Kahan p13 Example 2

Percentage Accurate: 100.0% → 100.0%

Time: 11.3s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 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 - ((2.0 / t) / (1.0 + (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 - ((2.0d0 / t) / (1.0d0 + (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 - ((2.0 / t) / (1.0 + (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 - ((2.0 / t) / (1.0 + (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(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + 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 - ((2.0 / t) / (1.0 + (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[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $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: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 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 - ((2.0 / t) / (1.0 + (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 - ((2.0d0 / t) / (1.0d0 + (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 - ((2.0 / t) / (1.0 + (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 - ((2.0 / t) / (1.0 + (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(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + 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 - ((2.0 / t) / (1.0 + (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[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 + \frac{1}{t}\\ t_2 := 2 - \frac{\frac{2}{t}}{t_1}\\ \frac{1 + \left(1 + \left({\left(2 + \frac{\frac{-2}{t}}{t_1}\right)}^{2} + -1\right)\right)}{2 + t_2 \cdot t_2} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double t) {
	double t_1 = 1.0 + (1.0 / t);
	double t_2 = 2.0 - ((2.0 / t) / t_1);
	return (1.0 + (1.0 + (Math.pow((2.0 + ((-2.0 / t) / t_1)), 2.0) + -1.0))) / (2.0 + (t_2 * t_2));
}

Python

def code(t):
	t_1 = 1.0 + (1.0 / t)
	t_2 = 2.0 - ((2.0 / t) / t_1)
	return (1.0 + (1.0 + (math.pow((2.0 + ((-2.0 / t) / t_1)), 2.0) + -1.0))) / (2.0 + (t_2 * t_2))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 98.7%

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

   3. log1p-udef 98.7%

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

   4. add-exp-log 100.0%

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

   5. pow2 100.0%

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

   6. sub-neg 100.0%

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

   7. distribute-neg-frac 100.0%

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

   8. distribute-neg-frac 100.0%

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

   9. metadata-eval 100.0%

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

3. Applied egg-rr 100.0%

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

   1. associate--l+ 100.0%

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

5. Simplified 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 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 - ((2.0 / t) / (1.0 + (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 - ((2.0d0 / t) / (1.0d0 + (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 - ((2.0 / t) / (1.0 + (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 - ((2.0 / t) / (1.0 + (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(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + 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 - ((2.0 / t) / (1.0 + (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[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $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]]]

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 3: 100.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ t_2 := \frac{-4}{1 + t}\\ \frac{1 + \left(4 + \left(t_2 + \frac{4 + t_2}{-1 - t}\right)\right)}{2 + t_1 \cdot t_1} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))) (t_2 (/ -4.0 (+ 1.0 t))))
   (/ (+ 1.0 (+ 4.0 (+ t_2 (/ (+ 4.0 t_2) (- -1.0 t))))) (+ 2.0 (* t_1 t_1)))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = -4.0 / (1.0 + t);
	return (1.0 + (4.0 + (t_2 + ((4.0 + t_2) / (-1.0 - t))))) / (2.0 + (t_1 * t_1));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    t_1 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))
    t_2 = (-4.0d0) / (1.0d0 + t)
    code = (1.0d0 + (4.0d0 + (t_2 + ((4.0d0 + t_2) / ((-1.0d0) - t))))) / (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)));
	double t_2 = -4.0 / (1.0 + t);
	return (1.0 + (4.0 + (t_2 + ((4.0 + t_2) / (-1.0 - t))))) / (2.0 + (t_1 * t_1));
}

Python

def code(t):
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))
	t_2 = -4.0 / (1.0 + t)
	return (1.0 + (4.0 + (t_2 + ((4.0 + t_2) / (-1.0 - t))))) / (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))))
	t_2 = Float64(-4.0 / Float64(1.0 + t))
	return Float64(Float64(1.0 + Float64(4.0 + Float64(t_2 + Float64(Float64(4.0 + t_2) / Float64(-1.0 - t))))) / 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)));
	t_2 = -4.0 / (1.0 + t);
	tmp = (1.0 + (4.0 + (t_2 + ((4.0 + t_2) / (-1.0 - t))))) / (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]}, Block[{t$95$2 = N[(-4.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 + N[(4.0 + N[(t$95$2 + N[(N[(4.0 + t$95$2), $MachinePrecision] / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 98.7%

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

   3. log1p-udef 98.7%

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

   4. add-exp-log 100.0%

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

   5. pow2 100.0%

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

   6. sub-neg 100.0%

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

   7. distribute-neg-frac 100.0%

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

   8. distribute-neg-frac 100.0%

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

   9. metadata-eval 100.0%

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

3. Applied egg-rr 100.0%

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

   1. associate--l+ 100.0%

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

5. Simplified 100.0%

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

   1. expm1-log1p-u 97.9%

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

   2. expm1-udef 97.9%

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

   3. associate-/l/ 97.9%

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

   4. *-commutative 97.9%

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

7. Applied egg-rr 97.9%

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

   1. expm1-def 97.9%

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

   2. expm1-log1p 100.0%

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

   3. distribute-rgt-in 100.0%

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

   4. lft-mult-inverse 100.0%

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

   5. *-lft-identity 100.0%

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

9. Simplified 100.0%

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

    1. associate-+r- 100.0%

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

    2. add-exp-log 98.6%

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

    3. log1p-udef 98.6%

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

    4. expm1-udef 100.0%

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

    5. expm1-log1p-u 100.0%

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

    6. unpow2 100.0%

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

11. Applied egg-rr 100.0%

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

    1. distribute-lft-in 100.0%

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

    2. *-commutative 100.0%

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

    3. distribute-lft-in 100.0%

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

    4. metadata-eval 100.0%

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

    5. associate-*r/ 100.0%

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

    6. metadata-eval 100.0%

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

    7. frac-2neg 100.0%

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

    8. metadata-eval 100.0%

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

    9. associate-*r/ 100.0%

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

    10. *-commutative 100.0%

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

    11. distribute-lft-in 100.0%

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

    12. metadata-eval 100.0%

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

    13. associate-*r/ 100.0%

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

    14. metadata-eval 100.0%

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

    15. +-commutative 100.0%

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

    16. distribute-neg-in 100.0%

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

    17. metadata-eval 100.0%

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

13. Applied egg-rr 100.0%

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

    1. associate-+l+ 100.0%

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

    2. unsub-neg 100.0%

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

15. Simplified 100.0%

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

16. Final simplification 100.0%

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

Alternative 4: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double t) {
	double t_1 = 2.0 + (-2.0 / (1.0 + t));
	double t_2 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	return (1.0 + (t_1 * t_1)) / (2.0 + (t_2 * 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 + ((-2.0d0) / (1.0d0 + t))
    t_2 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))
    code = (1.0d0 + (t_1 * t_1)) / (2.0d0 + (t_2 * t_2))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 98.7%

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

   3. log1p-udef 98.7%

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

   4. add-exp-log 100.0%

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

   5. pow2 100.0%

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

   6. sub-neg 100.0%

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

   7. distribute-neg-frac 100.0%

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

   8. distribute-neg-frac 100.0%

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

   9. metadata-eval 100.0%

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

3. Applied egg-rr 100.0%

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

   1. associate--l+ 100.0%

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

5. Simplified 100.0%

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

   1. expm1-log1p-u 97.9%

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

   2. expm1-udef 97.9%

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

   3. associate-/l/ 97.9%

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

   4. *-commutative 97.9%

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

7. Applied egg-rr 97.9%

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

   1. expm1-def 97.9%

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

   2. expm1-log1p 100.0%

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

   3. distribute-rgt-in 100.0%

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

   4. lft-mult-inverse 100.0%

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

   5. *-lft-identity 100.0%

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

9. Simplified 100.0%

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

    1. associate-+r- 100.0%

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

    2. add-exp-log 98.6%

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

    3. log1p-udef 98.6%

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

    4. expm1-udef 100.0%

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

    5. expm1-log1p-u 100.0%

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

    6. unpow2 100.0%

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

11. Applied egg-rr 100.0%

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

12. Final simplification 100.0%

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

Alternative 5: 100.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{4}{1 + t} + -8}{1 + t}\\ \frac{5 + t_1}{t_1 + 6} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ 4.0 (+ 1.0 t)) -8.0) (+ 1.0 t))))
   (/ (+ 5.0 t_1) (+ t_1 6.0))))

C

double code(double t) {
	double t_1 = ((4.0 / (1.0 + t)) + -8.0) / (1.0 + t);
	return (5.0 + t_1) / (t_1 + 6.0);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = ((4.0d0 / (1.0d0 + t)) + (-8.0d0)) / (1.0d0 + t)
    code = (5.0d0 + t_1) / (t_1 + 6.0d0)
end function

Java

public static double code(double t) {
	double t_1 = ((4.0 / (1.0 + t)) + -8.0) / (1.0 + t);
	return (5.0 + t_1) / (t_1 + 6.0);
}

Python

def code(t):
	t_1 = ((4.0 / (1.0 + t)) + -8.0) / (1.0 + t)
	return (5.0 + t_1) / (t_1 + 6.0)

Julia

function code(t)
	t_1 = Float64(Float64(Float64(4.0 / Float64(1.0 + t)) + -8.0) / Float64(1.0 + t))
	return Float64(Float64(5.0 + t_1) / Float64(t_1 + 6.0))
end

MATLAB

function tmp = code(t)
	t_1 = ((4.0 / (1.0 + t)) + -8.0) / (1.0 + t);
	tmp = (5.0 + t_1) / (t_1 + 6.0);
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(N[(4.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] + -8.0), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, N[(N[(5.0 + t$95$1), $MachinePrecision] / N[(t$95$1 + 6.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{5 + \frac{\frac{4}{1 + t} + -8}{1 + t}}{6 + \frac{\frac{4}{1 + t} + -8}{1 + t}}} \]

4. Final simplification 100.0%

   \[\leadsto \frac{5 + \frac{\frac{4}{1 + t} + -8}{1 + t}}{\frac{\frac{4}{1 + t} + -8}{1 + t} + 6} \]

Alternative 6: 99.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.82 \lor \neg \left(t \leq 0.235\right):\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.82) (not (<= t 0.235)))
   (+
    (/ 0.037037037037037035 (* t t))
    (- 0.8333333333333334 (/ 0.2222222222222222 t)))
   (+ (* t t) 0.5)))

C

double code(double t) {
	double tmp;
	if ((t <= -0.82) || !(t <= 0.235)) {
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 - (0.2222222222222222 / t));
	} else {
		tmp = (t * t) + 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.82d0)) .or. (.not. (t <= 0.235d0))) then
        tmp = (0.037037037037037035d0 / (t * t)) + (0.8333333333333334d0 - (0.2222222222222222d0 / t))
    else
        tmp = (t * t) + 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.82) || !(t <= 0.235)) {
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 - (0.2222222222222222 / t));
	} else {
		tmp = (t * t) + 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.82) or not (t <= 0.235):
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 - (0.2222222222222222 / t))
	else:
		tmp = (t * t) + 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.82) || !(t <= 0.235))
		tmp = Float64(Float64(0.037037037037037035 / Float64(t * t)) + Float64(0.8333333333333334 - Float64(0.2222222222222222 / t)));
	else
		tmp = Float64(Float64(t * t) + 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.82) || ~((t <= 0.235)))
		tmp = (0.037037037037037035 / (t * t)) + (0.8333333333333334 - (0.2222222222222222 / t));
	else
		tmp = (t * t) + 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.82], N[Not[LessEqual[t, 0.235]], $MachinePrecision]], N[(N[(0.037037037037037035 / N[(t * t), $MachinePrecision]), $MachinePrecision] + N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.819999999999999951 or 0.23499999999999999 < t

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 99.2%

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

      1. associate--l+ 99.2%

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

      2. associate-*r/ 99.2%

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

      3. metadata-eval 99.2%

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

      4. unpow2 99.2%

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

      5. associate-*r/ 99.2%

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

      6. metadata-eval 99.2%

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

   4. Simplified 99.2%

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

   if -0.819999999999999951 < t < 0.23499999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0 97.8%

      \[\leadsto \color{blue}{0.5 + {t}^{2}} \]
   3. Step-by-step derivation

      1. +-commutative 97.8%

         \[\leadsto \color{blue}{{t}^{2} + 0.5} \]

      2. unpow2 97.8%

         \[\leadsto \color{blue}{t \cdot t} + 0.5 \]

   4. Simplified 97.8%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.82 \lor \neg \left(t \leq 0.235\right):\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]

Alternative 7: 99.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.8 \lor \neg \left(t \leq 0.57\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.8) (not (<= t 0.57)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (+ (* t t) 0.5)))

C

double code(double t) {
	double tmp;
	if ((t <= -0.8) || !(t <= 0.57)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = (t * t) + 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.8d0)) .or. (.not. (t <= 0.57d0))) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else
        tmp = (t * t) + 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.8) || !(t <= 0.57)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = (t * t) + 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.8) or not (t <= 0.57):
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = (t * t) + 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.8) || !(t <= 0.57))
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = Float64(Float64(t * t) + 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.8) || ~((t <= 0.57)))
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	else
		tmp = (t * t) + 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.8], N[Not[LessEqual[t, 0.57]], $MachinePrecision]], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.80000000000000004 or 0.569999999999999951 < t

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 98.9%

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

      1. associate-*r/ 98.9%

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

      2. metadata-eval 98.9%

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

   4. Simplified 98.9%

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

   if -0.80000000000000004 < t < 0.569999999999999951

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0 97.8%

      \[\leadsto \color{blue}{0.5 + {t}^{2}} \]
   3. Step-by-step derivation

      1. +-commutative 97.8%

         \[\leadsto \color{blue}{{t}^{2} + 0.5} \]

      2. unpow2 97.8%

         \[\leadsto \color{blue}{t \cdot t} + 0.5 \]

   4. Simplified 97.8%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.8 \lor \neg \left(t \leq 0.57\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]

Alternative 8: 98.6% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.92:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.57:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.92)
   0.8333333333333334
   (if (<= t 0.57) (+ (* t t) 0.5) 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.92) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.57) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.92d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.57d0) then
        tmp = (t * t) + 0.5d0
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.92) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.57) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.92:
		tmp = 0.8333333333333334
	elif t <= 0.57:
		tmp = (t * t) + 0.5
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.92)
		tmp = 0.8333333333333334;
	elseif (t <= 0.57)
		tmp = Float64(Float64(t * t) + 0.5);
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.92)
		tmp = 0.8333333333333334;
	elseif (t <= 0.57)
		tmp = (t * t) + 0.5;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.92], 0.8333333333333334, If[LessEqual[t, 0.57], N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.92000000000000004 or 0.569999999999999951 < t

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 97.7%

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

   if -0.92000000000000004 < t < 0.569999999999999951

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0 97.8%

      \[\leadsto \color{blue}{0.5 + {t}^{2}} \]
   3. Step-by-step derivation

      1. +-commutative 97.8%

         \[\leadsto \color{blue}{{t}^{2} + 0.5} \]

      2. unpow2 97.8%

         \[\leadsto \color{blue}{t \cdot t} + 0.5 \]

   4. Simplified 97.8%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.92:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.57:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 9: 98.4% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.34:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.34) 0.8333333333333334 (if (<= t 1.0) 0.5 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.34) {
		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 <= (-0.34d0)) 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 <= -0.34) {
		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 <= -0.34:
		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 <= -0.34)
		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 <= -0.34)
		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, -0.34], 0.8333333333333334, If[LessEqual[t, 1.0], 0.5, 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.340000000000000024 or 1 < t

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 97.2%

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

   if -0.340000000000000024 < t < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0 97.7%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.34:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 10: 59.2% accurate, 51.0× 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%

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

2. Taylor expanded in t around 0 54.1%

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

3. Final simplification 54.1%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023257 
(FPCore (t)
  :name "Kahan p13 Example 2"
  :precision binary64
  (/ (+ 1.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))) (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))))