Kahan p13 Example 2

Percentage Accurate: 100.0% → 100.0%

Time: 4.4s

Alternatives: 13

Speedup: 1.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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) :: t_2
    t_1 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))
    t_2 = t_1 * t_1
    code = (1.0d0 + t_2) / (2.0d0 + t_2)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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) :: t_2
    t_1 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))
    t_2 = t_1 * t_1
    code = (1.0d0 + t_2) / (2.0d0 + t_2)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-/l/ N/A

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

   6. sub-to-fraction N/A

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

   7. associate-*l/ N/A

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

   8. lower-/.f64 N/A

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

3. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-/l/ N/A

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

   6. sub-to-fraction N/A

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

   7. associate-*l/ N/A

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

   8. lower-/.f64 N/A

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

5. Applied rewrites 99.9%

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

6. Taylor expanded in t around 0

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

   1. lower-*.f64 99.9%

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

8. Applied rewrites 99.9%

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

9. Taylor expanded in t around 0

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

    1. lower-*.f64 99.9%

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

11. Applied rewrites 99.9%

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

13. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

Alternative 3: 99.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 1:\\ \;\;\;\;\left(0.8333333333333334 \cdot \frac{\mathsf{fma}\left(1.2, \frac{0.2222222222222222 - \frac{\frac{0.04938271604938271}{t} - -0.037037037037037035}{t}}{t}, -1\right)}{-0.8333333333333334}\right) \cdot 0.8333333333333334\\ \mathbf{else}:\\ \;\;\;\;0.5 + {t}^{2} \cdot \left(1 + -2 \cdot t\right)\\ \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))) 1.0)
   (*
    (*
     0.8333333333333334
     (/
      (fma
       1.2
       (/
        (-
         0.2222222222222222
         (/ (- (/ 0.04938271604938271 t) -0.037037037037037035) t))
        t)
       -1.0)
      -0.8333333333333334))
    0.8333333333333334)
   (+ 0.5 (* (pow t 2.0) (+ 1.0 (* -2.0 t))))))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 1.0) {
		tmp = (0.8333333333333334 * (fma(1.2, ((0.2222222222222222 - (((0.04938271604938271 / t) - -0.037037037037037035) / t)) / t), -1.0) / -0.8333333333333334)) * 0.8333333333333334;
	} else {
		tmp = 0.5 + (pow(t, 2.0) * (1.0 + (-2.0 * t)));
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))) <= 1.0)
		tmp = Float64(Float64(0.8333333333333334 * Float64(fma(1.2, Float64(Float64(0.2222222222222222 - Float64(Float64(Float64(0.04938271604938271 / t) - -0.037037037037037035) / t)) / t), -1.0) / -0.8333333333333334)) * 0.8333333333333334);
	else
		tmp = Float64(0.5 + Float64((t ^ 2.0) * Float64(1.0 + Float64(-2.0 * t))));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], N[(N[(0.8333333333333334 * N[(N[(1.2 * N[(N[(0.2222222222222222 - N[(N[(N[(0.04938271604938271 / t), $MachinePrecision] - -0.037037037037037035), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + -1.0), $MachinePrecision] / -0.8333333333333334), $MachinePrecision]), $MachinePrecision] * 0.8333333333333334), $MachinePrecision], N[(0.5 + N[(N[Power[t, 2.0], $MachinePrecision] * N[(1.0 + N[(-2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in t around -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. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 50.9%

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

   4. Applied rewrites 50.9%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

         \[\leadsto \frac{5}{6} - \color{blue}{\left(\mathsf{neg}\left(-1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}\right)\right)} \]

      3. sub-to-mult N/A

         \[\leadsto \left(1 - \frac{\mathsf{neg}\left(-1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}\right)}{\frac{5}{6}}\right) \cdot \color{blue}{\frac{5}{6}} \]

      4. lower-unsound-*.f64 N/A

         \[\leadsto \left(1 - \frac{\mathsf{neg}\left(-1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}\right)}{\frac{5}{6}}\right) \cdot \color{blue}{\frac{5}{6}} \]

   6. Applied rewrites 50.9%

      \[\leadsto \left(1 - \frac{\frac{0.2222222222222222 - \frac{\frac{0.04938271604938271}{t} - -0.037037037037037035}{t}}{t}}{0.8333333333333334}\right) \cdot \color{blue}{0.8333333333333334} \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(1 - \frac{\frac{\frac{2}{9} - \frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t}}{t}}{\frac{5}{6}}\right) \cdot \frac{5}{6} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(1 - \frac{\frac{\frac{2}{9} - \frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t}}{t}}{\frac{5}{6}}\right) \cdot \frac{5}{6} \]

      3. sub-to-fraction N/A

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

      4. metadata-eval N/A

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

      5. sub-to-mult-rev N/A

         \[\leadsto \frac{\left(1 - \frac{\frac{\frac{2}{9} - \frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t}}{t}}{\frac{5}{6}}\right) \cdot \frac{5}{6}}{\frac{5}{6}} \cdot \frac{5}{6} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\left(1 - \frac{\frac{\frac{2}{9} - \frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t}}{t}}{\frac{5}{6}}\right) \cdot \frac{5}{6}}{\frac{5}{6}} \cdot \frac{5}{6} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{\left(1 - \frac{\frac{\frac{2}{9} - \frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t}}{t}}{\frac{5}{6}}\right) \cdot \frac{5}{6}}{\frac{5}{6}} \cdot \frac{5}{6} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{5}{6} \cdot \left(1 - \frac{\frac{\frac{2}{9} - \frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t}}{t}}{\frac{5}{6}}\right)}{\frac{5}{6}} \cdot \frac{5}{6} \]

      9. associate-/l* N/A

         \[\leadsto \left(\frac{5}{6} \cdot \frac{1 - \frac{\frac{\frac{2}{9} - \frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t}}{t}}{\frac{5}{6}}}{\frac{5}{6}}\right) \cdot \frac{5}{6} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{5}{6} \cdot \frac{1 - \frac{\frac{\frac{2}{9} - \frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t}}{t}}{\frac{5}{6}}}{\frac{5}{6}}\right) \cdot \frac{5}{6} \]

      11. lift--.f64 N/A

          \[\leadsto \left(\frac{5}{6} \cdot \frac{1 - \frac{\frac{\frac{2}{9} - \frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t}}{t}}{\frac{5}{6}}}{\frac{5}{6}}\right) \cdot \frac{5}{6} \]

      12. sub-negate-rev N/A

          \[\leadsto \left(\frac{5}{6} \cdot \frac{\mathsf{neg}\left(\left(\frac{\frac{\frac{2}{9} - \frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t}}{t}}{\frac{5}{6}} - 1\right)\right)}{\frac{5}{6}}\right) \cdot \frac{5}{6} \]

      13. metadata-eval N/A

          \[\leadsto \left(\frac{5}{6} \cdot \frac{\mathsf{neg}\left(\left(\frac{\frac{\frac{2}{9} - \frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t}}{t}}{\frac{5}{6}} - 1\right)\right)}{\mathsf{neg}\left(\frac{-5}{6}\right)}\right) \cdot \frac{5}{6} \]

      14. frac-2neg-rev N/A

          \[\leadsto \left(\frac{5}{6} \cdot \frac{\frac{\frac{\frac{2}{9} - \frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t}}{t}}{\frac{5}{6}} - 1}{\frac{-5}{6}}\right) \cdot \frac{5}{6} \]

   8. Applied rewrites 50.9%

      \[\leadsto \left(0.8333333333333334 \cdot \frac{\mathsf{fma}\left(1.2, \frac{0.2222222222222222 - \frac{\frac{0.04938271604938271}{t} - -0.037037037037037035}{t}}{t}, -1\right)}{-0.8333333333333334}\right) \cdot 0.8333333333333334 \]

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

   1. Initial program 100.0%

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

   2. 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. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 50.9%

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

   4. Applied rewrites 50.9%

      \[\leadsto \color{blue}{0.5 + {t}^{2} \cdot \left(1 + -2 \cdot t\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{1 + \frac{1}{t}} \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\frac{0.04938271604938271}{t} - -0.037037037037037035}{t} - 0.2222222222222222}{t}, 1.2, 1\right) \cdot 0.8333333333333334\\ \mathbf{else}:\\ \;\;\;\;0.5 + {t}^{2} \cdot \left(1 + -2 \cdot t\right)\\ \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))) 1.0)
   (*
    (fma
     (/
      (-
       (/ (- (/ 0.04938271604938271 t) -0.037037037037037035) t)
       0.2222222222222222)
      t)
     1.2
     1.0)
    0.8333333333333334)
   (+ 0.5 (* (pow t 2.0) (+ 1.0 (* -2.0 t))))))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / (1.0 + (1.0 / t))) <= 1.0) {
		tmp = fma((((((0.04938271604938271 / t) - -0.037037037037037035) / t) - 0.2222222222222222) / t), 1.2, 1.0) * 0.8333333333333334;
	} else {
		tmp = 0.5 + (pow(t, 2.0) * (1.0 + (-2.0 * t)));
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))) <= 1.0)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(Float64(0.04938271604938271 / t) - -0.037037037037037035) / t) - 0.2222222222222222) / t), 1.2, 1.0) * 0.8333333333333334);
	else
		tmp = Float64(0.5 + Float64((t ^ 2.0) * Float64(1.0 + Float64(-2.0 * t))));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], N[(N[(N[(N[(N[(N[(N[(0.04938271604938271 / t), $MachinePrecision] - -0.037037037037037035), $MachinePrecision] / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision] * 1.2 + 1.0), $MachinePrecision] * 0.8333333333333334), $MachinePrecision], N[(0.5 + N[(N[Power[t, 2.0], $MachinePrecision] * N[(1.0 + N[(-2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in t around -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. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 50.9%

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

   4. Applied rewrites 50.9%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

         \[\leadsto \frac{5}{6} - \color{blue}{\left(\mathsf{neg}\left(-1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}\right)\right)} \]

      3. sub-to-mult N/A

         \[\leadsto \left(1 - \frac{\mathsf{neg}\left(-1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}\right)}{\frac{5}{6}}\right) \cdot \color{blue}{\frac{5}{6}} \]

      4. lower-unsound-*.f64 N/A

         \[\leadsto \left(1 - \frac{\mathsf{neg}\left(-1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}\right)}{\frac{5}{6}}\right) \cdot \color{blue}{\frac{5}{6}} \]

   6. Applied rewrites 50.9%

      \[\leadsto \left(1 - \frac{\frac{0.2222222222222222 - \frac{\frac{0.04938271604938271}{t} - -0.037037037037037035}{t}}{t}}{0.8333333333333334}\right) \cdot \color{blue}{0.8333333333333334} \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(1 - \frac{\frac{\frac{2}{9} - \frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t}}{t}}{\frac{5}{6}}\right) \cdot \frac{5}{6} \]

      2. sub-flip N/A

         \[\leadsto \left(1 + \left(\mathsf{neg}\left(\frac{\frac{\frac{2}{9} - \frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t}}{t}}{\frac{5}{6}}\right)\right)\right) \cdot \frac{5}{6} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\frac{\frac{2}{9} - \frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t}}{t}}{\frac{5}{6}}\right)\right) + 1\right) \cdot \frac{5}{6} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\frac{\frac{2}{9} - \frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t}}{t}}{\frac{5}{6}}\right)\right) + 1\right) \cdot \frac{5}{6} \]

      5. mult-flip N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\frac{2}{9} - \frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t}}{t} \cdot \frac{1}{\frac{5}{6}}\right)\right) + 1\right) \cdot \frac{5}{6} \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\frac{2}{9} - \frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t}}{t}\right)\right) \cdot \frac{1}{\frac{5}{6}} + 1\right) \cdot \frac{5}{6} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\frac{2}{9} - \frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t}}{t}\right)\right) \cdot \frac{1}{\frac{5}{6}} + 1\right) \cdot \frac{5}{6} \]

      8. distribute-neg-frac N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left(\frac{2}{9} - \frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t}\right)\right)}{t} \cdot \frac{1}{\frac{5}{6}} + 1\right) \cdot \frac{5}{6} \]

      9. lift--.f64 N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left(\frac{2}{9} - \frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t}\right)\right)}{t} \cdot \frac{1}{\frac{5}{6}} + 1\right) \cdot \frac{5}{6} \]

      10. sub-negate-rev N/A

          \[\leadsto \left(\frac{\frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t} - \frac{2}{9}}{t} \cdot \frac{1}{\frac{5}{6}} + 1\right) \cdot \frac{5}{6} \]

      11. lift--.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t} - \frac{2}{9}}{t} \cdot \frac{1}{\frac{5}{6}} + 1\right) \cdot \frac{5}{6} \]

      12. lift-/.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t} - \frac{2}{9}}{t} \cdot \frac{1}{\frac{5}{6}} + 1\right) \cdot \frac{5}{6} \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t} - \frac{2}{9}}{t}, \frac{1}{\frac{5}{6}}, 1\right) \cdot \frac{5}{6} \]

      14. metadata-eval 50.9%

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{0.04938271604938271}{t} - -0.037037037037037035}{t} - 0.2222222222222222}{t}, 1.2, 1\right) \cdot 0.8333333333333334 \]

   8. Applied rewrites 50.9%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{0.04938271604938271}{t} - -0.037037037037037035}{t} - 0.2222222222222222}{t}, 1.2, 1\right) \cdot 0.8333333333333334 \]

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

   1. Initial program 100.0%

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

   2. 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. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 50.9%

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

   4. Applied rewrites 50.9%

      \[\leadsto \color{blue}{0.5 + {t}^{2} \cdot \left(1 + -2 \cdot t\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;\frac{1 + t\_2}{2 + t\_2} \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\frac{0.04938271604938271}{t} - -0.037037037037037035}{t} - 0.2222222222222222}{t}, 1.2, 1\right) \cdot 0.8333333333333334\\ \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))) (t_2 (* t_1 t_1)))
   (if (<= (/ (+ 1.0 t_2) (+ 2.0 t_2)) 0.6)
     (fma t t 0.5)
     (*
      (fma
       (/
        (-
         (/ (- (/ 0.04938271604938271 t) -0.037037037037037035) t)
         0.2222222222222222)
        t)
       1.2
       1.0)
      0.8333333333333334))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.6) {
		tmp = fma(t, t, 0.5);
	} else {
		tmp = fma((((((0.04938271604938271 / t) - -0.037037037037037035) / t) - 0.2222222222222222) / t), 1.2, 1.0) * 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))))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.6)
		tmp = fma(t, t, 0.5);
	else
		tmp = Float64(fma(Float64(Float64(Float64(Float64(Float64(0.04938271604938271 / t) - -0.037037037037037035) / t) - 0.2222222222222222) / t), 1.2, 1.0) * 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]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], 0.6], N[(t * t + 0.5), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(0.04938271604938271 / t), $MachinePrecision] - -0.037037037037037035), $MachinePrecision] / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision] * 1.2 + 1.0), $MachinePrecision] * 0.8333333333333334), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 #s(literal 1 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)))))) (+.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))))))) < 0.59999999999999998

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-pow.f64 51.8%

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

   4. Applied rewrites 51.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 51.8%

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

   6. Applied rewrites 51.8%

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

   if 0.59999999999999998 < (/.f64 (+.f64 #s(literal 1 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)))))) (+.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 100.0%

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

   2. Taylor expanded in t around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 50.9%

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

   4. Applied rewrites 50.9%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

         \[\leadsto \frac{5}{6} - \color{blue}{\left(\mathsf{neg}\left(-1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}\right)\right)} \]

      3. sub-to-mult N/A

         \[\leadsto \left(1 - \frac{\mathsf{neg}\left(-1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}\right)}{\frac{5}{6}}\right) \cdot \color{blue}{\frac{5}{6}} \]

      4. lower-unsound-*.f64 N/A

         \[\leadsto \left(1 - \frac{\mathsf{neg}\left(-1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}\right)}{\frac{5}{6}}\right) \cdot \color{blue}{\frac{5}{6}} \]

   6. Applied rewrites 50.9%

      \[\leadsto \left(1 - \frac{\frac{0.2222222222222222 - \frac{\frac{0.04938271604938271}{t} - -0.037037037037037035}{t}}{t}}{0.8333333333333334}\right) \cdot \color{blue}{0.8333333333333334} \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(1 - \frac{\frac{\frac{2}{9} - \frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t}}{t}}{\frac{5}{6}}\right) \cdot \frac{5}{6} \]

      2. sub-flip N/A

         \[\leadsto \left(1 + \left(\mathsf{neg}\left(\frac{\frac{\frac{2}{9} - \frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t}}{t}}{\frac{5}{6}}\right)\right)\right) \cdot \frac{5}{6} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\frac{\frac{2}{9} - \frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t}}{t}}{\frac{5}{6}}\right)\right) + 1\right) \cdot \frac{5}{6} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\frac{\frac{2}{9} - \frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t}}{t}}{\frac{5}{6}}\right)\right) + 1\right) \cdot \frac{5}{6} \]

      5. mult-flip N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\frac{2}{9} - \frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t}}{t} \cdot \frac{1}{\frac{5}{6}}\right)\right) + 1\right) \cdot \frac{5}{6} \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\frac{2}{9} - \frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t}}{t}\right)\right) \cdot \frac{1}{\frac{5}{6}} + 1\right) \cdot \frac{5}{6} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\frac{2}{9} - \frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t}}{t}\right)\right) \cdot \frac{1}{\frac{5}{6}} + 1\right) \cdot \frac{5}{6} \]

      8. distribute-neg-frac N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left(\frac{2}{9} - \frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t}\right)\right)}{t} \cdot \frac{1}{\frac{5}{6}} + 1\right) \cdot \frac{5}{6} \]

      9. lift--.f64 N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left(\frac{2}{9} - \frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t}\right)\right)}{t} \cdot \frac{1}{\frac{5}{6}} + 1\right) \cdot \frac{5}{6} \]

      10. sub-negate-rev N/A

          \[\leadsto \left(\frac{\frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t} - \frac{2}{9}}{t} \cdot \frac{1}{\frac{5}{6}} + 1\right) \cdot \frac{5}{6} \]

      11. lift--.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t} - \frac{2}{9}}{t} \cdot \frac{1}{\frac{5}{6}} + 1\right) \cdot \frac{5}{6} \]

      12. lift-/.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t} - \frac{2}{9}}{t} \cdot \frac{1}{\frac{5}{6}} + 1\right) \cdot \frac{5}{6} \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{4}{81}}{t} - \frac{-1}{27}}{t} - \frac{2}{9}}{t}, \frac{1}{\frac{5}{6}}, 1\right) \cdot \frac{5}{6} \]

      14. metadata-eval 50.9%

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{0.04938271604938271}{t} - -0.037037037037037035}{t} - 0.2222222222222222}{t}, 1.2, 1\right) \cdot 0.8333333333333334 \]

   8. Applied rewrites 50.9%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{0.04938271604938271}{t} - -0.037037037037037035}{t} - 0.2222222222222222}{t}, 1.2, 1\right) \cdot 0.8333333333333334 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;\frac{1 + t\_2}{2 + t\_2} \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{0.04938271604938271}{t} - -0.037037037037037035}{t} - 0.2222222222222222, \frac{1}{t}, 0.8333333333333334\right)\\ \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))) (t_2 (* t_1 t_1)))
   (if (<= (/ (+ 1.0 t_2) (+ 2.0 t_2)) 0.6)
     (fma t t 0.5)
     (fma
      (-
       (/ (- (/ 0.04938271604938271 t) -0.037037037037037035) t)
       0.2222222222222222)
      (/ 1.0 t)
      0.8333333333333334))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.6) {
		tmp = fma(t, t, 0.5);
	} else {
		tmp = fma(((((0.04938271604938271 / t) - -0.037037037037037035) / t) - 0.2222222222222222), (1.0 / 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))))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.6)
		tmp = fma(t, t, 0.5);
	else
		tmp = fma(Float64(Float64(Float64(Float64(0.04938271604938271 / t) - -0.037037037037037035) / t) - 0.2222222222222222), Float64(1.0 / 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]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], 0.6], N[(t * t + 0.5), $MachinePrecision], N[(N[(N[(N[(N[(0.04938271604938271 / t), $MachinePrecision] - -0.037037037037037035), $MachinePrecision] / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] * N[(1.0 / t), $MachinePrecision] + 0.8333333333333334), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 #s(literal 1 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)))))) (+.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))))))) < 0.59999999999999998

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-pow.f64 51.8%

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

   4. Applied rewrites 51.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 51.8%

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

   6. Applied rewrites 51.8%

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

   if 0.59999999999999998 < (/.f64 (+.f64 #s(literal 1 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)))))) (+.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 100.0%

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

   2. Taylor expanded in t around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 50.9%

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

   4. Applied rewrites 50.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}\right)\right) + \frac{5}{6} \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}\right)\right) + \frac{5}{6} \]

      6. mult-flip N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right) \cdot \frac{1}{t}\right)\right) + \frac{5}{6} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right) \cdot \frac{1}{t}\right)\right) + \frac{5}{6} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)\right)\right) \cdot \frac{1}{t} + \frac{5}{6} \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\left(\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)\right), \color{blue}{\frac{1}{t}}, \frac{5}{6}\right) \]

   6. Applied rewrites 50.9%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{0.04938271604938271}{t} - -0.037037037037037035}{t} - 0.2222222222222222, \color{blue}{\frac{1}{t}}, 0.8333333333333334\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;\frac{1 + t\_2}{2 + t\_2} \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.04938271604938271}{t} - -0.037037037037037035}{t} - 0.2222222222222222}{t} - -0.8333333333333334\\ \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))) (t_2 (* t_1 t_1)))
   (if (<= (/ (+ 1.0 t_2) (+ 2.0 t_2)) 0.6)
     (fma 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 t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.6) {
		tmp = fma(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))))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.6)
		tmp = fma(t, t, 0.5);
	else
		tmp = 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]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], 0.6], N[(t * t + 0.5), $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 (+.f64 #s(literal 1 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)))))) (+.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))))))) < 0.59999999999999998

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-pow.f64 51.8%

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

   4. Applied rewrites 51.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 51.8%

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

   6. Applied rewrites 51.8%

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

   if 0.59999999999999998 < (/.f64 (+.f64 #s(literal 1 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)))))) (+.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 100.0%

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

   2. Taylor expanded in t around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 50.9%

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

   4. Applied rewrites 50.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

         \[\leadsto -1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t} - \color{blue}{\left(\mathsf{neg}\left(\frac{5}{6}\right)\right)} \]

      4. lower--.f64 N/A

         \[\leadsto -1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t} - \color{blue}{\left(\mathsf{neg}\left(\frac{5}{6}\right)\right)} \]

   6. Applied rewrites 50.9%

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

Alternative 8: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;\frac{1 + t\_2}{2 + t\_2} \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{0.26666666666666666 - 0.044444444444444446 \cdot \frac{1}{t}}{t}\right) \cdot 0.8333333333333334\\ \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))) (t_2 (* t_1 t_1)))
   (if (<= (/ (+ 1.0 t_2) (+ 2.0 t_2)) 0.6)
     (fma t t 0.5)
     (*
      (- 1.0 (/ (- 0.26666666666666666 (* 0.044444444444444446 (/ 1.0 t))) t))
      0.8333333333333334))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.6) {
		tmp = fma(t, t, 0.5);
	} else {
		tmp = (1.0 - ((0.26666666666666666 - (0.044444444444444446 * (1.0 / t))) / 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))))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.6)
		tmp = fma(t, t, 0.5);
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(0.26666666666666666 - Float64(0.044444444444444446 * Float64(1.0 / t))) / 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]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], 0.6], N[(t * t + 0.5), $MachinePrecision], N[(N[(1.0 - N[(N[(0.26666666666666666 - N[(0.044444444444444446 * N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * 0.8333333333333334), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 #s(literal 1 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)))))) (+.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))))))) < 0.59999999999999998

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-pow.f64 51.8%

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

   4. Applied rewrites 51.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 51.8%

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

   6. Applied rewrites 51.8%

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

   if 0.59999999999999998 < (/.f64 (+.f64 #s(literal 1 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)))))) (+.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 100.0%

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

   2. Taylor expanded in t around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 50.9%

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

   4. Applied rewrites 50.9%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

         \[\leadsto \frac{5}{6} - \color{blue}{\left(\mathsf{neg}\left(-1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}\right)\right)} \]

      3. sub-to-mult N/A

         \[\leadsto \left(1 - \frac{\mathsf{neg}\left(-1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}\right)}{\frac{5}{6}}\right) \cdot \color{blue}{\frac{5}{6}} \]

      4. lower-unsound-*.f64 N/A

         \[\leadsto \left(1 - \frac{\mathsf{neg}\left(-1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}\right)}{\frac{5}{6}}\right) \cdot \color{blue}{\frac{5}{6}} \]

   6. Applied rewrites 50.9%

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

   7. Taylor expanded in t around inf

      \[\leadsto \left(1 - \frac{\frac{4}{15} - \frac{2}{45} \cdot \frac{1}{t}}{t}\right) \cdot 0.8333333333333334 \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(1 - \frac{\frac{4}{15} - \frac{2}{45} \cdot \frac{1}{t}}{t}\right) \cdot \frac{5}{6} \]

      2. lower--.f64 N/A

         \[\leadsto \left(1 - \frac{\frac{4}{15} - \frac{2}{45} \cdot \frac{1}{t}}{t}\right) \cdot \frac{5}{6} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(1 - \frac{\frac{4}{15} - \frac{2}{45} \cdot \frac{1}{t}}{t}\right) \cdot \frac{5}{6} \]

      4. lower-/.f64 51.8%

         \[\leadsto \left(1 - \frac{0.26666666666666666 - 0.044444444444444446 \cdot \frac{1}{t}}{t}\right) \cdot 0.8333333333333334 \]

   9. Applied rewrites 51.8%

      \[\leadsto \left(1 - \frac{0.26666666666666666 - 0.044444444444444446 \cdot \frac{1}{t}}{t}\right) \cdot 0.8333333333333334 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;\frac{1 + t\_2}{2 + t\_2} \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{\frac{-0.037037037037037035}{t} - -0.2222222222222222}{t}\\ \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))) (t_2 (* t_1 t_1)))
   (if (<= (/ (+ 1.0 t_2) (+ 2.0 t_2)) 0.6)
     (fma t t 0.5)
     (-
      0.8333333333333334
      (/ (- (/ -0.037037037037037035 t) -0.2222222222222222) t)))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.6) {
		tmp = fma(t, t, 0.5);
	} else {
		tmp = 0.8333333333333334 - (((-0.037037037037037035 / t) - -0.2222222222222222) / t);
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.6)
		tmp = fma(t, t, 0.5);
	else
		tmp = Float64(0.8333333333333334 - Float64(Float64(Float64(-0.037037037037037035 / t) - -0.2222222222222222) / t));
	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]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], 0.6], N[(t * t + 0.5), $MachinePrecision], N[(0.8333333333333334 - N[(N[(N[(-0.037037037037037035 / t), $MachinePrecision] - -0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 #s(literal 1 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)))))) (+.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))))))) < 0.59999999999999998

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-pow.f64 51.8%

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

   4. Applied rewrites 51.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 51.8%

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

   6. Applied rewrites 51.8%

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

   if 0.59999999999999998 < (/.f64 (+.f64 #s(literal 1 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)))))) (+.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 100.0%

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

   2. Taylor expanded in t around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 50.9%

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

   4. Applied rewrites 50.9%

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

   5. Taylor expanded in t around inf

      \[\leadsto 0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + \frac{\frac{-1}{27}}{t}}{t} \]
   6. Step-by-step derivation

      1. lower-/.f64 51.8%

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

   7. Applied rewrites 51.8%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

         \[\leadsto \frac{5}{6} - \color{blue}{\left(\mathsf{neg}\left(-1 \cdot \frac{\frac{2}{9} + \frac{\frac{-1}{27}}{t}}{t}\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{5}{6} - \color{blue}{\left(\mathsf{neg}\left(-1 \cdot \frac{\frac{2}{9} + \frac{\frac{-1}{27}}{t}}{t}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{5}{6} - \left(\mathsf{neg}\left(-1 \cdot \frac{\frac{2}{9} + \frac{\frac{-1}{27}}{t}}{t}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \frac{5}{6} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{2}{9} + \frac{\frac{-1}{27}}{t}}{t}\right)\right)\right)\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{5}{6} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{2}{9} + \frac{\frac{-1}{27}}{t}}{t}\right)\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \frac{5}{6} - \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(\frac{2}{9} + \frac{\frac{-1}{27}}{t}\right)\right)}{t}\right)\right) \]

      8. distribute-frac-neg2 N/A

         \[\leadsto \frac{5}{6} - \frac{\mathsf{neg}\left(\left(\frac{2}{9} + \frac{\frac{-1}{27}}{t}\right)\right)}{\color{blue}{\mathsf{neg}\left(t\right)}} \]

      9. frac-2neg N/A

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

      10. lift-/.f64 51.8%

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

   9. Applied rewrites 51.8%

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

Alternative 10: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;\frac{1 + t\_2}{2 + t\_2} \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))) (t_2 (* t_1 t_1)))
   (if (<= (/ (+ 1.0 t_2) (+ 2.0 t_2)) 0.6)
     (fma t t 0.5)
     (- 0.8333333333333334 (/ 0.2222222222222222 t)))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.6) {
		tmp = fma(t, t, 0.5);
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.6)
		tmp = fma(t, t, 0.5);
	else
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	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]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], 0.6], N[(t * t + 0.5), $MachinePrecision], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 #s(literal 1 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)))))) (+.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))))))) < 0.59999999999999998

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-pow.f64 51.8%

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

   4. Applied rewrites 51.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 51.8%

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

   6. Applied rewrites 51.8%

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

   if 0.59999999999999998 < (/.f64 (+.f64 #s(literal 1 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)))))) (+.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 100.0%

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

   2. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 51.2%

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

   4. Applied rewrites 51.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lower-/.f64 51.2%

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

   6. Applied rewrites 51.2%

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

Alternative 11: 98.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;\frac{1 + t\_2}{2 + t\_2} \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))) (t_2 (* t_1 t_1)))
   (if (<= (/ (+ 1.0 t_2) (+ 2.0 t_2)) 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 t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 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))))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 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]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], 0.6], N[(t * t + 0.5), $MachinePrecision], 0.8333333333333334]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 #s(literal 1 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)))))) (+.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))))))) < 0.59999999999999998

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-pow.f64 51.8%

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

   4. Applied rewrites 51.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 51.8%

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

   6. Applied rewrites 51.8%

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

   if 0.59999999999999998 < (/.f64 (+.f64 #s(literal 1 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)))))) (+.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 100.0%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 58.9%

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

Alternative 12: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;\frac{1 + t\_2}{2 + t\_2} \leq 0.59:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))) (t_2 (* t_1 t_1)))
   (if (<= (/ (+ 1.0 t_2) (+ 2.0 t_2)) 0.59) 0.5 0.8333333333333334)))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.59) {
		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) :: t_2
    real(8) :: tmp
    t_1 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))
    t_2 = t_1 * t_1
    if (((1.0d0 + t_2) / (2.0d0 + t_2)) <= 0.59d0) 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 t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.59) {
		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)))
	t_2 = t_1 * t_1
	tmp = 0
	if ((1.0 + t_2) / (2.0 + t_2)) <= 0.59:
		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))))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.59)
		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)));
	t_2 = t_1 * t_1;
	tmp = 0.0;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.59)
		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]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], 0.59], 0.5, 0.8333333333333334]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 #s(literal 1 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)))))) (+.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))))))) < 0.58999999999999997

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 59.3%

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

   if 0.58999999999999997 < (/.f64 (+.f64 #s(literal 1 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)))))) (+.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 100.0%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 58.9%

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

Alternative 13: 59.3% accurate, 82.4× speedup

Math

\[0.5 \]

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 100.0%

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

2. Taylor expanded in t around 0

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

4. Applied rewrites 59.3%

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

Reproduce

herbie shell --seed 2025204 
(FPCore (t)
  :name "Kahan p13 Example 2"
  :precision binary64
  (/ (+ 1.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))) (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))))