Kahan p13 Example 2

Percentage Accurate: 100.0% → 100.0%

Time: 9.4s

Alternatives: 14

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

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

FPCore

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

C

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

Fortran

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, 1.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. 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. unpow1 N/A

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

   3. metadata-eval N/A

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

   4. sqrt-pow1 N/A

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

   5. pow2 N/A

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

   6. rem-sqrt-square-rev N/A

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

   7. lift--.f64 N/A

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

   8. fabs-sub N/A

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

   9. unpow1 N/A

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

   10. metadata-eval N/A

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

   11. sqrt-pow1 N/A

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

   12. pow2 N/A

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

   13. rem-sqrt-square-rev N/A

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

   14. lift--.f64 N/A

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

   15. fabs-sub N/A

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

4. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. sqr-abs-rev N/A

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

   3. lift--.f64 N/A

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

   4. fabs-sub N/A

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

   5. lift--.f64 N/A

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

   6. fabs-sub N/A

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

   7. sqr-abs N/A

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

6. Applied rewrites 100.0%

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

Alternative 2: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right) \cdot t - 8, t, 4\right) \cdot t\right) \cdot t\\ t_2 := 2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\\ t_3 := t\_2 \cdot t\_2\\ \mathbf{if}\;\frac{1 + t\_3}{2 + t\_3} \leq 0.83:\\ \;\;\;\;\frac{1 + t\_1}{2 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{\frac{0.04938271604938271}{t} + 0.037037037037037035}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (* (* (fma (- (* (fma -16.0 t 12.0) t) 8.0) t 4.0) t) t))
        (t_2 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (pow t -1.0)))))
        (t_3 (* t_2 t_2)))
   (if (<= (/ (+ 1.0 t_3) (+ 2.0 t_3)) 0.83)
     (/ (+ 1.0 t_1) (+ 2.0 t_1))
     (-
      0.8333333333333334
      (/
       (-
        0.2222222222222222
        (/ (+ (/ 0.04938271604938271 t) 0.037037037037037035) t))
       t)))))

C

double code(double t) {
	double t_1 = (fma(((fma(-16.0, t, 12.0) * t) - 8.0), t, 4.0) * t) * t;
	double t_2 = 2.0 - ((2.0 / t) / (1.0 + pow(t, -1.0)));
	double t_3 = t_2 * t_2;
	double tmp;
	if (((1.0 + t_3) / (2.0 + t_3)) <= 0.83) {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	} else {
		tmp = 0.8333333333333334 - ((0.2222222222222222 - (((0.04938271604938271 / t) + 0.037037037037037035) / t)) / t);
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(Float64(fma(Float64(Float64(fma(-16.0, t, 12.0) * t) - 8.0), t, 4.0) * t) * t)
	t_2 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + (t ^ -1.0))))
	t_3 = Float64(t_2 * t_2)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_3) / Float64(2.0 + t_3)) <= 0.83)
		tmp = Float64(Float64(1.0 + t_1) / Float64(2.0 + t_1));
	else
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 - Float64(Float64(Float64(0.04938271604938271 / t) + 0.037037037037037035) / t)) / t));
	end
	return tmp
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(-16.0 * t + 12.0), $MachinePrecision] * t), $MachinePrecision] - 8.0), $MachinePrecision] * t + 4.0), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[Power[t, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * t$95$2), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 + t$95$3), $MachinePrecision] / N[(2.0 + t$95$3), $MachinePrecision]), $MachinePrecision], 0.83], N[(N[(1.0 + t$95$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 - N[(N[(0.2222222222222222 - N[(N[(N[(0.04938271604938271 / t), $MachinePrecision] + 0.037037037037037035), $MachinePrecision] / t), $MachinePrecision]), $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))))))) < 0.82999999999999996

   1. Initial program 100.0%

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

      1. 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. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. pow2 N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. lift--.f64 N/A

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

      8. fabs-sub N/A

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

      9. unpow1 N/A

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

      10. metadata-eval N/A

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

      11. sqrt-pow1 N/A

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

      12. pow2 N/A

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

      13. rem-sqrt-square-rev N/A

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

      14. lift--.f64 N/A

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

      15. fabs-sub N/A

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. sqr-abs-rev N/A

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

      3. lift--.f64 N/A

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

      4. fabs-sub N/A

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

      5. lift--.f64 N/A

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

      6. fabs-sub N/A

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

      7. sqr-abs N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in t around 0

      \[\leadsto \frac{1 + \left(\frac{2}{1 + t} - 2\right) \cdot \left(\frac{2}{1 + t} - 2\right)}{2 + \color{blue}{{t}^{2} \cdot \left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1 + \left(\frac{2}{1 + t} - 2\right) \cdot \left(\frac{2}{1 + t} - 2\right)}{2 + \color{blue}{\left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right) \cdot {t}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{1 + \left(\frac{2}{1 + t} - 2\right) \cdot \left(\frac{2}{1 + t} - 2\right)}{2 + \left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{1 + \left(\frac{2}{1 + t} - 2\right) \cdot \left(\frac{2}{1 + t} - 2\right)}{2 + \color{blue}{\left(\left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right) \cdot t\right) \cdot t}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1 + \left(\frac{2}{1 + t} - 2\right) \cdot \left(\frac{2}{1 + t} - 2\right)}{2 + \color{blue}{\left(\left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right) \cdot t\right) \cdot t}} \]

   9. Applied rewrites 98.6%

      \[\leadsto \frac{1 + \left(\frac{2}{1 + t} - 2\right) \cdot \left(\frac{2}{1 + t} - 2\right)}{2 + \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right) \cdot t - 8, t, 4\right) \cdot t\right) \cdot t}} \]

   10. Taylor expanded in t around 0

       \[\leadsto \frac{1 + \color{blue}{{t}^{2} \cdot \left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)}}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right) \cdot t - 8, t, 4\right) \cdot t\right) \cdot t} \]
   11. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{1 + \color{blue}{\left(t \cdot t\right)} \cdot \left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right) \cdot t - 8, t, 4\right) \cdot t\right) \cdot t} \]

       2. associate-*l* N/A

          \[\leadsto \frac{1 + \color{blue}{t \cdot \left(t \cdot \left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)\right)}}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right) \cdot t - 8, t, 4\right) \cdot t\right) \cdot t} \]

       3. *-commutative N/A

          \[\leadsto \frac{1 + \color{blue}{\left(t \cdot \left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)\right) \cdot t}}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right) \cdot t - 8, t, 4\right) \cdot t\right) \cdot t} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{1 + \color{blue}{\left(t \cdot \left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)\right) \cdot t}}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right) \cdot t - 8, t, 4\right) \cdot t\right) \cdot t} \]

       5. *-commutative N/A

          \[\leadsto \frac{1 + \color{blue}{\left(\left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right) \cdot t\right)} \cdot t}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right) \cdot t - 8, t, 4\right) \cdot t\right) \cdot t} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{1 + \color{blue}{\left(\left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right) \cdot t\right)} \cdot t}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right) \cdot t - 8, t, 4\right) \cdot t\right) \cdot t} \]

       7. +-commutative N/A

          \[\leadsto \frac{1 + \left(\color{blue}{\left(t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right) + 4\right)} \cdot t\right) \cdot t}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right) \cdot t - 8, t, 4\right) \cdot t\right) \cdot t} \]

       8. *-commutative N/A

          \[\leadsto \frac{1 + \left(\left(\color{blue}{\left(t \cdot \left(12 + -16 \cdot t\right) - 8\right) \cdot t} + 4\right) \cdot t\right) \cdot t}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right) \cdot t - 8, t, 4\right) \cdot t\right) \cdot t} \]

       9. lower-fma.f64 N/A

          \[\leadsto \frac{1 + \left(\color{blue}{\mathsf{fma}\left(t \cdot \left(12 + -16 \cdot t\right) - 8, t, 4\right)} \cdot t\right) \cdot t}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right) \cdot t - 8, t, 4\right) \cdot t\right) \cdot t} \]

       10. lower--.f64 N/A

           \[\leadsto \frac{1 + \left(\mathsf{fma}\left(\color{blue}{t \cdot \left(12 + -16 \cdot t\right) - 8}, t, 4\right) \cdot t\right) \cdot t}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right) \cdot t - 8, t, 4\right) \cdot t\right) \cdot t} \]

       11. *-commutative N/A

           \[\leadsto \frac{1 + \left(\mathsf{fma}\left(\color{blue}{\left(12 + -16 \cdot t\right) \cdot t} - 8, t, 4\right) \cdot t\right) \cdot t}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right) \cdot t - 8, t, 4\right) \cdot t\right) \cdot t} \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{1 + \left(\mathsf{fma}\left(\color{blue}{\left(12 + -16 \cdot t\right) \cdot t} - 8, t, 4\right) \cdot t\right) \cdot t}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right) \cdot t - 8, t, 4\right) \cdot t\right) \cdot t} \]

       13. +-commutative N/A

           \[\leadsto \frac{1 + \left(\mathsf{fma}\left(\color{blue}{\left(-16 \cdot t + 12\right)} \cdot t - 8, t, 4\right) \cdot t\right) \cdot t}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right) \cdot t - 8, t, 4\right) \cdot t\right) \cdot t} \]

       14. lower-fma.f64 98.6

           \[\leadsto \frac{1 + \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-16, t, 12\right)} \cdot t - 8, t, 4\right) \cdot t\right) \cdot t}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right) \cdot t - 8, t, 4\right) \cdot t\right) \cdot t} \]

   12. Applied rewrites 98.6%

       \[\leadsto \frac{1 + \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right) \cdot t - 8, t, 4\right) \cdot t\right) \cdot t}}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right) \cdot t - 8, t, 4\right) \cdot t\right) \cdot t} \]

   if 0.82999999999999996 < (/.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. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right) + \frac{5}{6}\right)} - \frac{2}{9} \cdot \frac{1}{t} \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right) + \left(\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}\right)} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}\right) + \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)} \]

      4. associate--r- N/A

         \[\leadsto \color{blue}{\frac{5}{6} - \left(\frac{2}{9} \cdot \frac{1}{t} - \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{5}{6} - \left(\color{blue}{\frac{\frac{2}{9} \cdot 1}{t}} - \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \frac{5}{6} - \left(\frac{\color{blue}{\frac{2}{9}}}{t} - \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right) \]

      7. cube-mult N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. div-add-rev N/A

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

      12. unpow2 N/A

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

      13. associate-/l/ N/A

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

      14. div-sub N/A

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right)} \leq 0.83:\\ \;\;\;\;\frac{1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right) \cdot t - 8, t, 4\right) \cdot t\right) \cdot t}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right) \cdot t - 8, t, 4\right) \cdot t\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{\frac{0.04938271604938271}{t} + 0.037037037037037035}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;\frac{1 + t\_2}{2 + t\_2} \leq 0.83:\\ \;\;\;\;\frac{1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right) \cdot t - 8, t, 4\right) \cdot t\right) \cdot t}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{\frac{0.04938271604938271}{t} + 0.037037037037037035}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (pow t -1.0))))) (t_2 (* t_1 t_1)))
   (if (<= (/ (+ 1.0 t_2) (+ 2.0 t_2)) 0.83)
     (/
      (+ 1.0 (* (* (fma (- (* (fma -16.0 t 12.0) t) 8.0) t 4.0) t) t))
      (fma (fma (- (* 12.0 t) 8.0) t 4.0) (* t t) 2.0))
     (-
      0.8333333333333334
      (/
       (-
        0.2222222222222222
        (/ (+ (/ 0.04938271604938271 t) 0.037037037037037035) t))
       t)))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + pow(t, -1.0)));
	double t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.83) {
		tmp = (1.0 + ((fma(((fma(-16.0, t, 12.0) * t) - 8.0), t, 4.0) * t) * t)) / fma(fma(((12.0 * t) - 8.0), t, 4.0), (t * t), 2.0);
	} else {
		tmp = 0.8333333333333334 - ((0.2222222222222222 - (((0.04938271604938271 / t) + 0.037037037037037035) / t)) / t);
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + (t ^ -1.0))))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.83)
		tmp = Float64(Float64(1.0 + Float64(Float64(fma(Float64(Float64(fma(-16.0, t, 12.0) * t) - 8.0), t, 4.0) * t) * t)) / fma(fma(Float64(Float64(12.0 * t) - 8.0), t, 4.0), Float64(t * t), 2.0));
	else
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 - Float64(Float64(Float64(0.04938271604938271 / t) + 0.037037037037037035) / t)) / 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[Power[t, -1.0], $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], 0.83], N[(N[(1.0 + N[(N[(N[(N[(N[(N[(-16.0 * t + 12.0), $MachinePrecision] * t), $MachinePrecision] - 8.0), $MachinePrecision] * t + 4.0), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(12.0 * t), $MachinePrecision] - 8.0), $MachinePrecision] * t + 4.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 - N[(N[(0.2222222222222222 - N[(N[(N[(0.04938271604938271 / t), $MachinePrecision] + 0.037037037037037035), $MachinePrecision] / t), $MachinePrecision]), $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))))))) < 0.82999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

      \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{2 + {t}^{2} \cdot \left(4 + t \cdot \left(12 \cdot t - 8\right)\right)}} \]
   4. Step-by-step derivation

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

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{2 - \left(\mathsf{neg}\left({t}^{2}\right)\right) \cdot \left(4 + t \cdot \left(12 \cdot t - 8\right)\right)}} \]

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

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{2 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({t}^{2}\right)\right)\right)\right) \cdot \left(4 + t \cdot \left(12 \cdot t - 8\right)\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({t}^{2}\right)\right)\right)\right) \cdot \left(4 + t \cdot \left(12 \cdot t - 8\right)\right) + 2}} \]

      4. remove-double-neg N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{{t}^{2}} \cdot \left(4 + t \cdot \left(12 \cdot t - 8\right)\right) + 2} \]

      5. *-commutative N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{\left(4 + t \cdot \left(12 \cdot t - 8\right)\right) \cdot {t}^{2}} + 2} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{\mathsf{fma}\left(4 + t \cdot \left(12 \cdot t - 8\right), {t}^{2}, 2\right)}} \]

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

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\mathsf{fma}\left(\color{blue}{4 - \left(\mathsf{neg}\left(t\right)\right) \cdot \left(12 \cdot t - 8\right)}, {t}^{2}, 2\right)} \]

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

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\mathsf{fma}\left(\color{blue}{4 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) \cdot \left(12 \cdot t - 8\right)}, {t}^{2}, 2\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) \cdot \left(12 \cdot t - 8\right) + 4}, {t}^{2}, 2\right)} \]

      10. remove-double-neg N/A

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\mathsf{fma}\left(\color{blue}{t} \cdot \left(12 \cdot t - 8\right) + 4, {t}^{2}, 2\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\mathsf{fma}\left(\color{blue}{\left(12 \cdot t - 8\right) \cdot t} + 4, {t}^{2}, 2\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(12 \cdot t - 8, t, 4\right)}, {t}^{2}, 2\right)} \]

      13. lower--.f64 N/A

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{12 \cdot t - 8}, t, 4\right), {t}^{2}, 2\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{12 \cdot t} - 8, t, 4\right), {t}^{2}, 2\right)} \]

      15. unpow2 N/A

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), \color{blue}{t \cdot t}, 2\right)} \]

      16. lower-*.f64 98.5

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), \color{blue}{t \cdot t}, 2\right)} \]

   5. Applied rewrites 98.5%

      \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{1 + \color{blue}{{t}^{2} \cdot \left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{1 + \color{blue}{\left(t \cdot t\right)} \cdot \left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

      2. associate-*l* N/A

         \[\leadsto \frac{1 + \color{blue}{t \cdot \left(t \cdot \left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{1 + \color{blue}{\left(t \cdot \left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)\right) \cdot t}}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1 + \color{blue}{\left(t \cdot \left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)\right) \cdot t}}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

   8. Applied rewrites 98.5%

      \[\leadsto \frac{1 + \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right) \cdot t - 8, t, 4\right) \cdot t\right) \cdot t}}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

   if 0.82999999999999996 < (/.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. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right) + \frac{5}{6}\right)} - \frac{2}{9} \cdot \frac{1}{t} \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right) + \left(\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}\right)} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}\right) + \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)} \]

      4. associate--r- N/A

         \[\leadsto \color{blue}{\frac{5}{6} - \left(\frac{2}{9} \cdot \frac{1}{t} - \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{5}{6} - \left(\color{blue}{\frac{\frac{2}{9} \cdot 1}{t}} - \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \frac{5}{6} - \left(\frac{\color{blue}{\frac{2}{9}}}{t} - \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right) \]

      7. cube-mult N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. div-add-rev N/A

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

      12. unpow2 N/A

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

      13. associate-/l/ N/A

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

      14. div-sub N/A

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right)} \leq 0.83:\\ \;\;\;\;\frac{1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right) \cdot t - 8, t, 4\right) \cdot t\right) \cdot t}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{\frac{0.04938271604938271}{t} + 0.037037037037037035}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(12 \cdot t - 8, t, 4\right)\\ t_2 := 2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\\ t_3 := t\_2 \cdot t\_2\\ \mathbf{if}\;\frac{1 + t\_3}{2 + t\_3} \leq 0.83:\\ \;\;\;\;\frac{1 + \left(t\_1 \cdot t\right) \cdot t}{\mathsf{fma}\left(t\_1, t \cdot t, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{\frac{0.04938271604938271}{t} + 0.037037037037037035}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (fma (- (* 12.0 t) 8.0) t 4.0))
        (t_2 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (pow t -1.0)))))
        (t_3 (* t_2 t_2)))
   (if (<= (/ (+ 1.0 t_3) (+ 2.0 t_3)) 0.83)
     (/ (+ 1.0 (* (* t_1 t) t)) (fma t_1 (* t t) 2.0))
     (-
      0.8333333333333334
      (/
       (-
        0.2222222222222222
        (/ (+ (/ 0.04938271604938271 t) 0.037037037037037035) t))
       t)))))

C

double code(double t) {
	double t_1 = fma(((12.0 * t) - 8.0), t, 4.0);
	double t_2 = 2.0 - ((2.0 / t) / (1.0 + pow(t, -1.0)));
	double t_3 = t_2 * t_2;
	double tmp;
	if (((1.0 + t_3) / (2.0 + t_3)) <= 0.83) {
		tmp = (1.0 + ((t_1 * t) * t)) / fma(t_1, (t * t), 2.0);
	} else {
		tmp = 0.8333333333333334 - ((0.2222222222222222 - (((0.04938271604938271 / t) + 0.037037037037037035) / t)) / t);
	}
	return tmp;
}

Julia

function code(t)
	t_1 = fma(Float64(Float64(12.0 * t) - 8.0), t, 4.0)
	t_2 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + (t ^ -1.0))))
	t_3 = Float64(t_2 * t_2)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_3) / Float64(2.0 + t_3)) <= 0.83)
		tmp = Float64(Float64(1.0 + Float64(Float64(t_1 * t) * t)) / fma(t_1, Float64(t * t), 2.0));
	else
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 - Float64(Float64(Float64(0.04938271604938271 / t) + 0.037037037037037035) / t)) / t));
	end
	return tmp
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(N[(12.0 * t), $MachinePrecision] - 8.0), $MachinePrecision] * t + 4.0), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[Power[t, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * t$95$2), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 + t$95$3), $MachinePrecision] / N[(2.0 + t$95$3), $MachinePrecision]), $MachinePrecision], 0.83], N[(N[(1.0 + N[(N[(t$95$1 * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(t * t), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 - N[(N[(0.2222222222222222 - N[(N[(N[(0.04938271604938271 / t), $MachinePrecision] + 0.037037037037037035), $MachinePrecision] / t), $MachinePrecision]), $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))))))) < 0.82999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

      \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{2 + {t}^{2} \cdot \left(4 + t \cdot \left(12 \cdot t - 8\right)\right)}} \]
   4. Step-by-step derivation

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

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{2 - \left(\mathsf{neg}\left({t}^{2}\right)\right) \cdot \left(4 + t \cdot \left(12 \cdot t - 8\right)\right)}} \]

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

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{2 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({t}^{2}\right)\right)\right)\right) \cdot \left(4 + t \cdot \left(12 \cdot t - 8\right)\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({t}^{2}\right)\right)\right)\right) \cdot \left(4 + t \cdot \left(12 \cdot t - 8\right)\right) + 2}} \]

      4. remove-double-neg N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{{t}^{2}} \cdot \left(4 + t \cdot \left(12 \cdot t - 8\right)\right) + 2} \]

      5. *-commutative N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{\left(4 + t \cdot \left(12 \cdot t - 8\right)\right) \cdot {t}^{2}} + 2} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{\mathsf{fma}\left(4 + t \cdot \left(12 \cdot t - 8\right), {t}^{2}, 2\right)}} \]

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

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\mathsf{fma}\left(\color{blue}{4 - \left(\mathsf{neg}\left(t\right)\right) \cdot \left(12 \cdot t - 8\right)}, {t}^{2}, 2\right)} \]

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

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\mathsf{fma}\left(\color{blue}{4 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) \cdot \left(12 \cdot t - 8\right)}, {t}^{2}, 2\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) \cdot \left(12 \cdot t - 8\right) + 4}, {t}^{2}, 2\right)} \]

      10. remove-double-neg N/A

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\mathsf{fma}\left(\color{blue}{t} \cdot \left(12 \cdot t - 8\right) + 4, {t}^{2}, 2\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\mathsf{fma}\left(\color{blue}{\left(12 \cdot t - 8\right) \cdot t} + 4, {t}^{2}, 2\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(12 \cdot t - 8, t, 4\right)}, {t}^{2}, 2\right)} \]

      13. lower--.f64 N/A

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{12 \cdot t - 8}, t, 4\right), {t}^{2}, 2\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{12 \cdot t} - 8, t, 4\right), {t}^{2}, 2\right)} \]

      15. unpow2 N/A

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), \color{blue}{t \cdot t}, 2\right)} \]

      16. lower-*.f64 98.5

          \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), \color{blue}{t \cdot t}, 2\right)} \]

   5. Applied rewrites 98.5%

      \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{1 + \color{blue}{{t}^{2} \cdot \left(4 + t \cdot \left(12 \cdot t - 8\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{1 + \color{blue}{\left(t \cdot t\right)} \cdot \left(4 + t \cdot \left(12 \cdot t - 8\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

      2. associate-*l* N/A

         \[\leadsto \frac{1 + \color{blue}{t \cdot \left(t \cdot \left(4 + t \cdot \left(12 \cdot t - 8\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{1 + \color{blue}{\left(t \cdot \left(4 + t \cdot \left(12 \cdot t - 8\right)\right)\right) \cdot t}}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1 + \color{blue}{\left(t \cdot \left(4 + t \cdot \left(12 \cdot t - 8\right)\right)\right) \cdot t}}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{1 + \color{blue}{\left(\left(4 + t \cdot \left(12 \cdot t - 8\right)\right) \cdot t\right)} \cdot t}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1 + \color{blue}{\left(\left(4 + t \cdot \left(12 \cdot t - 8\right)\right) \cdot t\right)} \cdot t}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

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

         \[\leadsto \frac{1 + \left(\color{blue}{\left(4 - \left(\mathsf{neg}\left(t\right)\right) \cdot \left(12 \cdot t - 8\right)\right)} \cdot t\right) \cdot t}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

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

         \[\leadsto \frac{1 + \left(\color{blue}{\left(4 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) \cdot \left(12 \cdot t - 8\right)\right)} \cdot t\right) \cdot t}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{1 + \left(\color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) \cdot \left(12 \cdot t - 8\right) + 4\right)} \cdot t\right) \cdot t}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

      10. remove-double-neg N/A

          \[\leadsto \frac{1 + \left(\left(\color{blue}{t} \cdot \left(12 \cdot t - 8\right) + 4\right) \cdot t\right) \cdot t}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{1 + \left(\left(\color{blue}{\left(12 \cdot t - 8\right) \cdot t} + 4\right) \cdot t\right) \cdot t}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{1 + \left(\color{blue}{\mathsf{fma}\left(12 \cdot t - 8, t, 4\right)} \cdot t\right) \cdot t}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

      13. lower--.f64 N/A

          \[\leadsto \frac{1 + \left(\mathsf{fma}\left(\color{blue}{12 \cdot t - 8}, t, 4\right) \cdot t\right) \cdot t}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

      14. lower-*.f64 98.5

          \[\leadsto \frac{1 + \left(\mathsf{fma}\left(\color{blue}{12 \cdot t} - 8, t, 4\right) \cdot t\right) \cdot t}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

   8. Applied rewrites 98.5%

      \[\leadsto \frac{1 + \color{blue}{\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right) \cdot t\right) \cdot t}}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)} \]

   if 0.82999999999999996 < (/.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. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right) + \frac{5}{6}\right)} - \frac{2}{9} \cdot \frac{1}{t} \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right) + \left(\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}\right)} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}\right) + \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)} \]

      4. associate--r- N/A

         \[\leadsto \color{blue}{\frac{5}{6} - \left(\frac{2}{9} \cdot \frac{1}{t} - \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{5}{6} - \left(\color{blue}{\frac{\frac{2}{9} \cdot 1}{t}} - \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \frac{5}{6} - \left(\frac{\color{blue}{\frac{2}{9}}}{t} - \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right) \]

      7. cube-mult N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. div-add-rev N/A

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

      12. unpow2 N/A

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

      13. associate-/l/ N/A

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

      14. div-sub N/A

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right)} \leq 0.83:\\ \;\;\;\;\frac{1 + \left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right) \cdot t\right) \cdot t}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{\frac{0.04938271604938271}{t} + 0.037037037037037035}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;\frac{1 + t\_2}{2 + t\_2} \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{\frac{0.04938271604938271}{t} + 0.037037037037037035}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (pow t -1.0))))) (t_2 (* t_1 t_1)))
   (if (<= (/ (+ 1.0 t_2) (+ 2.0 t_2)) 0.8)
     (fma (fma (- t 2.0) t 1.0) (* t t) 0.5)
     (-
      0.8333333333333334
      (/
       (-
        0.2222222222222222
        (/ (+ (/ 0.04938271604938271 t) 0.037037037037037035) t))
       t)))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + pow(t, -1.0)));
	double t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.8) {
		tmp = fma(fma((t - 2.0), t, 1.0), (t * t), 0.5);
	} else {
		tmp = 0.8333333333333334 - ((0.2222222222222222 - (((0.04938271604938271 / t) + 0.037037037037037035) / t)) / t);
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + (t ^ -1.0))))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.8)
		tmp = fma(fma(Float64(t - 2.0), t, 1.0), Float64(t * t), 0.5);
	else
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 - Float64(Float64(Float64(0.04938271604938271 / t) + 0.037037037037037035) / t)) / 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[Power[t, -1.0], $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], 0.8], N[(N[(N[(t - 2.0), $MachinePrecision] * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(0.8333333333333334 - N[(N[(0.2222222222222222 - N[(N[(N[(0.04938271604938271 / t), $MachinePrecision] + 0.037037037037037035), $MachinePrecision] / t), $MachinePrecision]), $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))))))) < 0.80000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if 0.80000000000000004 < (/.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. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right) + \frac{5}{6}\right)} - \frac{2}{9} \cdot \frac{1}{t} \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right) + \left(\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}\right)} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}\right) + \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)} \]

      4. associate--r- N/A

         \[\leadsto \color{blue}{\frac{5}{6} - \left(\frac{2}{9} \cdot \frac{1}{t} - \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{5}{6} - \left(\color{blue}{\frac{\frac{2}{9} \cdot 1}{t}} - \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \frac{5}{6} - \left(\frac{\color{blue}{\frac{2}{9}}}{t} - \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right) \]

      7. cube-mult N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. div-add-rev N/A

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

      12. unpow2 N/A

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

      13. associate-/l/ N/A

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

      14. div-sub N/A

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

   5. Applied rewrites 99.2%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right)} \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{\frac{0.04938271604938271}{t} + 0.037037037037037035}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;\frac{1 + t\_2}{2 + t\_2} \leq 0.83:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right) + \frac{0.037037037037037035}{t \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (pow t -1.0))))) (t_2 (* t_1 t_1)))
   (if (<= (/ (+ 1.0 t_2) (+ 2.0 t_2)) 0.83)
     (fma (fma (- t 2.0) t 1.0) (* t t) 0.5)
     (+
      (- 0.8333333333333334 (/ 0.2222222222222222 t))
      (/ 0.037037037037037035 (* t t))))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + pow(t, -1.0)));
	double t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.83) {
		tmp = fma(fma((t - 2.0), t, 1.0), (t * t), 0.5);
	} else {
		tmp = (0.8333333333333334 - (0.2222222222222222 / t)) + (0.037037037037037035 / (t * t));
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + (t ^ -1.0))))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.83)
		tmp = fma(fma(Float64(t - 2.0), t, 1.0), Float64(t * t), 0.5);
	else
		tmp = Float64(Float64(0.8333333333333334 - Float64(0.2222222222222222 / t)) + Float64(0.037037037037037035 / Float64(t * 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[Power[t, -1.0], $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], 0.83], N[(N[(N[(t - 2.0), $MachinePrecision] * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision] + N[(0.037037037037037035 / N[(t * t), $MachinePrecision]), $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))))))) < 0.82999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if 0.82999999999999996 < (/.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. Add Preprocessing
   3. 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. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. pow2 N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. lift--.f64 N/A

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

      8. fabs-sub N/A

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

      9. unpow1 N/A

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

      10. metadata-eval N/A

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

      11. sqrt-pow1 N/A

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

      12. pow2 N/A

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

      13. rem-sqrt-square-rev N/A

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

      14. lift--.f64 N/A

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

      15. fabs-sub N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. div-sub N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 99.6

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

   7. Applied rewrites 99.6%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035}{t}}{t}} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.6%

      \[\leadsto \left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right) + \color{blue}{\frac{\frac{0.037037037037037035}{t}}{t}} \]
   10. Step-by-step derivation

   11. Applied rewrites 99.6%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right)} \leq 0.83:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right) + \frac{0.037037037037037035}{t \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;\frac{1 + t\_2}{2 + t\_2} \leq 0.83:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (pow t -1.0))))) (t_2 (* t_1 t_1)))
   (if (<= (/ (+ 1.0 t_2) (+ 2.0 t_2)) 0.83)
     (fma (fma (- t 2.0) t 1.0) (* t t) 0.5)
     (-
      0.8333333333333334
      (/ (- 0.2222222222222222 (/ 0.037037037037037035 t)) t)))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + pow(t, -1.0)));
	double t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.83) {
		tmp = fma(fma((t - 2.0), t, 1.0), (t * t), 0.5);
	} else {
		tmp = 0.8333333333333334 - ((0.2222222222222222 - (0.037037037037037035 / t)) / t);
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + (t ^ -1.0))))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.83)
		tmp = fma(fma(Float64(t - 2.0), t, 1.0), Float64(t * t), 0.5);
	else
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 - Float64(0.037037037037037035 / t)) / 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[Power[t, -1.0], $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], 0.83], N[(N[(N[(t - 2.0), $MachinePrecision] * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(0.8333333333333334 - N[(N[(0.2222222222222222 - N[(0.037037037037037035 / t), $MachinePrecision]), $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))))))) < 0.82999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if 0.82999999999999996 < (/.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. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. div-sub N/A

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

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

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

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

      15. *-commutative N/A

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

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

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

      17. distribute-lft-neg-out N/A

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

      18. distribute-rgt-neg-in N/A

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

      19. metadata-eval N/A

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

      20. *-rgt-identity N/A

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

   5. Applied rewrites 99.6%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right)} \leq 0.83:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;\frac{1 + t\_2}{2 + t\_2} \leq 0.83:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (pow t -1.0))))) (t_2 (* t_1 t_1)))
   (if (<= (/ (+ 1.0 t_2) (+ 2.0 t_2)) 0.83)
     (fma (fma (- t 2.0) t 1.0) (* t t) 0.5)
     (- 0.8333333333333334 (/ 0.2222222222222222 t)))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + pow(t, -1.0)));
	double t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.83) {
		tmp = fma(fma((t - 2.0), t, 1.0), (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 + (t ^ -1.0))))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.83)
		tmp = fma(fma(Float64(t - 2.0), t, 1.0), Float64(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[Power[t, -1.0], $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], 0.83], N[(N[(N[(t - 2.0), $MachinePrecision] * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 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))))))) < 0.82999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if 0.82999999999999996 < (/.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. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 99.3

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

   5. Applied rewrites 99.3%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right)} \leq 0.83:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;\frac{1 + t\_2}{2 + t\_2} \leq 0.83:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (pow t -1.0))))) (t_2 (* t_1 t_1)))
   (if (<= (/ (+ 1.0 t_2) (+ 2.0 t_2)) 0.83)
     (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 + pow(t, -1.0)));
	double t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.83) {
		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 + (t ^ -1.0))))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.83)
		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[Power[t, -1.0], $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], 0.83], 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))))))) < 0.82999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 98.4

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

   5. Applied rewrites 98.4%

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

   if 0.82999999999999996 < (/.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. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 99.3

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

   5. Applied rewrites 99.3%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right)} \leq 0.83:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;\frac{1 + t\_2}{2 + t\_2} \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (pow t -1.0))))) (t_2 (* t_1 t_1)))
   (if (<= (/ (+ 1.0 t_2) (+ 2.0 t_2)) 0.8) (fma t t 0.5) 0.8333333333333334)))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + pow(t, -1.0)));
	double t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.8) {
		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 + (t ^ -1.0))))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.8)
		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[Power[t, -1.0], $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], 0.8], 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))))))) < 0.80000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if 0.80000000000000004 < (/.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. Add Preprocessing

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 97.6%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right)} \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;\frac{1 + t\_2}{2 + t\_2} \leq 0.68:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (pow t -1.0))))) (t_2 (* t_1 t_1)))
   (if (<= (/ (+ 1.0 t_2) (+ 2.0 t_2)) 0.68) 0.5 0.8333333333333334)))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + pow(t, -1.0)));
	double t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.68) {
		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 + (t ** (-1.0d0))))
    t_2 = t_1 * t_1
    if (((1.0d0 + t_2) / (2.0d0 + t_2)) <= 0.68d0) 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 + Math.pow(t, -1.0)));
	double t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.68) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	t_1 = 2.0 - ((2.0 / t) / (1.0 + math.pow(t, -1.0)))
	t_2 = t_1 * t_1
	tmp = 0
	if ((1.0 + t_2) / (2.0 + t_2)) <= 0.68:
		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 + (t ^ -1.0))))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.68)
		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 + (t ^ -1.0)));
	t_2 = t_1 * t_1;
	tmp = 0.0;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.68)
		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[Power[t, -1.0], $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], 0.68], 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))))))) < 0.680000000000000049

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 98.7%

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

   if 0.680000000000000049 < (/.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. Add Preprocessing

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 97.6%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + {t}^{-1}}\right)} \leq 0.68:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 100.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]}, N[(N[(1.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $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. Add Preprocessing
3. 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. unpow1 N/A

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

   3. metadata-eval N/A

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

   4. sqrt-pow1 N/A

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

   5. pow2 N/A

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

   6. rem-sqrt-square-rev N/A

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

   7. lift--.f64 N/A

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

   8. fabs-sub N/A

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

   9. unpow1 N/A

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

   10. metadata-eval N/A

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

   11. sqrt-pow1 N/A

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

   12. pow2 N/A

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

   13. rem-sqrt-square-rev N/A

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

   14. lift--.f64 N/A

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

   15. fabs-sub N/A

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

4. Applied rewrites 100.0%

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

6. Applied rewrites 100.0%

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

Alternative 13: 100.0% accurate, 2.0× speedup

Math

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

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. Add Preprocessing
3. 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. unpow1 N/A

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

   3. metadata-eval N/A

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

   4. sqrt-pow1 N/A

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

   5. pow2 N/A

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

   6. rem-sqrt-square-rev N/A

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

   7. lift--.f64 N/A

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

   8. fabs-sub N/A

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

   9. unpow1 N/A

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

   10. metadata-eval N/A

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

   11. sqrt-pow1 N/A

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

   12. pow2 N/A

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

   13. rem-sqrt-square-rev N/A

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

   14. lift--.f64 N/A

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

   15. fabs-sub N/A

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

4. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. sqr-abs-rev N/A

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

   3. lift--.f64 N/A

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

   4. fabs-sub N/A

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

   5. lift--.f64 N/A

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

   6. fabs-sub N/A

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

   7. sqr-abs N/A

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

6. Applied rewrites 100.0%

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

8. Applied rewrites 100.0%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-+.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift--.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. lift--.f64 N/A

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

10. Applied rewrites 100.0%

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

Alternative 14: 59.6% accurate, 184.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (t) :precision binary64 0.5)

C

double code(double t) {
	return 0.5;
}

Fortran

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. Add Preprocessing

3. Taylor expanded in t around 0

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

5. Applied rewrites 62.6%

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

Reproduce

herbie shell --seed 2024347 
(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))))))))