Kahan p13 Example 2

Percentage Accurate: 100.0% → 100.0%

Time: 39.7s

Alternatives: 5

Speedup: 1.4×

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, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 + \frac{-2}{t + 1}\\ t_2 := 2 + \frac{2}{-1 - t}\\ \frac{-1 + \left(2 + t\_1 \cdot t\_1\right)}{2 + t\_2 \cdot t\_2} \end{array} \end{array} \]

FPCore

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

C

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

Python

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

Julia

function code(t)
	t_1 = Float64(2.0 + Float64(-2.0 / Float64(t + 1.0)))
	t_2 = Float64(2.0 + Float64(2.0 / Float64(-1.0 - t)))
	return Float64(Float64(-1.0 + Float64(2.0 + 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 / (t + 1.0));
	t_2 = 2.0 + (2.0 / (-1.0 - t));
	tmp = (-1.0 + (2.0 + (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[(t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[(2.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(-1.0 + N[(2.0 + N[(t$95$1 * t$95$1), $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. Add Preprocessing
3. Step-by-step derivation

   1. expm1-log1p-u 100.0%

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

   2. expm1-undefine 100.0%

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

   3. pow2 100.0%

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

   4. sub-neg 100.0%

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

   5. distribute-neg-frac 100.0%

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

   6. distribute-neg-frac 100.0%

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

   7. metadata-eval 100.0%

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

4. Applied egg-rr 100.0%

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

   1. sub-neg 100.0%

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

   2. log1p-undefine 100.0%

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

   3. rem-exp-log 100.0%

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

   4. associate-+r+ 100.0%

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

   5. metadata-eval 100.0%

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

   6. metadata-eval 100.0%

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

   7. +-commutative 100.0%

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

   8. associate-/r* 100.0%

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

   9. distribute-lft-in 100.0%

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

   10. *-rgt-identity 100.0%

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

   11. rgt-mult-inverse 100.0%

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

6. Simplified 100.0%

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

   1. unpow2 100.0%

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

8. Applied egg-rr 100.0%

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

   1. div-inv 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. times-frac 100.0%

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

   4. metadata-eval 100.0%

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

10. Applied egg-rr 100.0%

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

    1. associate-/l/ 100.0%

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

    2. associate-*r/ 100.0%

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

    3. metadata-eval 100.0%

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

    4. *-commutative 100.0%

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

    5. distribute-rgt-in 100.0%

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

    6. *-lft-identity 100.0%

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

    7. lft-mult-inverse 100.0%

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

12. Simplified 100.0%

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

    1. div-inv 100.0%

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

    2. *-un-lft-identity 100.0%

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

    3. times-frac 100.0%

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

    4. metadata-eval 100.0%

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

14. Applied egg-rr 100.0%

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

    1. associate-/l/ 100.0%

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

    2. associate-*r/ 100.0%

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

    3. metadata-eval 100.0%

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

    4. *-commutative 100.0%

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

    5. distribute-rgt-in 100.0%

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

    6. *-lft-identity 100.0%

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

    7. lft-mult-inverse 100.0%

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

16. Simplified 100.0%

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

17. Final simplification 100.0%

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

Alternative 2: 98.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t \cdot \left(4 + \frac{-4}{t + 1}\right)}{2 + t \cdot \left(t \cdot 4\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))) 2e-8)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (/ (+ 1.0 (* t (+ 4.0 (/ -4.0 (+ t 1.0))))) (+ 2.0 (* t (* t 4.0))))))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 2e-8) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = (1.0 + (t * (4.0 + (-4.0 / (t + 1.0))))) / (2.0 + (t * (t * 4.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((2.0d0 / t) / (1.0d0 + (1.0d0 / t))) <= 2d-8) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else
        tmp = (1.0d0 + (t * (4.0d0 + ((-4.0d0) / (t + 1.0d0))))) / (2.0d0 + (t * (t * 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 2e-8) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = (1.0 + (t * (4.0 + (-4.0 / (t + 1.0))))) / (2.0 + (t * (t * 4.0)));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if ((2.0 / t) / (1.0 + (1.0 / t))) <= 2e-8:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = (1.0 + (t * (4.0 + (-4.0 / (t + 1.0))))) / (2.0 + (t * (t * 4.0)))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))) <= 2e-8)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = Float64(Float64(1.0 + Float64(t * Float64(4.0 + Float64(-4.0 / Float64(t + 1.0))))) / Float64(2.0 + Float64(t * Float64(t * 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 2e-8)
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	else
		tmp = (1.0 + (t * (4.0 + (-4.0 / (t + 1.0))))) / (2.0 + (t * (t * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-8], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(t * N[(4.0 + N[(-4.0 / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(t * N[(t * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 2e-8

   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. Add Preprocessing

   3. Taylor expanded in t around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in t around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   if 2e-8 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) 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. Add Preprocessing

   3. Taylor expanded in t around 0 99.4%

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

   4. Taylor expanded in t around 0 99.4%

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

      1. *-un-lft-identity 99.4%

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

      2. +-commutative 99.4%

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

      3. fma-define 99.4%

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

      4. +-commutative 99.4%

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

      5. fma-define 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. *-lft-identity 99.4%

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

   8. Simplified 99.4%

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

   9. Taylor expanded in t around 0 99.8%

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

       1. *-commutative 99.8%

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

   11. Simplified 99.8%

       \[\leadsto \frac{1 + \left(4 + \frac{-4}{t + 1}\right) \cdot t}{2 + \color{blue}{\left(t \cdot 4\right)} \cdot t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t \cdot \left(4 + \frac{-4}{t + 1}\right)}{2 + t \cdot \left(t \cdot 4\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.34:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.65:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.34)
   0.8333333333333334
   (if (<= t 0.65) 0.5 (- 0.8333333333333334 (/ 0.2222222222222222 t)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.34) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.65) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.34d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.65d0) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.34) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.65) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.34:
		tmp = 0.8333333333333334
	elif t <= 0.65:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.34)
		tmp = 0.8333333333333334;
	elseif (t <= 0.65)
		tmp = 0.5;
	else
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.34)
		tmp = 0.8333333333333334;
	elseif (t <= 0.65)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.34], 0.8333333333333334, If[LessEqual[t, 0.65], 0.5, N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.340000000000000024

   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. Add Preprocessing

   3. Taylor expanded in t around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in t around inf 100.0%

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

   if -0.340000000000000024 < t < 0.650000000000000022

   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. Add Preprocessing

   3. Taylor expanded in t around 0 98.8%

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

   4. Taylor expanded in t around 0 98.9%

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

   if 0.650000000000000022 < 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. Add Preprocessing

   3. Taylor expanded in t around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in t around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.34:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.65:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 4.6× 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. Add Preprocessing

   3. Taylor expanded in t around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in t around inf 99.3%

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

   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. Add Preprocessing

   3. Taylor expanded in t around 0 98.8%

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

   4. Taylor expanded in t around 0 98.9%

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

4. Final simplification 99.1%

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

Alternative 5: 58.8% 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. Add Preprocessing

3. Taylor expanded in t around 0 58.5%

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

4. Taylor expanded in t around 0 59.9%

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

5. Final simplification 59.9%

   \[\leadsto 0.5 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024066 
(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))))))))