Kahan p13 Example 1

Percentage Accurate: 99.9% → 100.0%

Time: 3.2s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* 2.0 t) (+ 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 * t) / (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 * t) / (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 * t) / (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 * t) / (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(Float64(2.0 * t) / 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 * t) / (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[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $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: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* 2.0 t) (+ 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 * t) / (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 * t) / (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 * t) / (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 * t) / (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(Float64(2.0 * t) / 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 * t) / (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[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $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, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \frac{t}{t + 1}\\ t_2 := \frac{t}{1 + t} \cdot 2\\ \frac{\mathsf{fma}\left(t\_1, t\_1, 1\right)}{\mathsf{fma}\left(t\_2, t\_2, 2\right)} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. associate-/l* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. associate-/l* N/A

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

   17. lower-*.f64 N/A

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

   18. lower-/.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 99.9

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

3. Applied rewrites 99.9%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. +-commutative N/A

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

   10. pow2 N/A

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

   11. +-commutative N/A

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

   12. associate-*r/ N/A

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

   13. pow2 N/A

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

   14. lower-fma.f64 N/A

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

5. Applied rewrites 100.0%

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

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \frac{t}{t + 1}\\ t_2 := \frac{2 \cdot t}{1 + t}\\ t_3 := t\_2 \cdot t\_2\\ \mathbf{if}\;\frac{1 + t\_3}{2 + t\_3} \leq 0.6:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, t\_1, 1\right)}{2 + \mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right) \cdot t - 8, t, 4\right) \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.037037037037037035, t, 0.04938271604938271\right)}{t \cdot t}, -1, 0.2222222222222222\right)}{t}, -1, 0.8333333333333334\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (* 2.0 (/ t (+ t 1.0))))
        (t_2 (/ (* 2.0 t) (+ 1.0 t)))
        (t_3 (* t_2 t_2)))
   (if (<= (/ (+ 1.0 t_3) (+ 2.0 t_3)) 0.6)
     (/
      (fma t_1 t_1 1.0)
      (+ 2.0 (* (fma (- (* (fma -16.0 t 12.0) t) 8.0) t 4.0) (* t t))))
     (fma
      (/
       (fma
        (/ (fma 0.037037037037037035 t 0.04938271604938271) (* t t))
        -1.0
        0.2222222222222222)
       t)
      -1.0
      0.8333333333333334))))

C

double code(double t) {
	double t_1 = 2.0 * (t / (t + 1.0));
	double t_2 = (2.0 * t) / (1.0 + t);
	double t_3 = t_2 * t_2;
	double tmp;
	if (((1.0 + t_3) / (2.0 + t_3)) <= 0.6) {
		tmp = fma(t_1, t_1, 1.0) / (2.0 + (fma(((fma(-16.0, t, 12.0) * t) - 8.0), t, 4.0) * (t * t)));
	} else {
		tmp = fma((fma((fma(0.037037037037037035, t, 0.04938271604938271) / (t * t)), -1.0, 0.2222222222222222) / t), -1.0, 0.8333333333333334);
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(2.0 * Float64(t / Float64(t + 1.0)))
	t_2 = Float64(Float64(2.0 * t) / Float64(1.0 + t))
	t_3 = Float64(t_2 * t_2)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_3) / Float64(2.0 + t_3)) <= 0.6)
		tmp = Float64(fma(t_1, t_1, 1.0) / Float64(2.0 + Float64(fma(Float64(Float64(fma(-16.0, t, 12.0) * t) - 8.0), t, 4.0) * Float64(t * t))));
	else
		tmp = fma(Float64(fma(Float64(fma(0.037037037037037035, t, 0.04938271604938271) / Float64(t * t)), -1.0, 0.2222222222222222) / t), -1.0, 0.8333333333333334);
	end
	return tmp
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 * N[(t / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $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], 0.6], N[(N[(t$95$1 * t$95$1 + 1.0), $MachinePrecision] / N[(2.0 + N[(N[(N[(N[(N[(-16.0 * t + 12.0), $MachinePrecision] * t), $MachinePrecision] - 8.0), $MachinePrecision] * t + 4.0), $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.037037037037037035 * t + 0.04938271604938271), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision] * -1.0 + 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision] * -1.0 + 0.8333333333333334), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. associate-/l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.7

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

   6. Applied rewrites 99.7%

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

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

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

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

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

      10. flip-- N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in t around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.3

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

   9. Applied rewrites 99.3%

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

Alternative 3: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 \cdot t}{1 + t}\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;\frac{1 + t\_2}{2 + t\_2} \leq 0.6:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{t}{1 + t} \cdot 2, \mathsf{fma}\left(t \cdot 2 - 2, t, 2\right) \cdot t, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.037037037037037035, t, 0.04938271604938271\right)}{t \cdot t}, -1, 0.2222222222222222\right)}{t}, -1, 0.8333333333333334\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* 2.0 t) (+ 1.0 t))) (t_2 (* t_1 t_1)))
   (if (<= (/ (+ 1.0 t_2) (+ 2.0 t_2)) 0.6)
     (/
      (fma (* (/ t (+ 1.0 t)) 2.0) (* (fma (- (* t 2.0) 2.0) t 2.0) t) 1.0)
      (fma (fma (- (* 12.0 t) 8.0) t 4.0) (* t t) 2.0))
     (fma
      (/
       (fma
        (/ (fma 0.037037037037037035 t 0.04938271604938271) (* t t))
        -1.0
        0.2222222222222222)
       t)
      -1.0
      0.8333333333333334))))

C

double code(double t) {
	double t_1 = (2.0 * t) / (1.0 + t);
	double t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.6) {
		tmp = fma(((t / (1.0 + t)) * 2.0), (fma(((t * 2.0) - 2.0), t, 2.0) * t), 1.0) / fma(fma(((12.0 * t) - 8.0), t, 4.0), (t * t), 2.0);
	} else {
		tmp = fma((fma((fma(0.037037037037037035, t, 0.04938271604938271) / (t * t)), -1.0, 0.2222222222222222) / t), -1.0, 0.8333333333333334);
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(Float64(2.0 * t) / Float64(1.0 + t))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.6)
		tmp = Float64(fma(Float64(Float64(t / Float64(1.0 + t)) * 2.0), Float64(fma(Float64(Float64(t * 2.0) - 2.0), t, 2.0) * t), 1.0) / fma(fma(Float64(Float64(12.0 * t) - 8.0), t, 4.0), Float64(t * t), 2.0));
	else
		tmp = fma(Float64(fma(Float64(fma(0.037037037037037035, t, 0.04938271604938271) / Float64(t * t)), -1.0, 0.2222222222222222) / t), -1.0, 0.8333333333333334);
	end
	return tmp
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], 0.6], N[(N[(N[(N[(t / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(N[(N[(t * 2.0), $MachinePrecision] - 2.0), $MachinePrecision] * t + 2.0), $MachinePrecision] * t), $MachinePrecision] + 1.0), $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[(N[(N[(N[(N[(0.037037037037037035 * t + 0.04938271604938271), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision] * -1.0 + 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision] * -1.0 + 0.8333333333333334), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.6

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 99.6%

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

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

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

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

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

      10. flip-- N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in t around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.3

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

   9. Applied rewrites 99.3%

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

Alternative 4: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 \cdot t}{1 + t}\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;\frac{1 + t\_2}{2 + t\_2} \leq 0.6:\\ \;\;\;\;\frac{1 + \mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right) \cdot t - 8, t, 4\right) \cdot \left(t \cdot t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(12 \cdot t - 8, t, 4\right), t \cdot t, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.037037037037037035, t, 0.04938271604938271\right)}{t \cdot t}, -1, 0.2222222222222222\right)}{t}, -1, 0.8333333333333334\right)\\ \end{array} \end{array} \]

FPCore

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

C

double code(double t) {
	double t_1 = (2.0 * t) / (1.0 + t);
	double t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.6) {
		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 = fma((fma((fma(0.037037037037037035, t, 0.04938271604938271) / (t * t)), -1.0, 0.2222222222222222) / t), -1.0, 0.8333333333333334);
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(Float64(2.0 * t) / Float64(1.0 + t))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.6)
		tmp = Float64(Float64(1.0 + Float64(fma(Float64(Float64(fma(-16.0, t, 12.0) * t) - 8.0), t, 4.0) * Float64(t * t))) / fma(fma(Float64(Float64(12.0 * t) - 8.0), t, 4.0), Float64(t * t), 2.0));
	else
		tmp = fma(Float64(fma(Float64(fma(0.037037037037037035, t, 0.04938271604938271) / Float64(t * t)), -1.0, 0.2222222222222222) / t), -1.0, 0.8333333333333334);
	end
	return tmp
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], 0.6], N[(N[(1.0 + N[(N[(N[(N[(N[(-16.0 * t + 12.0), $MachinePrecision] * t), $MachinePrecision] - 8.0), $MachinePrecision] * t + 4.0), $MachinePrecision] * N[(t * t), $MachinePrecision]), $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[(N[(N[(N[(N[(0.037037037037037035 * t + 0.04938271604938271), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision] * -1.0 + 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision] * -1.0 + 0.8333333333333334), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 99.6

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

   7. Applied rewrites 99.6%

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

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

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

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

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

      10. flip-- N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in t around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.3

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

   9. Applied rewrites 99.3%

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

Alternative 5: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(12 \cdot t - 8, t, 4\right)\\ t_2 := \frac{2 \cdot t}{1 + t}\\ t_3 := t\_2 \cdot t\_2\\ \mathbf{if}\;\frac{1 + t\_3}{2 + t\_3} \leq 0.6:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1 \cdot t, t, 1\right)}{\mathsf{fma}\left(t\_1, t \cdot t, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.037037037037037035, t, 0.04938271604938271\right)}{t \cdot t}, -1, 0.2222222222222222\right)}{t}, -1, 0.8333333333333334\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (fma (- (* 12.0 t) 8.0) t 4.0))
        (t_2 (/ (* 2.0 t) (+ 1.0 t)))
        (t_3 (* t_2 t_2)))
   (if (<= (/ (+ 1.0 t_3) (+ 2.0 t_3)) 0.6)
     (/ (fma (* t_1 t) t 1.0) (fma t_1 (* t t) 2.0))
     (fma
      (/
       (fma
        (/ (fma 0.037037037037037035 t 0.04938271604938271) (* t t))
        -1.0
        0.2222222222222222)
       t)
      -1.0
      0.8333333333333334))))

C

double code(double t) {
	double t_1 = fma(((12.0 * t) - 8.0), t, 4.0);
	double t_2 = (2.0 * t) / (1.0 + t);
	double t_3 = t_2 * t_2;
	double tmp;
	if (((1.0 + t_3) / (2.0 + t_3)) <= 0.6) {
		tmp = fma((t_1 * t), t, 1.0) / fma(t_1, (t * t), 2.0);
	} else {
		tmp = fma((fma((fma(0.037037037037037035, t, 0.04938271604938271) / (t * t)), -1.0, 0.2222222222222222) / t), -1.0, 0.8333333333333334);
	}
	return tmp;
}

Julia

function code(t)
	t_1 = fma(Float64(Float64(12.0 * t) - 8.0), t, 4.0)
	t_2 = Float64(Float64(2.0 * t) / Float64(1.0 + t))
	t_3 = Float64(t_2 * t_2)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_3) / Float64(2.0 + t_3)) <= 0.6)
		tmp = Float64(fma(Float64(t_1 * t), t, 1.0) / fma(t_1, Float64(t * t), 2.0));
	else
		tmp = fma(Float64(fma(Float64(fma(0.037037037037037035, t, 0.04938271604938271) / Float64(t * t)), -1.0, 0.2222222222222222) / t), -1.0, 0.8333333333333334);
	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[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $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], 0.6], N[(N[(N[(t$95$1 * t), $MachinePrecision] * t + 1.0), $MachinePrecision] / N[(t$95$1 * N[(t * t), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.037037037037037035 * t + 0.04938271604938271), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision] * -1.0 + 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision] * -1.0 + 0.8333333333333334), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-fma.f64 99.6

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

   7. Applied rewrites 99.6%

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

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

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

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

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

      10. flip-- N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in t around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.3

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

   9. Applied rewrites 99.3%

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

Alternative 6: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double t) {
	double t_1 = (2.0 * t) / (1.0 + t);
	double t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.6) {
		tmp = fma(fma((t - 2.0), t, 1.0), (t * t), 0.5);
	} else {
		tmp = fma((fma((fma(0.037037037037037035, t, 0.04938271604938271) / (t * t)), -1.0, 0.2222222222222222) / t), -1.0, 0.8333333333333334);
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(Float64(2.0 * t) / Float64(1.0 + t))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.6)
		tmp = fma(fma(Float64(t - 2.0), t, 1.0), Float64(t * t), 0.5);
	else
		tmp = fma(Float64(fma(Float64(fma(0.037037037037037035, t, 0.04938271604938271) / Float64(t * t)), -1.0, 0.2222222222222222) / t), -1.0, 0.8333333333333334);
	end
	return tmp
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], 0.6], N[(N[(N[(t - 2.0), $MachinePrecision] * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(N[(N[(0.037037037037037035 * t + 0.04938271604938271), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision] * -1.0 + 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision] * -1.0 + 0.8333333333333334), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

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

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

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

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

      10. flip-- N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in t around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.3

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

   9. Applied rewrites 99.3%

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

Alternative 7: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(t)
	t_1 = Float64(Float64(2.0 * t) / Float64(1.0 + t))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.6)
		tmp = fma(fma(Float64(t - 2.0), t, 1.0), Float64(t * t), 0.5);
	else
		tmp = fma(Float64(Float64(Float64(0.2222222222222222 * t) - 0.037037037037037035) / Float64(t * t)), -1.0, 0.8333333333333334);
	end
	return tmp
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], 0.6], N[(N[(N[(t - 2.0), $MachinePrecision] * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(N[(0.2222222222222222 * t), $MachinePrecision] - 0.037037037037037035), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision] * -1.0 + 0.8333333333333334), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

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

   1. Initial program 99.8%

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

   2. Taylor expanded in t around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 99.1

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

   7. Applied rewrites 99.1%

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

Alternative 8: 99.2% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* 2.0 t) (+ 1.0 t))) (t_2 (* t_1 t_1)))
   (if (<= (/ (+ 1.0 t_2) (+ 2.0 t_2)) 0.6)
     (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 * t) / (1.0 + t);
	double t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.6) {
		tmp = fma(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(Float64(2.0 * t) / Float64(1.0 + t))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.6)
		tmp = fma(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[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], 0.6], N[(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 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)))) (+.f64 #s(literal 2 binary64) (*.f64 (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t))))) < 0.599999999999999978

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

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

   1. Initial program 99.8%

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

   2. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 98.8

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

   4. Applied rewrites 98.8%

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

Alternative 9: 99.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double t) {
	double t_1 = (2.0 * t) / (1.0 + t);
	double t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.6) {
		tmp = fma(fma(-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(Float64(2.0 * t) / Float64(1.0 + t))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.6)
		tmp = fma(fma(-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[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], 0.6], N[(N[(-2.0 * 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 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)))) (+.f64 #s(literal 2 binary64) (*.f64 (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t))))) < 0.599999999999999978

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

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

   1. Initial program 99.8%

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

   2. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 98.8

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

   4. Applied rewrites 98.8%

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

Alternative 10: 99.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(t)
	t_1 = Float64(Float64(2.0 * t) / Float64(1.0 + t))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.6)
		tmp = fma(t, t, 0.5);
	else
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	end
	return tmp
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], 0.6], N[(t * t + 0.5), $MachinePrecision], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 99.3

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

   4. Applied rewrites 99.3%

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

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

   1. Initial program 99.8%

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

   2. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 98.8

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

   4. Applied rewrites 98.8%

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

Alternative 11: 98.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(t)
	t_1 = Float64(Float64(2.0 * t) / Float64(1.0 + t))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.6)
		tmp = fma(t, t, 0.5);
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], 0.6], N[(t * t + 0.5), $MachinePrecision], 0.8333333333333334]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 99.3

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

   4. Applied rewrites 99.3%

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

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

   1. Initial program 99.8%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 97.8%

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

Alternative 12: 98.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 \cdot t}{1 + t}\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;\frac{1 + t\_2}{2 + t\_2} \leq 0.66:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* 2.0 t) (+ 1.0 t))) (t_2 (* t_1 t_1)))
   (if (<= (/ (+ 1.0 t_2) (+ 2.0 t_2)) 0.66) 0.5 0.8333333333333334)))

C

double code(double t) {
	double t_1 = (2.0 * t) / (1.0 + t);
	double t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.66) {
		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 * t) / (1.0d0 + t)
    t_2 = t_1 * t_1
    if (((1.0d0 + t_2) / (2.0d0 + t_2)) <= 0.66d0) 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 * t) / (1.0 + t);
	double t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.66) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	t_1 = (2.0 * t) / (1.0 + t)
	t_2 = t_1 * t_1
	tmp = 0
	if ((1.0 + t_2) / (2.0 + t_2)) <= 0.66:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	t_1 = Float64(Float64(2.0 * t) / Float64(1.0 + t))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.66)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = (2.0 * t) / (1.0 + t);
	t_2 = t_1 * t_1;
	tmp = 0.0;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.66)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $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], 0.66], 0.5, 0.8333333333333334]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 #s(literal 1 binary64) (*.f64 (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)))) (+.f64 #s(literal 2 binary64) (*.f64 (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t))))) < 0.660000000000000031

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 98.9%

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

   if 0.660000000000000031 < (/.f64 (+.f64 #s(literal 1 binary64) (*.f64 (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)))) (+.f64 #s(literal 2 binary64) (*.f64 (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)))))

   1. Initial program 99.8%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 97.8%

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

Alternative 13: 59.4% accurate, 104.0× 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 99.9%

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

2. Taylor expanded in t around 0

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

4. Applied rewrites 59.4%

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

Reproduce

herbie shell --seed 2025093 
(FPCore (t)
  :name "Kahan p13 Example 1"
  :precision binary64
  (/ (+ 1.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t)))) (+ 2.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t))))))