Kahan p13 Example 2

Percentage Accurate: 100.0% → 100.0%

Time: 11.1s

Alternatives: 7

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 + N[(-2.0 / N[(t + 1.0), $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$1 * N[(2.0 + N[(2.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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-mult-inverse 100.0%

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

   11. *-rgt-identity 100.0%

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

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. associate-/l* 100.0%

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

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-/r* 100.0%

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

    2. associate-*r/ 100.0%

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

    3. metadata-eval 100.0%

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

    4. distribute-lft-in 100.0%

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

    5. rgt-mult-inverse 100.0%

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

    6. *-rgt-identity 100.0%

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

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. sub-neg 100.0%

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

    2. distribute-neg-frac 100.0%

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

    3. distribute-neg-frac 100.0%

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

    4. metadata-eval 100.0%

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

14. Applied egg-rr 100.0%

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

    1. associate-/r* 100.0%

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

    2. distribute-lft-in 100.0%

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

    3. rgt-mult-inverse 100.0%

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

    4. *-rgt-identity 100.0%

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

16. Simplified 100.0%

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

17. Final simplification 100.0%

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

Alternative 2: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.6:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{elif}\;t \leq 0.72:\\ \;\;\;\;\frac{1 + t \cdot \left(4 + \frac{-4}{t + 1}\right)}{2 + t \cdot \left(t \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.6)
   (-
    0.8333333333333334
    (/ (+ 0.2222222222222222 (/ -0.037037037037037035 t)) t))
   (if (<= t 0.72)
     (/ (+ 1.0 (* t (+ 4.0 (/ -4.0 (+ t 1.0))))) (+ 2.0 (* t (* t 4.0))))
     (+
      0.8333333333333334
      (/
       (-
        (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
        0.2222222222222222)
       t)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.6) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else if (t <= 0.72) {
		tmp = (1.0 + (t * (4.0 + (-4.0 / (t + 1.0))))) / (2.0 + (t * (t * 4.0)));
	} else {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.6d0)) then
        tmp = 0.8333333333333334d0 - ((0.2222222222222222d0 + ((-0.037037037037037035d0) / t)) / t)
    else if (t <= 0.72d0) then
        tmp = (1.0d0 + (t * (4.0d0 + ((-4.0d0) / (t + 1.0d0))))) / (2.0d0 + (t * (t * 4.0d0)))
    else
        tmp = 0.8333333333333334d0 + ((((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t) - 0.2222222222222222d0) / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.6) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else if (t <= 0.72) {
		tmp = (1.0 + (t * (4.0 + (-4.0 / (t + 1.0))))) / (2.0 + (t * (t * 4.0)));
	} else {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.6:
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t)
	elif t <= 0.72:
		tmp = (1.0 + (t * (4.0 + (-4.0 / (t + 1.0))))) / (2.0 + (t * (t * 4.0)))
	else:
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.6)
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 + Float64(-0.037037037037037035 / t)) / t));
	elseif (t <= 0.72)
		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))));
	else
		tmp = Float64(0.8333333333333334 + Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) - 0.2222222222222222) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.6)
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	elseif (t <= 0.72)
		tmp = (1.0 + (t * (4.0 + (-4.0 / (t + 1.0))))) / (2.0 + (t * (t * 4.0)));
	else
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.6], N[(0.8333333333333334 - N[(N[(0.2222222222222222 + N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.72], 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], N[(0.8333333333333334 + N[(N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.599999999999999978

   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%

      \[\leadsto \color{blue}{0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 - 0.037037037037037035 \cdot \frac{1}{t}}{t}} \]
   4. Step-by-step derivation

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. associate-*r/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. distribute-neg-frac 100.0%

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

      7. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   if -0.599999999999999978 < t < 0.71999999999999997

   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%

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

   4. Taylor expanded in t around 0 98.8%

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

      1. *-un-lft-identity 98.8%

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

      2. +-commutative 98.8%

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

      3. fma-define 98.8%

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

      4. associate-/l/ 98.8%

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

      5. +-commutative 98.8%

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

      6. fma-define 98.8%

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

      7. associate-/l/ 98.8%

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

   6. Applied egg-rr 98.8%

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

      1. *-lft-identity 98.8%

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

   8. Simplified 98.8%

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

   9. Taylor expanded in t around 0 99.1%

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

       1. *-commutative 99.1%

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

   11. Simplified 99.1%

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

   if 0.71999999999999997 < 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 99.4%

      \[\leadsto \color{blue}{0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}} \]
   4. Step-by-step derivation

      1. mul-1-neg 99.4%

         \[\leadsto 0.8333333333333334 + \color{blue}{\left(-\frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}\right)} \]

      2. unsub-neg 99.4%

         \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}} \]

      3. mul-1-neg 99.4%

         \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222 + \color{blue}{\left(-\frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}\right)}}{t} \]

      4. unsub-neg 99.4%

         \[\leadsto 0.8333333333333334 - \frac{\color{blue}{0.2222222222222222 - \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}}{t} \]

      5. associate-*r/ 99.4%

         \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \color{blue}{\frac{0.04938271604938271 \cdot 1}{t}}}{t}}{t} \]

      6. metadata-eval 99.4%

         \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{\color{blue}{0.04938271604938271}}{t}}{t}}{t} \]

   5. Simplified 99.4%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.6:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{elif}\;t \leq 0.72:\\ \;\;\;\;\frac{1 + t \cdot \left(4 + \frac{-4}{t + 1}\right)}{2 + t \cdot \left(t \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.52:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{elif}\;t \leq 0.68:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.52)
   (-
    0.8333333333333334
    (/ (+ 0.2222222222222222 (/ -0.037037037037037035 t)) t))
   (if (<= t 0.68)
     0.5
     (+
      0.8333333333333334
      (/
       (-
        (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
        0.2222222222222222)
       t)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.52) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else if (t <= 0.68) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.52d0)) then
        tmp = 0.8333333333333334d0 - ((0.2222222222222222d0 + ((-0.037037037037037035d0) / t)) / t)
    else if (t <= 0.68d0) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0 + ((((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t) - 0.2222222222222222d0) / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.52) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else if (t <= 0.68) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.52:
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t)
	elif t <= 0.68:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.52)
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 + Float64(-0.037037037037037035 / t)) / t));
	elseif (t <= 0.68)
		tmp = 0.5;
	else
		tmp = Float64(0.8333333333333334 + Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) - 0.2222222222222222) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.52)
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	elseif (t <= 0.68)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.52], N[(0.8333333333333334 - N[(N[(0.2222222222222222 + N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.68], 0.5, N[(0.8333333333333334 + N[(N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.52000000000000002

   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%

      \[\leadsto \color{blue}{0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 - 0.037037037037037035 \cdot \frac{1}{t}}{t}} \]
   4. Step-by-step derivation

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. associate-*r/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. distribute-neg-frac 100.0%

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

      7. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   if -0.52000000000000002 < t < 0.680000000000000049

   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 98.4%

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

   if 0.680000000000000049 < 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 99.4%

      \[\leadsto \color{blue}{0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}} \]
   4. Step-by-step derivation

      1. mul-1-neg 99.4%

         \[\leadsto 0.8333333333333334 + \color{blue}{\left(-\frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}\right)} \]

      2. unsub-neg 99.4%

         \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}} \]

      3. mul-1-neg 99.4%

         \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222 + \color{blue}{\left(-\frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}\right)}}{t} \]

      4. unsub-neg 99.4%

         \[\leadsto 0.8333333333333334 - \frac{\color{blue}{0.2222222222222222 - \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}}{t} \]

      5. associate-*r/ 99.4%

         \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \color{blue}{\frac{0.04938271604938271 \cdot 1}{t}}}{t}}{t} \]

      6. metadata-eval 99.4%

         \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{\color{blue}{0.04938271604938271}}{t}}{t}}{t} \]

   5. Simplified 99.4%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.52:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{elif}\;t \leq 0.68:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.52 \lor \neg \left(t \leq 0.23\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.52) (not (<= t 0.23)))
   (-
    0.8333333333333334
    (/ (+ 0.2222222222222222 (/ -0.037037037037037035 t)) t))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.52) || !(t <= 0.23)) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.52d0)) .or. (.not. (t <= 0.23d0))) then
        tmp = 0.8333333333333334d0 - ((0.2222222222222222d0 + ((-0.037037037037037035d0) / t)) / t)
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.52) || !(t <= 0.23)) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.52) or not (t <= 0.23):
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.52) || !(t <= 0.23))
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 + Float64(-0.037037037037037035 / t)) / t));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.52) || ~((t <= 0.23)))
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.52], N[Not[LessEqual[t, 0.23]], $MachinePrecision]], N[(0.8333333333333334 - N[(N[(0.2222222222222222 + N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if t < -0.52000000000000002 or 0.23000000000000001 < 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 99.6%

      \[\leadsto \color{blue}{0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 - 0.037037037037037035 \cdot \frac{1}{t}}{t}} \]
   4. Step-by-step derivation

      1. mul-1-neg 99.6%

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

      2. unsub-neg 99.6%

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

      3. sub-neg 99.6%

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

      4. associate-*r/ 99.6%

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

      5. metadata-eval 99.6%

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

      6. distribute-neg-frac 99.6%

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

      7. metadata-eval 99.6%

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

   5. Simplified 99.6%

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

   if -0.52000000000000002 < t < 0.23000000000000001

   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 98.4%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.52 \lor \neg \left(t \leq 0.23\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.68\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.49) (not (<= t 0.68)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.49) || !(t <= 0.68)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.49d0)) .or. (.not. (t <= 0.68d0))) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.49) || !(t <= 0.68)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.49) or not (t <= 0.68):
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.49) || !(t <= 0.68))
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.49) || ~((t <= 0.68)))
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.49], N[Not[LessEqual[t, 0.68]], $MachinePrecision]], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if t < -0.48999999999999999 or 0.680000000000000049 < 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 99.4%

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

      1. associate-*r/ 99.4%

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

      2. metadata-eval 99.4%

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

   5. Simplified 99.4%

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

   if -0.48999999999999999 < t < 0.680000000000000049

   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 98.4%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.68\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.5% 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 98.2%

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

   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. Add Preprocessing

   3. Taylor expanded in t around 0 98.4%

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

Alternative 7: 59.3% 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 55.3%

   \[\leadsto \color{blue}{0.5} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024135 
(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))))))))