Kahan p13 Example 3

Percentage Accurate: 99.9% → 100.0%

Time: 3.0s

Alternatives: 13

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t)
use fmin_fmax_functions
    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: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. lift-/.f64 N/A

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

3. Applied rewrites 100.0%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-fma.f64 N/A

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

5. Applied rewrites 100.0%

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

   1. lift-fma.f64 N/A

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

   2. *-lft-identity N/A

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

   3. lower-+.f64 100.0

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

7. Applied rewrites 100.0%

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

   1. lift-fma.f64 N/A

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

   2. *-lft-identity N/A

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

   3. lower-+.f64 100.0

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

9. Applied rewrites 100.0%

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

Alternative 2: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ \mathbf{if}\;1 - \frac{1}{2 + t\_1 \cdot t\_1} \leq 0.6:\\ \;\;\;\;1 - \mathsf{fma}\left(\left(2 - t\right) \cdot t - 1, t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\left(-\frac{\left(\frac{0.04938271604938271}{t \cdot t} - \frac{-0.037037037037037035}{t}\right) - 0.2222222222222222}{t}\right) + 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))
   (if (<= (- 1.0 (/ 1.0 (+ 2.0 (* t_1 t_1)))) 0.6)
     (- 1.0 (fma (- (* (- 2.0 t) t) 1.0) (* t t) 0.5))
     (-
      1.0
      (+
       (-
        (/
         (-
          (- (/ 0.04938271604938271 (* t t)) (/ -0.037037037037037035 t))
          0.2222222222222222)
         t))
       0.16666666666666666)))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double tmp;
	if ((1.0 - (1.0 / (2.0 + (t_1 * t_1)))) <= 0.6) {
		tmp = 1.0 - fma((((2.0 - t) * t) - 1.0), (t * t), 0.5);
	} else {
		tmp = 1.0 - (-((((0.04938271604938271 / (t * t)) - (-0.037037037037037035 / t)) - 0.2222222222222222) / t) + 0.16666666666666666);
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	tmp = 0.0
	if (Float64(1.0 - Float64(1.0 / Float64(2.0 + Float64(t_1 * t_1)))) <= 0.6)
		tmp = Float64(1.0 - fma(Float64(Float64(Float64(2.0 - t) * t) - 1.0), Float64(t * t), 0.5));
	else
		tmp = Float64(1.0 - Float64(Float64(-Float64(Float64(Float64(Float64(0.04938271604938271 / Float64(t * t)) - Float64(-0.037037037037037035 / t)) - 0.2222222222222222) / t)) + 0.16666666666666666));
	end
	return tmp
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]}, If[LessEqual[N[(1.0 - N[(1.0 / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.6], N[(1.0 - N[(N[(N[(N[(2.0 - t), $MachinePrecision] * t), $MachinePrecision] - 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[((-N[(N[(N[(N[(0.04938271604938271 / N[(t * t), $MachinePrecision]), $MachinePrecision] - N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]) + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 2 binary64) (*.f64 (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))) (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))))))) < 0.599999999999999978

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-/.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 99.6

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

   6. Applied rewrites 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. mul-1-neg N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 99.6

         \[\leadsto 1 - \mathsf{fma}\left(\left(2 - t\right) \cdot t - 1, t \cdot t, 0.5\right) \]

   8. Applied rewrites 99.6%

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

   if 0.599999999999999978 < (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 2 binary64) (*.f64 (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))) (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))))))))

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-/.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in t around -inf

      \[\leadsto 1 - \color{blue}{\left(\frac{1}{6} + -1 \cdot \frac{\left(\frac{\frac{4}{81}}{{t}^{2}} + \frac{1}{27} \cdot \frac{1}{t}\right) - \frac{2}{9}}{t}\right)} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 1 - \left(-1 \cdot \frac{\left(\frac{\frac{4}{81}}{{t}^{2}} + \frac{1}{27} \cdot \frac{1}{t}\right) - \frac{2}{9}}{t} + \color{blue}{\frac{1}{6}}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto 1 - \left(-1 \cdot \frac{\left(\frac{\frac{4}{81}}{{t}^{2}} + \frac{1}{27} \cdot \frac{1}{t}\right) - \frac{2}{9}}{t} + \color{blue}{\frac{1}{6}}\right) \]

   6. Applied rewrites 99.4%

      \[\leadsto 1 - \color{blue}{\left(\left(-\frac{\left(\frac{0.04938271604938271}{t \cdot t} - \frac{-0.037037037037037035}{t}\right) - 0.2222222222222222}{t}\right) + 0.16666666666666666\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ \mathbf{if}\;1 - \frac{1}{2 + t\_1 \cdot t\_1} \leq 0.6:\\ \;\;\;\;1 - \mathsf{fma}\left(\left(2 - t\right) \cdot t - 1, t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\left(-\frac{\frac{0.04938271604938271}{t} + 0.037037037037037035}{t}\right) + 0.2222222222222222}{t}\right) + 0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))
   (if (<= (- 1.0 (/ 1.0 (+ 2.0 (* t_1 t_1)))) 0.6)
     (- 1.0 (fma (- (* (- 2.0 t) t) 1.0) (* t t) 0.5))
     (+
      (-
       (/
        (+
         (- (/ (+ (/ 0.04938271604938271 t) 0.037037037037037035) t))
         0.2222222222222222)
        t))
      0.8333333333333334))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double tmp;
	if ((1.0 - (1.0 / (2.0 + (t_1 * t_1)))) <= 0.6) {
		tmp = 1.0 - fma((((2.0 - t) * t) - 1.0), (t * t), 0.5);
	} else {
		tmp = -((-(((0.04938271604938271 / t) + 0.037037037037037035) / t) + 0.2222222222222222) / t) + 0.8333333333333334;
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	tmp = 0.0
	if (Float64(1.0 - Float64(1.0 / Float64(2.0 + Float64(t_1 * t_1)))) <= 0.6)
		tmp = Float64(1.0 - fma(Float64(Float64(Float64(2.0 - t) * t) - 1.0), Float64(t * t), 0.5));
	else
		tmp = Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(Float64(0.04938271604938271 / t) + 0.037037037037037035) / t)) + 0.2222222222222222) / t)) + 0.8333333333333334);
	end
	return tmp
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]}, If[LessEqual[N[(1.0 - N[(1.0 / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.6], N[(1.0 - N[(N[(N[(N[(2.0 - t), $MachinePrecision] * t), $MachinePrecision] - 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[((-N[(N[((-N[(N[(N[(0.04938271604938271 / t), $MachinePrecision] + 0.037037037037037035), $MachinePrecision] / t), $MachinePrecision]) + 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]) + 0.8333333333333334), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 2 binary64) (*.f64 (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))) (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))))))) < 0.599999999999999978

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-/.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 99.6

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

   6. Applied rewrites 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. mul-1-neg N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 99.6

         \[\leadsto 1 - \mathsf{fma}\left(\left(2 - t\right) \cdot t - 1, t \cdot t, 0.5\right) \]

   8. Applied rewrites 99.6%

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

   if 0.599999999999999978 < (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 2 binary64) (*.f64 (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))) (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))))))))

   1. Initial program 99.9%

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

   2. Taylor expanded in t around -inf

      \[\leadsto \color{blue}{\frac{5}{6} + -1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto -1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t} + \color{blue}{\frac{5}{6}} \]

      2. lower-+.f64 N/A

         \[\leadsto -1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t} + \color{blue}{\frac{5}{6}} \]

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\left(-\frac{\left(-\frac{\frac{0.04938271604938271}{t} + 0.037037037037037035}{t}\right) + 0.2222222222222222}{t}\right) + 0.8333333333333334} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ \mathbf{if}\;1 - \frac{1}{2 + t\_1 \cdot t\_1} \leq 0.6:\\ \;\;\;\;1 - \mathsf{fma}\left(\left(2 - t\right) \cdot t - 1, t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222 - \frac{0.037037037037037035}{t}}{t} + 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))
   (if (<= (- 1.0 (/ 1.0 (+ 2.0 (* t_1 t_1)))) 0.6)
     (- 1.0 (fma (- (* (- 2.0 t) t) 1.0) (* t t) 0.5))
     (-
      1.0
      (+
       (/ (- 0.2222222222222222 (/ 0.037037037037037035 t)) t)
       0.16666666666666666)))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double tmp;
	if ((1.0 - (1.0 / (2.0 + (t_1 * t_1)))) <= 0.6) {
		tmp = 1.0 - fma((((2.0 - t) * t) - 1.0), (t * t), 0.5);
	} else {
		tmp = 1.0 - (((0.2222222222222222 - (0.037037037037037035 / t)) / t) + 0.16666666666666666);
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	tmp = 0.0
	if (Float64(1.0 - Float64(1.0 / Float64(2.0 + Float64(t_1 * t_1)))) <= 0.6)
		tmp = Float64(1.0 - fma(Float64(Float64(Float64(2.0 - t) * t) - 1.0), Float64(t * t), 0.5));
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(0.2222222222222222 - Float64(0.037037037037037035 / t)) / t) + 0.16666666666666666));
	end
	return tmp
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]}, If[LessEqual[N[(1.0 - N[(1.0 / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.6], N[(1.0 - N[(N[(N[(N[(2.0 - t), $MachinePrecision] * t), $MachinePrecision] - 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(N[(0.2222222222222222 - N[(0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 2 binary64) (*.f64 (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))) (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))))))) < 0.599999999999999978

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-/.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 99.6

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

   6. Applied rewrites 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. mul-1-neg N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 99.6

         \[\leadsto 1 - \mathsf{fma}\left(\left(2 - t\right) \cdot t - 1, t \cdot t, 0.5\right) \]

   8. Applied rewrites 99.6%

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

   if 0.599999999999999978 < (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 2 binary64) (*.f64 (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))) (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))))))))

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-/.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in t around -inf

      \[\leadsto 1 - \color{blue}{\left(\frac{1}{6} + -1 \cdot \frac{\frac{1}{27} \cdot \frac{1}{t} - \frac{2}{9}}{t}\right)} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 1 - \left(-1 \cdot \frac{\frac{1}{27} \cdot \frac{1}{t} - \frac{2}{9}}{t} + \color{blue}{\frac{1}{6}}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto 1 - \left(-1 \cdot \frac{\frac{1}{27} \cdot \frac{1}{t} - \frac{2}{9}}{t} + \color{blue}{\frac{1}{6}}\right) \]

      3. mul-1-neg N/A

         \[\leadsto 1 - \left(\left(\mathsf{neg}\left(\frac{\frac{1}{27} \cdot \frac{1}{t} - \frac{2}{9}}{t}\right)\right) + \frac{1}{6}\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto 1 - \left(\left(-\frac{\frac{1}{27} \cdot \frac{1}{t} - \frac{2}{9}}{t}\right) + \frac{1}{6}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto 1 - \left(\left(-\frac{\frac{1}{27} \cdot \frac{1}{t} - \frac{2}{9}}{t}\right) + \frac{1}{6}\right) \]

      6. lower--.f64 N/A

         \[\leadsto 1 - \left(\left(-\frac{\frac{1}{27} \cdot \frac{1}{t} - \frac{2}{9}}{t}\right) + \frac{1}{6}\right) \]

      7. associate-*r/ N/A

         \[\leadsto 1 - \left(\left(-\frac{\frac{\frac{1}{27} \cdot 1}{t} - \frac{2}{9}}{t}\right) + \frac{1}{6}\right) \]

      8. metadata-eval N/A

         \[\leadsto 1 - \left(\left(-\frac{\frac{\frac{1}{27}}{t} - \frac{2}{9}}{t}\right) + \frac{1}{6}\right) \]

      9. lower-/.f64 99.2

         \[\leadsto 1 - \left(\left(-\frac{\frac{0.037037037037037035}{t} - 0.2222222222222222}{t}\right) + 0.16666666666666666\right) \]

   6. Applied rewrites 99.2%

      \[\leadsto 1 - \color{blue}{\left(\left(-\frac{\frac{0.037037037037037035}{t} - 0.2222222222222222}{t}\right) + 0.16666666666666666\right)} \]

   7. Taylor expanded in t around inf

      \[\leadsto 1 - \left(\frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{t} + \frac{1}{6}\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 1 - \left(\frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{t} + \frac{1}{6}\right) \]

      2. associate-*r/ N/A

         \[\leadsto 1 - \left(\frac{\frac{2}{9} - \frac{\frac{1}{27} \cdot 1}{t}}{t} + \frac{1}{6}\right) \]

      3. metadata-eval N/A

         \[\leadsto 1 - \left(\frac{\frac{2}{9} - \frac{\frac{1}{27}}{t}}{t} + \frac{1}{6}\right) \]

      4. lower--.f64 N/A

         \[\leadsto 1 - \left(\frac{\frac{2}{9} - \frac{\frac{1}{27}}{t}}{t} + \frac{1}{6}\right) \]

      5. lift-/.f64 99.2

         \[\leadsto 1 - \left(\frac{0.2222222222222222 - \frac{0.037037037037037035}{t}}{t} + 0.16666666666666666\right) \]

   9. Applied rewrites 99.2%

      \[\leadsto 1 - \left(\frac{0.2222222222222222 - \frac{0.037037037037037035}{t}}{t} + 0.16666666666666666\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ \mathbf{if}\;1 - \frac{1}{2 + t\_1 \cdot t\_1} \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222 - \frac{0.037037037037037035}{t}}{t} + 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))
   (if (<= (- 1.0 (/ 1.0 (+ 2.0 (* t_1 t_1)))) 0.6)
     (fma (fma (- t 2.0) t 1.0) (* t t) 0.5)
     (-
      1.0
      (+
       (/ (- 0.2222222222222222 (/ 0.037037037037037035 t)) t)
       0.16666666666666666)))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double tmp;
	if ((1.0 - (1.0 / (2.0 + (t_1 * t_1)))) <= 0.6) {
		tmp = fma(fma((t - 2.0), t, 1.0), (t * t), 0.5);
	} else {
		tmp = 1.0 - (((0.2222222222222222 - (0.037037037037037035 / t)) / t) + 0.16666666666666666);
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	tmp = 0.0
	if (Float64(1.0 - Float64(1.0 / Float64(2.0 + Float64(t_1 * t_1)))) <= 0.6)
		tmp = fma(fma(Float64(t - 2.0), t, 1.0), Float64(t * t), 0.5);
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(0.2222222222222222 - Float64(0.037037037037037035 / t)) / t) + 0.16666666666666666));
	end
	return tmp
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]}, If[LessEqual[N[(1.0 - N[(1.0 / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.6], N[(N[(N[(t - 2.0), $MachinePrecision] * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(1.0 - N[(N[(N[(0.2222222222222222 - N[(0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 2 binary64) (*.f64 (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))) (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))))))) < 0.599999999999999978

   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. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), {t}^{2}, \frac{1}{2}\right) \]

      8. unpow2 N/A

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

      9. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   if 0.599999999999999978 < (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 2 binary64) (*.f64 (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))) (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))))))))

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-/.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in t around -inf

      \[\leadsto 1 - \color{blue}{\left(\frac{1}{6} + -1 \cdot \frac{\frac{1}{27} \cdot \frac{1}{t} - \frac{2}{9}}{t}\right)} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 1 - \left(-1 \cdot \frac{\frac{1}{27} \cdot \frac{1}{t} - \frac{2}{9}}{t} + \color{blue}{\frac{1}{6}}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto 1 - \left(-1 \cdot \frac{\frac{1}{27} \cdot \frac{1}{t} - \frac{2}{9}}{t} + \color{blue}{\frac{1}{6}}\right) \]

      3. mul-1-neg N/A

         \[\leadsto 1 - \left(\left(\mathsf{neg}\left(\frac{\frac{1}{27} \cdot \frac{1}{t} - \frac{2}{9}}{t}\right)\right) + \frac{1}{6}\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto 1 - \left(\left(-\frac{\frac{1}{27} \cdot \frac{1}{t} - \frac{2}{9}}{t}\right) + \frac{1}{6}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto 1 - \left(\left(-\frac{\frac{1}{27} \cdot \frac{1}{t} - \frac{2}{9}}{t}\right) + \frac{1}{6}\right) \]

      6. lower--.f64 N/A

         \[\leadsto 1 - \left(\left(-\frac{\frac{1}{27} \cdot \frac{1}{t} - \frac{2}{9}}{t}\right) + \frac{1}{6}\right) \]

      7. associate-*r/ N/A

         \[\leadsto 1 - \left(\left(-\frac{\frac{\frac{1}{27} \cdot 1}{t} - \frac{2}{9}}{t}\right) + \frac{1}{6}\right) \]

      8. metadata-eval N/A

         \[\leadsto 1 - \left(\left(-\frac{\frac{\frac{1}{27}}{t} - \frac{2}{9}}{t}\right) + \frac{1}{6}\right) \]

      9. lower-/.f64 99.2

         \[\leadsto 1 - \left(\left(-\frac{\frac{0.037037037037037035}{t} - 0.2222222222222222}{t}\right) + 0.16666666666666666\right) \]

   6. Applied rewrites 99.2%

      \[\leadsto 1 - \color{blue}{\left(\left(-\frac{\frac{0.037037037037037035}{t} - 0.2222222222222222}{t}\right) + 0.16666666666666666\right)} \]

   7. Taylor expanded in t around inf

      \[\leadsto 1 - \left(\frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{t} + \frac{1}{6}\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 1 - \left(\frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{t} + \frac{1}{6}\right) \]

      2. associate-*r/ N/A

         \[\leadsto 1 - \left(\frac{\frac{2}{9} - \frac{\frac{1}{27} \cdot 1}{t}}{t} + \frac{1}{6}\right) \]

      3. metadata-eval N/A

         \[\leadsto 1 - \left(\frac{\frac{2}{9} - \frac{\frac{1}{27}}{t}}{t} + \frac{1}{6}\right) \]

      4. lower--.f64 N/A

         \[\leadsto 1 - \left(\frac{\frac{2}{9} - \frac{\frac{1}{27}}{t}}{t} + \frac{1}{6}\right) \]

      5. lift-/.f64 99.2

         \[\leadsto 1 - \left(\frac{0.2222222222222222 - \frac{0.037037037037037035}{t}}{t} + 0.16666666666666666\right) \]

   9. Applied rewrites 99.2%

      \[\leadsto 1 - \left(\frac{0.2222222222222222 - \frac{0.037037037037037035}{t}}{t} + 0.16666666666666666\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ \mathbf{if}\;1 - \frac{1}{2 + t\_1 \cdot t\_1} \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222 - \frac{0.037037037037037035}{t}}{t} + 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))
   (if (<= (- 1.0 (/ 1.0 (+ 2.0 (* t_1 t_1)))) 0.6)
     (fma (fma -2.0 t 1.0) (* t t) 0.5)
     (-
      1.0
      (+
       (/ (- 0.2222222222222222 (/ 0.037037037037037035 t)) t)
       0.16666666666666666)))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double tmp;
	if ((1.0 - (1.0 / (2.0 + (t_1 * t_1)))) <= 0.6) {
		tmp = fma(fma(-2.0, t, 1.0), (t * t), 0.5);
	} else {
		tmp = 1.0 - (((0.2222222222222222 - (0.037037037037037035 / t)) / t) + 0.16666666666666666);
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	tmp = 0.0
	if (Float64(1.0 - Float64(1.0 / Float64(2.0 + Float64(t_1 * t_1)))) <= 0.6)
		tmp = fma(fma(-2.0, t, 1.0), Float64(t * t), 0.5);
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(0.2222222222222222 - Float64(0.037037037037037035 / t)) / t) + 0.16666666666666666));
	end
	return tmp
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]}, If[LessEqual[N[(1.0 - N[(1.0 / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.6], N[(N[(-2.0 * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(1.0 - N[(N[(N[(0.2222222222222222 - N[(0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 2 binary64) (*.f64 (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))) (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))))))) < 0.599999999999999978

   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. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

   if 0.599999999999999978 < (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 2 binary64) (*.f64 (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))) (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))))))))

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-/.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in t around -inf

      \[\leadsto 1 - \color{blue}{\left(\frac{1}{6} + -1 \cdot \frac{\frac{1}{27} \cdot \frac{1}{t} - \frac{2}{9}}{t}\right)} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 1 - \left(-1 \cdot \frac{\frac{1}{27} \cdot \frac{1}{t} - \frac{2}{9}}{t} + \color{blue}{\frac{1}{6}}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto 1 - \left(-1 \cdot \frac{\frac{1}{27} \cdot \frac{1}{t} - \frac{2}{9}}{t} + \color{blue}{\frac{1}{6}}\right) \]

      3. mul-1-neg N/A

         \[\leadsto 1 - \left(\left(\mathsf{neg}\left(\frac{\frac{1}{27} \cdot \frac{1}{t} - \frac{2}{9}}{t}\right)\right) + \frac{1}{6}\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto 1 - \left(\left(-\frac{\frac{1}{27} \cdot \frac{1}{t} - \frac{2}{9}}{t}\right) + \frac{1}{6}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto 1 - \left(\left(-\frac{\frac{1}{27} \cdot \frac{1}{t} - \frac{2}{9}}{t}\right) + \frac{1}{6}\right) \]

      6. lower--.f64 N/A

         \[\leadsto 1 - \left(\left(-\frac{\frac{1}{27} \cdot \frac{1}{t} - \frac{2}{9}}{t}\right) + \frac{1}{6}\right) \]

      7. associate-*r/ N/A

         \[\leadsto 1 - \left(\left(-\frac{\frac{\frac{1}{27} \cdot 1}{t} - \frac{2}{9}}{t}\right) + \frac{1}{6}\right) \]

      8. metadata-eval N/A

         \[\leadsto 1 - \left(\left(-\frac{\frac{\frac{1}{27}}{t} - \frac{2}{9}}{t}\right) + \frac{1}{6}\right) \]

      9. lower-/.f64 99.2

         \[\leadsto 1 - \left(\left(-\frac{\frac{0.037037037037037035}{t} - 0.2222222222222222}{t}\right) + 0.16666666666666666\right) \]

   6. Applied rewrites 99.2%

      \[\leadsto 1 - \color{blue}{\left(\left(-\frac{\frac{0.037037037037037035}{t} - 0.2222222222222222}{t}\right) + 0.16666666666666666\right)} \]

   7. Taylor expanded in t around inf

      \[\leadsto 1 - \left(\frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{t} + \frac{1}{6}\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 1 - \left(\frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{t} + \frac{1}{6}\right) \]

      2. associate-*r/ N/A

         \[\leadsto 1 - \left(\frac{\frac{2}{9} - \frac{\frac{1}{27} \cdot 1}{t}}{t} + \frac{1}{6}\right) \]

      3. metadata-eval N/A

         \[\leadsto 1 - \left(\frac{\frac{2}{9} - \frac{\frac{1}{27}}{t}}{t} + \frac{1}{6}\right) \]

      4. lower--.f64 N/A

         \[\leadsto 1 - \left(\frac{\frac{2}{9} - \frac{\frac{1}{27}}{t}}{t} + \frac{1}{6}\right) \]

      5. lift-/.f64 99.2

         \[\leadsto 1 - \left(\frac{0.2222222222222222 - \frac{0.037037037037037035}{t}}{t} + 0.16666666666666666\right) \]

   9. Applied rewrites 99.2%

      \[\leadsto 1 - \left(\frac{0.2222222222222222 - \frac{0.037037037037037035}{t}}{t} + 0.16666666666666666\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ \mathbf{if}\;1 - \frac{1}{2 + t\_1 \cdot t\_1} \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.037037037037037035}{t} - 0.2222222222222222}{t} + 0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))
   (if (<= (- 1.0 (/ 1.0 (+ 2.0 (* t_1 t_1)))) 0.6)
     (fma (fma -2.0 t 1.0) (* t t) 0.5)
     (+
      (/ (- (/ 0.037037037037037035 t) 0.2222222222222222) t)
      0.8333333333333334))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double tmp;
	if ((1.0 - (1.0 / (2.0 + (t_1 * t_1)))) <= 0.6) {
		tmp = fma(fma(-2.0, t, 1.0), (t * t), 0.5);
	} else {
		tmp = (((0.037037037037037035 / t) - 0.2222222222222222) / t) + 0.8333333333333334;
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	tmp = 0.0
	if (Float64(1.0 - Float64(1.0 / Float64(2.0 + Float64(t_1 * t_1)))) <= 0.6)
		tmp = fma(fma(-2.0, t, 1.0), Float64(t * t), 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(0.037037037037037035 / t) - 0.2222222222222222) / t) + 0.8333333333333334);
	end
	return tmp
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]}, If[LessEqual[N[(1.0 - N[(1.0 / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.6], N[(N[(-2.0 * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(N[(0.037037037037037035 / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision] + 0.8333333333333334), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 2 binary64) (*.f64 (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))) (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))))))) < 0.599999999999999978

   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. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

   if 0.599999999999999978 < (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 2 binary64) (*.f64 (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))) (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))))))))

   1. Initial program 99.9%

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

   2. Taylor expanded in t around -inf

      \[\leadsto \color{blue}{\frac{5}{6} + -1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto -1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t} + \color{blue}{\frac{5}{6}} \]

      2. lower-+.f64 N/A

         \[\leadsto -1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t} + \color{blue}{\frac{5}{6}} \]

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in t around inf

      \[\leadsto \frac{\frac{1}{27} \cdot \frac{1}{t} - \frac{2}{9}}{t} + \frac{5}{6} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{27} \cdot \frac{1}{t} - \frac{2}{9}}{t} + \frac{5}{6} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\frac{1}{27} \cdot \frac{1}{t} - \frac{2}{9}}{t} + \frac{5}{6} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{\frac{1}{27} \cdot 1}{t} - \frac{2}{9}}{t} + \frac{5}{6} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\frac{\frac{1}{27}}{t} - \frac{2}{9}}{t} + \frac{5}{6} \]

      5. lower-/.f64 99.2

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

   7. Applied rewrites 99.2%

      \[\leadsto \frac{\frac{0.037037037037037035}{t} - 0.2222222222222222}{t} + 0.8333333333333334 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ \mathbf{if}\;1 - \frac{1}{2 + t\_1 \cdot t\_1} \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.037037037037037035}{t} - 0.2222222222222222}{t} + 0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))
   (if (<= (- 1.0 (/ 1.0 (+ 2.0 (* t_1 t_1)))) 0.6)
     (fma t t 0.5)
     (+
      (/ (- (/ 0.037037037037037035 t) 0.2222222222222222) t)
      0.8333333333333334))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double tmp;
	if ((1.0 - (1.0 / (2.0 + (t_1 * t_1)))) <= 0.6) {
		tmp = fma(t, t, 0.5);
	} else {
		tmp = (((0.037037037037037035 / t) - 0.2222222222222222) / t) + 0.8333333333333334;
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	tmp = 0.0
	if (Float64(1.0 - Float64(1.0 / Float64(2.0 + Float64(t_1 * t_1)))) <= 0.6)
		tmp = fma(t, t, 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(0.037037037037037035 / t) - 0.2222222222222222) / t) + 0.8333333333333334);
	end
	return tmp
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]}, If[LessEqual[N[(1.0 - N[(1.0 / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.6], N[(t * t + 0.5), $MachinePrecision], N[(N[(N[(N[(0.037037037037037035 / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision] + 0.8333333333333334), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 2 binary64) (*.f64 (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))) (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))))))) < 0.599999999999999978

   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. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 99.3

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{t}, 0.5\right) \]

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, t, 0.5\right)} \]

   if 0.599999999999999978 < (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 2 binary64) (*.f64 (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))) (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))))))))

   1. Initial program 99.9%

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

   2. Taylor expanded in t around -inf

      \[\leadsto \color{blue}{\frac{5}{6} + -1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto -1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t} + \color{blue}{\frac{5}{6}} \]

      2. lower-+.f64 N/A

         \[\leadsto -1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t} + \color{blue}{\frac{5}{6}} \]

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in t around inf

      \[\leadsto \frac{\frac{1}{27} \cdot \frac{1}{t} - \frac{2}{9}}{t} + \frac{5}{6} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{27} \cdot \frac{1}{t} - \frac{2}{9}}{t} + \frac{5}{6} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\frac{1}{27} \cdot \frac{1}{t} - \frac{2}{9}}{t} + \frac{5}{6} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{\frac{1}{27} \cdot 1}{t} - \frac{2}{9}}{t} + \frac{5}{6} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\frac{\frac{1}{27}}{t} - \frac{2}{9}}{t} + \frac{5}{6} \]

      5. lower-/.f64 99.2

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

   7. Applied rewrites 99.2%

      \[\leadsto \frac{\frac{0.037037037037037035}{t} - 0.2222222222222222}{t} + 0.8333333333333334 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ \mathbf{if}\;\frac{1}{2 + t\_1 \cdot t\_1} \leq 0.2:\\ \;\;\;\;0.8333333333333334 - \frac{1}{t} \cdot 0.2222222222222222\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))
   (if (<= (/ 1.0 (+ 2.0 (* t_1 t_1))) 0.2)
     (- 0.8333333333333334 (* (/ 1.0 t) 0.2222222222222222))
     (fma t t 0.5))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double tmp;
	if ((1.0 / (2.0 + (t_1 * t_1))) <= 0.2) {
		tmp = 0.8333333333333334 - ((1.0 / t) * 0.2222222222222222);
	} else {
		tmp = fma(t, t, 0.5);
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	tmp = 0.0
	if (Float64(1.0 / Float64(2.0 + Float64(t_1 * t_1))) <= 0.2)
		tmp = Float64(0.8333333333333334 - Float64(Float64(1.0 / t) * 0.2222222222222222));
	else
		tmp = fma(t, t, 0.5);
	end
	return tmp
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]}, If[LessEqual[N[(1.0 / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.2], N[(0.8333333333333334 - N[(N[(1.0 / t), $MachinePrecision] * 0.2222222222222222), $MachinePrecision]), $MachinePrecision], N[(t * t + 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 2 binary64) (*.f64 (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))) (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))))))) < 0.20000000000000001

   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. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

         \[\leadsto \frac{5}{6} - \color{blue}{\frac{2}{9} \cdot \frac{1}{t}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{5}{6} - \frac{\frac{2}{9} \cdot 1}{\color{blue}{t}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{5}{6} - \frac{\frac{2}{9}}{t} \]

      4. lower-/.f64 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift-/.f64 N/A

         \[\leadsto \frac{5}{6} - \frac{\frac{2}{9}}{\color{blue}{t}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{5}{6} - \frac{\frac{2}{9} \cdot 1}{t} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{5}{6} - \frac{2}{9} \cdot \color{blue}{\frac{1}{t}} \]

      4. *-commutative N/A

         \[\leadsto \frac{5}{6} - \frac{1}{t} \cdot \color{blue}{\frac{2}{9}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{5}{6} - \frac{1}{t} \cdot \color{blue}{\frac{2}{9}} \]

      6. lift-/.f64 99.1

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

   6. Applied rewrites 99.1%

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

   if 0.20000000000000001 < (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 2 binary64) (*.f64 (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))) (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))))))

   1. Initial program 99.9%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 99.2

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{t}, 0.5\right) \]

   4. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, t, 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ \mathbf{if}\;\frac{1}{2 + t\_1 \cdot t\_1} \leq 0.2:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))
   (if (<= (/ 1.0 (+ 2.0 (* t_1 t_1))) 0.2)
     (- 0.8333333333333334 (/ 0.2222222222222222 t))
     (fma t t 0.5))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double tmp;
	if ((1.0 / (2.0 + (t_1 * t_1))) <= 0.2) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = fma(t, t, 0.5);
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	tmp = 0.0
	if (Float64(1.0 / Float64(2.0 + Float64(t_1 * t_1))) <= 0.2)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = fma(t, t, 0.5);
	end
	return tmp
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]}, If[LessEqual[N[(1.0 / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.2], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], N[(t * t + 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 2 binary64) (*.f64 (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))) (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))))))) < 0.20000000000000001

   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. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

         \[\leadsto \frac{5}{6} - \color{blue}{\frac{2}{9} \cdot \frac{1}{t}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{5}{6} - \frac{\frac{2}{9} \cdot 1}{\color{blue}{t}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{5}{6} - \frac{\frac{2}{9}}{t} \]

      4. lower-/.f64 99.1

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

   4. Applied rewrites 99.1%

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

   if 0.20000000000000001 < (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 2 binary64) (*.f64 (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))) (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))))))

   1. Initial program 99.9%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 99.2

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{t}, 0.5\right) \]

   4. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, t, 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 98.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ \mathbf{if}\;1 - \frac{1}{2 + t\_1 \cdot t\_1} \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))
   (if (<= (- 1.0 (/ 1.0 (+ 2.0 (* t_1 t_1)))) 0.6)
     (fma t t 0.5)
     0.8333333333333334)))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double tmp;
	if ((1.0 - (1.0 / (2.0 + (t_1 * t_1)))) <= 0.6) {
		tmp = fma(t, t, 0.5);
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	tmp = 0.0
	if (Float64(1.0 - Float64(1.0 / Float64(2.0 + Float64(t_1 * t_1)))) <= 0.6)
		tmp = fma(t, t, 0.5);
	else
		tmp = 0.8333333333333334;
	end
	return tmp
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]}, If[LessEqual[N[(1.0 - N[(1.0 / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.6], N[(t * t + 0.5), $MachinePrecision], 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 2 binary64) (*.f64 (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))) (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))))))) < 0.599999999999999978

   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. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 99.3

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{t}, 0.5\right) \]

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, t, 0.5\right)} \]

   if 0.599999999999999978 < (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 2 binary64) (*.f64 (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))) (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))))))))

   1. Initial program 99.9%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 97.9%

      \[\leadsto \color{blue}{0.8333333333333334} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 98.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ \mathbf{if}\;1 - \frac{1}{2 + t\_1 \cdot t\_1} \leq 0.65:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))
   (if (<= (- 1.0 (/ 1.0 (+ 2.0 (* t_1 t_1)))) 0.65) 0.5 0.8333333333333334)))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double tmp;
	if ((1.0 - (1.0 / (2.0 + (t_1 * t_1)))) <= 0.65) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))
    if ((1.0d0 - (1.0d0 / (2.0d0 + (t_1 * t_1)))) <= 0.65d0) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double tmp;
	if ((1.0 - (1.0 / (2.0 + (t_1 * t_1)))) <= 0.65) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))
	tmp = 0
	if (1.0 - (1.0 / (2.0 + (t_1 * t_1)))) <= 0.65:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	tmp = 0.0
	if (Float64(1.0 - Float64(1.0 / Float64(2.0 + Float64(t_1 * t_1)))) <= 0.65)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	tmp = 0.0;
	if ((1.0 - (1.0 / (2.0 + (t_1 * t_1)))) <= 0.65)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
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]}, If[LessEqual[N[(1.0 - N[(1.0 / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.65], 0.5, 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 2 binary64) (*.f64 (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))) (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))))))) < 0.650000000000000022

   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. Taylor expanded in t around 0

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

   4. Applied rewrites 99.0%

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

   if 0.650000000000000022 < (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 2 binary64) (*.f64 (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))) (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))))))))

   1. Initial program 99.9%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 97.9%

      \[\leadsto \color{blue}{0.8333333333333334} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 59.6% accurate, 43.5× 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t)
use fmin_fmax_functions
    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 99.9%

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

2. Taylor expanded in t around 0

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

4. Applied rewrites 59.6%

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

Reproduce

herbie shell --seed 2025116 
(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)))))))))