

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

      6. associate-*r* 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 98.3%

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

   5. Taylor expanded in t around 0 99.6%

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

   if 3.99999999999999976e-11 < (/.f64 (*.f64 2 t) (+.f64 1 t))

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   2. Taylor expanded in t around inf 97.9%

      \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{\color{blue}{6 - 8 \cdot \frac{1}{t}}} \]
   3. Step-by-step derivation

      1. associate-*r/ 97.9%

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

      2. metadata-eval 97.9%

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

   4. Simplified 97.9%

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

   5. Taylor expanded in t around inf 98.4%

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

      1. associate-*r/ 98.4%

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

      2. metadata-eval 98.4%

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

   7. Simplified 98.4%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t \cdot 2}{1 + t} \leq 4 \cdot 10^{-11}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \]

Alternative 5: 98.4% accurate, 6.8× 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%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]

   2. Taylor expanded in t around inf 96.2%

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

   3. Taylor expanded in t around inf 96.5%

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

   if -0.330000000000000016 < t < 1

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. associate-*r/ 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. swap-sqr 100.0%

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

      6. associate-*r* 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 98.3%

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

   5. Taylor expanded in t around 0 99.6%

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

4. Final simplification 97.9%

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

Alternative 6: 59.4% accurate, 35.0× 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%

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

   1. associate-*l/ 99.6%

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

   2. associate-*r/ 73.8%

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

   3. *-commutative 73.8%

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

   4. *-commutative 73.8%

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

   5. swap-sqr 73.8%

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

   6. associate-*r* 73.8%

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

   7. metadata-eval 73.8%

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

   8. associate-*l/ 73.8%

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

3. Simplified 73.0%

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

4. Taylor expanded in t around inf 71.1%

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

5. Taylor expanded in t around 0 57.1%

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

6. Final simplification 57.1%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023320 
(FPCore (t)
  :name "Kahan p13 Example 1"
  :precision binary64
  (/ (+ 1.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t)))) (+ 2.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t))))))