Kahan p13 Example 3

Percentage Accurate: 99.9% → 99.9%

Time: 8.0s

Alternatives: 10

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 99.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.46:\\ \;\;\;\;1 + \frac{-1}{6 + \frac{-8}{t}}\\ \mathbf{elif}\;t \leq 0.75:\\ \;\;\;\;0.5 + \left(t \cdot t\right) \cdot \left(1 + t \cdot \left(t + -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.46)
   (+ 1.0 (/ -1.0 (+ 6.0 (/ -8.0 t))))
   (if (<= t 0.75)
     (+ 0.5 (* (* t t) (+ 1.0 (* t (+ t -2.0)))))
     (- 1.0 (+ (/ 0.2222222222222222 t) 0.16666666666666666)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.46) {
		tmp = 1.0 + (-1.0 / (6.0 + (-8.0 / t)));
	} else if (t <= 0.75) {
		tmp = 0.5 + ((t * t) * (1.0 + (t * (t + -2.0))));
	} else {
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.46d0)) then
        tmp = 1.0d0 + ((-1.0d0) / (6.0d0 + ((-8.0d0) / t)))
    else if (t <= 0.75d0) then
        tmp = 0.5d0 + ((t * t) * (1.0d0 + (t * (t + (-2.0d0)))))
    else
        tmp = 1.0d0 - ((0.2222222222222222d0 / t) + 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.46) {
		tmp = 1.0 + (-1.0 / (6.0 + (-8.0 / t)));
	} else if (t <= 0.75) {
		tmp = 0.5 + ((t * t) * (1.0 + (t * (t + -2.0))));
	} else {
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.46:
		tmp = 1.0 + (-1.0 / (6.0 + (-8.0 / t)))
	elif t <= 0.75:
		tmp = 0.5 + ((t * t) * (1.0 + (t * (t + -2.0))))
	else:
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.46)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(6.0 + Float64(-8.0 / t))));
	elseif (t <= 0.75)
		tmp = Float64(0.5 + Float64(Float64(t * t) * Float64(1.0 + Float64(t * Float64(t + -2.0)))));
	else
		tmp = Float64(1.0 - Float64(Float64(0.2222222222222222 / t) + 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.46)
		tmp = 1.0 + (-1.0 / (6.0 + (-8.0 / t)));
	elseif (t <= 0.75)
		tmp = 0.5 + ((t * t) * (1.0 + (t * (t + -2.0))));
	else
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.46], N[(1.0 + N[(-1.0 / N[(6.0 + N[(-8.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.75], N[(0.5 + N[(N[(t * t), $MachinePrecision] * N[(1.0 + N[(t * N[(t + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(0.2222222222222222 / t), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.46000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(6 - 8 \cdot \frac{1}{t}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \left(6 + \color{blue}{\left(\mathsf{neg}\left(8 \cdot \frac{1}{t}\right)\right)}\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(6, \color{blue}{\left(\mathsf{neg}\left(8 \cdot \frac{1}{t}\right)\right)}\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(6, \left(\mathsf{neg}\left(\frac{8 \cdot 1}{t}\right)\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(6, \left(\mathsf{neg}\left(\frac{8}{t}\right)\right)\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(6, \left(\frac{\mathsf{neg}\left(8\right)}{\color{blue}{t}}\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(6, \left(\frac{-8}{t}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 98.6%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(6, \mathsf{/.f64}\left(-8, \color{blue}{t}\right)\right)\right)\right) \]

   5. Simplified 98.6%

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

   if -0.46000000000000002 < t < 0.75

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{1}{-6 + \frac{2}{1 + t} \cdot \left(\frac{2}{-1 - t} + 4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({t}^{2} \cdot \left(1 + t \cdot \left(t - 2\right)\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left(1 + t \cdot \left(t - 2\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{1} + t \cdot \left(t - 2\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{1} + t \cdot \left(t - 2\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(1, \color{blue}{\left(t \cdot \left(t - 2\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \color{blue}{\left(t - 2\right)}\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(t + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(t + -2\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 99.5%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(t, \color{blue}{-2}\right)\right)\right)\right)\right) \]

   7. Simplified 99.5%

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

   if 0.75 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{1}{6} + \frac{2}{9} \cdot \frac{1}{t}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\frac{2}{9} \cdot \frac{1}{t} + \color{blue}{\frac{1}{6}}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{2}{9} \cdot \frac{1}{t}\right), \color{blue}{\frac{1}{6}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{\frac{2}{9} \cdot 1}{t}\right), \frac{1}{6}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{\frac{2}{9}}{t}\right), \frac{1}{6}\right)\right) \]

      5. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{2}{9}, t\right), \frac{1}{6}\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto 1 - \color{blue}{\left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.46:\\ \;\;\;\;1 + \frac{-1}{6 + \frac{-8}{t}}\\ \mathbf{elif}\;t \leq 0.75:\\ \;\;\;\;0.5 + \left(t \cdot t\right) \cdot \left(1 + t \cdot \left(t + -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;1 + \frac{-1}{6 + \frac{-8}{t}}\\ \mathbf{elif}\;t \leq 0.56:\\ \;\;\;\;0.5 + t \cdot \left(t \cdot \left(1 + t \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.49)
   (+ 1.0 (/ -1.0 (+ 6.0 (/ -8.0 t))))
   (if (<= t 0.56)
     (+ 0.5 (* t (* t (+ 1.0 (* t -2.0)))))
     (- 1.0 (+ (/ 0.2222222222222222 t) 0.16666666666666666)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.49) {
		tmp = 1.0 + (-1.0 / (6.0 + (-8.0 / t)));
	} else if (t <= 0.56) {
		tmp = 0.5 + (t * (t * (1.0 + (t * -2.0))));
	} else {
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.49d0)) then
        tmp = 1.0d0 + ((-1.0d0) / (6.0d0 + ((-8.0d0) / t)))
    else if (t <= 0.56d0) then
        tmp = 0.5d0 + (t * (t * (1.0d0 + (t * (-2.0d0)))))
    else
        tmp = 1.0d0 - ((0.2222222222222222d0 / t) + 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.49) {
		tmp = 1.0 + (-1.0 / (6.0 + (-8.0 / t)));
	} else if (t <= 0.56) {
		tmp = 0.5 + (t * (t * (1.0 + (t * -2.0))));
	} else {
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.49:
		tmp = 1.0 + (-1.0 / (6.0 + (-8.0 / t)))
	elif t <= 0.56:
		tmp = 0.5 + (t * (t * (1.0 + (t * -2.0))))
	else:
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.49)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(6.0 + Float64(-8.0 / t))));
	elseif (t <= 0.56)
		tmp = Float64(0.5 + Float64(t * Float64(t * Float64(1.0 + Float64(t * -2.0)))));
	else
		tmp = Float64(1.0 - Float64(Float64(0.2222222222222222 / t) + 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.49)
		tmp = 1.0 + (-1.0 / (6.0 + (-8.0 / t)));
	elseif (t <= 0.56)
		tmp = 0.5 + (t * (t * (1.0 + (t * -2.0))));
	else
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.49], N[(1.0 + N[(-1.0 / N[(6.0 + N[(-8.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.56], N[(0.5 + N[(t * N[(t * N[(1.0 + N[(t * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(0.2222222222222222 / t), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.48999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(6 - 8 \cdot \frac{1}{t}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \left(6 + \color{blue}{\left(\mathsf{neg}\left(8 \cdot \frac{1}{t}\right)\right)}\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(6, \color{blue}{\left(\mathsf{neg}\left(8 \cdot \frac{1}{t}\right)\right)}\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(6, \left(\mathsf{neg}\left(\frac{8 \cdot 1}{t}\right)\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(6, \left(\mathsf{neg}\left(\frac{8}{t}\right)\right)\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(6, \left(\frac{\mathsf{neg}\left(8\right)}{\color{blue}{t}}\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(6, \left(\frac{-8}{t}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 98.6%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(6, \mathsf{/.f64}\left(-8, \color{blue}{t}\right)\right)\right)\right) \]

   5. Simplified 98.6%

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

   if -0.48999999999999999 < t < 0.56000000000000005

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{1}{-6 + \frac{2}{1 + t} \cdot \left(\frac{2}{-1 - t} + 4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({t}^{2} \cdot \left(1 + -2 \cdot t\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(t \cdot t\right) \cdot \left(\color{blue}{1} + -2 \cdot t\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(t \cdot \color{blue}{\left(t \cdot \left(1 + -2 \cdot t\right)\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(t \cdot \left(\left(1 + -2 \cdot t\right) \cdot \color{blue}{t}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \color{blue}{\left(\left(1 + -2 \cdot t\right) \cdot t\right)}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{\left(1 + -2 \cdot t\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{\left(1 + -2 \cdot t\right)}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(1, \color{blue}{\left(-2 \cdot t\right)}\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(1, \left(t \cdot \color{blue}{-2}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 99.5%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \color{blue}{-2}\right)\right)\right)\right)\right) \]

   7. Simplified 99.5%

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

   if 0.56000000000000005 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{1}{6} + \frac{2}{9} \cdot \frac{1}{t}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\frac{2}{9} \cdot \frac{1}{t} + \color{blue}{\frac{1}{6}}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{2}{9} \cdot \frac{1}{t}\right), \color{blue}{\frac{1}{6}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{\frac{2}{9} \cdot 1}{t}\right), \frac{1}{6}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{\frac{2}{9}}{t}\right), \frac{1}{6}\right)\right) \]

      5. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{2}{9}, t\right), \frac{1}{6}\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto 1 - \color{blue}{\left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;1 + \frac{-1}{6 + \frac{-8}{t}}\\ \mathbf{elif}\;t \leq 0.56:\\ \;\;\;\;0.5 + t \cdot \left(t \cdot \left(1 + t \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{1}{-6 + \frac{2}{1 + t} \cdot \left(\frac{2}{-1 - t} + 4\right)} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (+ 1.0 (/ 1.0 (+ -6.0 (* (/ 2.0 (+ 1.0 t)) (+ (/ 2.0 (- -1.0 t)) 4.0))))))

C

double code(double t) {
	return 1.0 + (1.0 / (-6.0 + ((2.0 / (1.0 + t)) * ((2.0 / (-1.0 - t)) + 4.0))));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = 1.0d0 + (1.0d0 / ((-6.0d0) + ((2.0d0 / (1.0d0 + t)) * ((2.0d0 / ((-1.0d0) - t)) + 4.0d0))))
end function

Java

public static double code(double t) {
	return 1.0 + (1.0 / (-6.0 + ((2.0 / (1.0 + t)) * ((2.0 / (-1.0 - t)) + 4.0))));
}

Python

def code(t):
	return 1.0 + (1.0 / (-6.0 + ((2.0 / (1.0 + t)) * ((2.0 / (-1.0 - t)) + 4.0))))

Julia

function code(t)
	return Float64(1.0 + Float64(1.0 / Float64(-6.0 + Float64(Float64(2.0 / Float64(1.0 + t)) * Float64(Float64(2.0 / Float64(-1.0 - t)) + 4.0)))))
end

MATLAB

function tmp = code(t)
	tmp = 1.0 + (1.0 / (-6.0 + ((2.0 / (1.0 + t)) * ((2.0 / (-1.0 - t)) + 4.0))));
end

Wolfram

code[t_] := N[(1.0 + N[(1.0 / N[(-6.0 + N[(N[(2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{1 + \frac{1}{-6 + \frac{2}{1 + t} \cdot \left(\frac{2}{-1 - t} + 4\right)}} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 5: 99.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.55:\\ \;\;\;\;1 + \frac{-1}{6 + \frac{-8}{t}}\\ \mathbf{elif}\;t \leq 0.56:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.55)
   (+ 1.0 (/ -1.0 (+ 6.0 (/ -8.0 t))))
   (if (<= t 0.56)
     (+ 0.5 (* t t))
     (- 1.0 (+ (/ 0.2222222222222222 t) 0.16666666666666666)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.55) {
		tmp = 1.0 + (-1.0 / (6.0 + (-8.0 / t)));
	} else if (t <= 0.56) {
		tmp = 0.5 + (t * t);
	} else {
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.55d0)) then
        tmp = 1.0d0 + ((-1.0d0) / (6.0d0 + ((-8.0d0) / t)))
    else if (t <= 0.56d0) then
        tmp = 0.5d0 + (t * t)
    else
        tmp = 1.0d0 - ((0.2222222222222222d0 / t) + 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.55) {
		tmp = 1.0 + (-1.0 / (6.0 + (-8.0 / t)));
	} else if (t <= 0.56) {
		tmp = 0.5 + (t * t);
	} else {
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.55:
		tmp = 1.0 + (-1.0 / (6.0 + (-8.0 / t)))
	elif t <= 0.56:
		tmp = 0.5 + (t * t)
	else:
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.55)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(6.0 + Float64(-8.0 / t))));
	elseif (t <= 0.56)
		tmp = Float64(0.5 + Float64(t * t));
	else
		tmp = Float64(1.0 - Float64(Float64(0.2222222222222222 / t) + 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.55)
		tmp = 1.0 + (-1.0 / (6.0 + (-8.0 / t)));
	elseif (t <= 0.56)
		tmp = 0.5 + (t * t);
	else
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.55], N[(1.0 + N[(-1.0 / N[(6.0 + N[(-8.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.56], N[(0.5 + N[(t * t), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(0.2222222222222222 / t), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.55000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(6 - 8 \cdot \frac{1}{t}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \left(6 + \color{blue}{\left(\mathsf{neg}\left(8 \cdot \frac{1}{t}\right)\right)}\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(6, \color{blue}{\left(\mathsf{neg}\left(8 \cdot \frac{1}{t}\right)\right)}\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(6, \left(\mathsf{neg}\left(\frac{8 \cdot 1}{t}\right)\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(6, \left(\mathsf{neg}\left(\frac{8}{t}\right)\right)\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(6, \left(\frac{\mathsf{neg}\left(8\right)}{\color{blue}{t}}\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(6, \left(\frac{-8}{t}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 98.6%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(6, \mathsf{/.f64}\left(-8, \color{blue}{t}\right)\right)\right)\right) \]

   5. Simplified 98.6%

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

   if -0.55000000000000004 < t < 0.56000000000000005

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{1}{-6 + \frac{2}{1 + t} \cdot \left(\frac{2}{-1 - t} + 4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({t}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(t \cdot \color{blue}{t}\right)\right) \]

      3. *-lowering-*.f64 99.5%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right) \]

   7. Simplified 99.5%

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

   if 0.56000000000000005 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{1}{6} + \frac{2}{9} \cdot \frac{1}{t}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\frac{2}{9} \cdot \frac{1}{t} + \color{blue}{\frac{1}{6}}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{2}{9} \cdot \frac{1}{t}\right), \color{blue}{\frac{1}{6}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{\frac{2}{9} \cdot 1}{t}\right), \frac{1}{6}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{\frac{2}{9}}{t}\right), \frac{1}{6}\right)\right) \]

      5. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{2}{9}, t\right), \frac{1}{6}\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto 1 - \color{blue}{\left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.55:\\ \;\;\;\;1 + \frac{-1}{6 + \frac{-8}{t}}\\ \mathbf{elif}\;t \leq 0.56:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.79:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.56:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.79)
   (+ 0.8333333333333334 (/ -0.2222222222222222 t))
   (if (<= t 0.56)
     (+ 0.5 (* t t))
     (- 1.0 (+ (/ 0.2222222222222222 t) 0.16666666666666666)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.79) {
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t);
	} else if (t <= 0.56) {
		tmp = 0.5 + (t * t);
	} else {
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.79d0)) then
        tmp = 0.8333333333333334d0 + ((-0.2222222222222222d0) / t)
    else if (t <= 0.56d0) then
        tmp = 0.5d0 + (t * t)
    else
        tmp = 1.0d0 - ((0.2222222222222222d0 / t) + 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.79) {
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t);
	} else if (t <= 0.56) {
		tmp = 0.5 + (t * t);
	} else {
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.79:
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t)
	elif t <= 0.56:
		tmp = 0.5 + (t * t)
	else:
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.79)
		tmp = Float64(0.8333333333333334 + Float64(-0.2222222222222222 / t));
	elseif (t <= 0.56)
		tmp = Float64(0.5 + Float64(t * t));
	else
		tmp = Float64(1.0 - Float64(Float64(0.2222222222222222 / t) + 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.79)
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t);
	elseif (t <= 0.56)
		tmp = 0.5 + (t * t);
	else
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.79], N[(0.8333333333333334 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.56], N[(0.5 + N[(t * t), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(0.2222222222222222 / t), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.79000000000000004

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{1}{-6 + \frac{2}{1 + t} \cdot \left(\frac{2}{-1 - t} + 4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{5}{6} + \color{blue}{\left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \color{blue}{\left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\mathsf{neg}\left(\frac{\frac{2}{9} \cdot 1}{t}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\mathsf{neg}\left(\frac{\frac{2}{9}}{t}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\mathsf{neg}\left(\frac{2}{9}\right)}{\color{blue}{t}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{2}{9}\right)\right), \color{blue}{t}\right)\right) \]

      7. metadata-eval 100.0%

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\frac{-2}{9}, t\right)\right) \]

   7. Simplified 100.0%

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

   if -0.79000000000000004 < t < 0.56000000000000005

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{1}{-6 + \frac{2}{1 + t} \cdot \left(\frac{2}{-1 - t} + 4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({t}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(t \cdot \color{blue}{t}\right)\right) \]

      3. *-lowering-*.f64 98.9%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right) \]

   7. Simplified 98.9%

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

   if 0.56000000000000005 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{1}{6} + \frac{2}{9} \cdot \frac{1}{t}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\frac{2}{9} \cdot \frac{1}{t} + \color{blue}{\frac{1}{6}}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{2}{9} \cdot \frac{1}{t}\right), \color{blue}{\frac{1}{6}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{\frac{2}{9} \cdot 1}{t}\right), \frac{1}{6}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{\frac{2}{9}}{t}\right), \frac{1}{6}\right)\right) \]

      5. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{2}{9}, t\right), \frac{1}{6}\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto 1 - \color{blue}{\left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 99.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.79:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.56:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (+ 0.8333333333333334 (/ -0.2222222222222222 t))))
   (if (<= t -0.79) t_1 (if (<= t 0.56) (+ 0.5 (* t t)) t_1))))

C

double code(double t) {
	double t_1 = 0.8333333333333334 + (-0.2222222222222222 / t);
	double tmp;
	if (t <= -0.79) {
		tmp = t_1;
	} else if (t <= 0.56) {
		tmp = 0.5 + (t * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.8333333333333334d0 + ((-0.2222222222222222d0) / t)
    if (t <= (-0.79d0)) then
        tmp = t_1
    else if (t <= 0.56d0) then
        tmp = 0.5d0 + (t * t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 0.8333333333333334 + (-0.2222222222222222 / t);
	double tmp;
	if (t <= -0.79) {
		tmp = t_1;
	} else if (t <= 0.56) {
		tmp = 0.5 + (t * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(t):
	t_1 = 0.8333333333333334 + (-0.2222222222222222 / t)
	tmp = 0
	if t <= -0.79:
		tmp = t_1
	elif t <= 0.56:
		tmp = 0.5 + (t * t)
	else:
		tmp = t_1
	return tmp

Julia

function code(t)
	t_1 = Float64(0.8333333333333334 + Float64(-0.2222222222222222 / t))
	tmp = 0.0
	if (t <= -0.79)
		tmp = t_1;
	elseif (t <= 0.56)
		tmp = Float64(0.5 + Float64(t * t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 0.8333333333333334 + (-0.2222222222222222 / t);
	tmp = 0.0;
	if (t <= -0.79)
		tmp = t_1;
	elseif (t <= 0.56)
		tmp = 0.5 + (t * t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(0.8333333333333334 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.79], t$95$1, If[LessEqual[t, 0.56], N[(0.5 + N[(t * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.79000000000000004 or 0.56000000000000005 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{1}{-6 + \frac{2}{1 + t} \cdot \left(\frac{2}{-1 - t} + 4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{5}{6} + \color{blue}{\left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \color{blue}{\left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\mathsf{neg}\left(\frac{\frac{2}{9} \cdot 1}{t}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\mathsf{neg}\left(\frac{\frac{2}{9}}{t}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\mathsf{neg}\left(\frac{2}{9}\right)}{\color{blue}{t}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{2}{9}\right)\right), \color{blue}{t}\right)\right) \]

      7. metadata-eval 100.0%

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\frac{-2}{9}, t\right)\right) \]

   7. Simplified 100.0%

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

   if -0.79000000000000004 < t < 0.56000000000000005

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{1}{-6 + \frac{2}{1 + t} \cdot \left(\frac{2}{-1 - t} + 4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({t}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(t \cdot \color{blue}{t}\right)\right) \]

      3. *-lowering-*.f64 98.9%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right) \]

   7. Simplified 98.9%

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

Alternative 8: 98.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.9:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.56:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.9)
   0.8333333333333334
   (if (<= t 0.56) (+ 0.5 (* t t)) 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.9) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.56) {
		tmp = 0.5 + (t * t);
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.9d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.56d0) then
        tmp = 0.5d0 + (t * t)
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.9) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.56) {
		tmp = 0.5 + (t * t);
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.9:
		tmp = 0.8333333333333334
	elif t <= 0.56:
		tmp = 0.5 + (t * t)
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.9)
		tmp = 0.8333333333333334;
	elseif (t <= 0.56)
		tmp = Float64(0.5 + Float64(t * t));
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.9)
		tmp = 0.8333333333333334;
	elseif (t <= 0.56)
		tmp = 0.5 + (t * t);
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.9], 0.8333333333333334, If[LessEqual[t, 0.56], N[(0.5 + N[(t * t), $MachinePrecision]), $MachinePrecision], 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.900000000000000022 or 0.56000000000000005 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{1}{-6 + \frac{2}{1 + t} \cdot \left(\frac{2}{-1 - t} + 4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{5}{6}} \]
   6. Step-by-step derivation

   7. Simplified 99.4%

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

   if -0.900000000000000022 < t < 0.56000000000000005

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{1}{-6 + \frac{2}{1 + t} \cdot \left(\frac{2}{-1 - t} + 4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({t}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(t \cdot \color{blue}{t}\right)\right) \]

      3. *-lowering-*.f64 98.9%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right) \]

   7. Simplified 98.9%

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

Alternative 9: 98.3% accurate, 2.6× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.33) 0.8333333333333334 (if (<= t 1.0) 0.5 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.33) {
		tmp = 0.8333333333333334;
	} else if (t <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.33d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 1.0d0) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.33) {
		tmp = 0.8333333333333334;
	} else if (t <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.33:
		tmp = 0.8333333333333334
	elif t <= 1.0:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.33)
		tmp = 0.8333333333333334;
	elseif (t <= 1.0)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.33)
		tmp = 0.8333333333333334;
	elseif (t <= 1.0)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.33], 0.8333333333333334, If[LessEqual[t, 1.0], 0.5, 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.330000000000000016 or 1 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{1}{-6 + \frac{2}{1 + t} \cdot \left(\frac{2}{-1 - t} + 4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{5}{6}} \]
   6. Step-by-step derivation

   7. Simplified 98.8%

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

   if -0.330000000000000016 < t < 1

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{1}{-6 + \frac{2}{1 + t} \cdot \left(\frac{2}{-1 - t} + 4\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   6. Step-by-step derivation

   7. Simplified 99.4%

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

Alternative 10: 59.3% accurate, 29.0× 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%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{1 + \frac{1}{-6 + \frac{2}{1 + t} \cdot \left(\frac{2}{-1 - t} + 4\right)}} \]
4. Add Preprocessing

5. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\frac{1}{2}} \]
6. Step-by-step derivation

7. Simplified 59.5%

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

Reproduce

herbie shell --seed 2024138 
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  (- 1.0 (/ 1.0 (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))))))