Kahan p13 Example 1

Percentage Accurate: 99.9% → 100.0%

Time: 6.3s

Alternatives: 10

Speedup: 1.2×

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, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{t - -1}\\ \frac{\mathsf{fma}\left(4, t\_1 \cdot t\_1, 1\right)}{\mathsf{fma}\left(t\_1, t\_1 \cdot 4, 2\right)} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ t (- t -1.0))))
   (/ (fma 4.0 (* t_1 t_1) 1.0) (fma t_1 (* t_1 4.0) 2.0))))

C

double code(double t) {
	double t_1 = t / (t - -1.0);
	return fma(4.0, (t_1 * t_1), 1.0) / fma(t_1, (t_1 * 4.0), 2.0);
}

Julia

function code(t)
	t_1 = Float64(t / Float64(t - -1.0))
	return Float64(fma(4.0, Float64(t_1 * t_1), 1.0) / fma(t_1, Float64(t_1 * 4.0), 2.0))
end

Wolfram

code[t_] := Block[{t$95$1 = N[(t / N[(t - -1.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(4.0 * N[(t$95$1 * t$95$1), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$1 * N[(t$95$1 * 4.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.6%

   \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
2. Add Preprocessing
3. 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. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/l* N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/l* N/A

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

   10. swap-sqr N/A

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

   11. lower-fma.f64 N/A

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

4. Applied rewrites 99.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

6. Applied rewrites 100.0%

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

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 \cdot t}{1 + t}\\ t_2 := t\_1 \cdot t\_1\\ t_3 := \frac{t}{t - -1}\\ \mathbf{if}\;\frac{1 + t\_2}{2 + t\_2} \leq 0.8:\\ \;\;\;\;\frac{\mathsf{fma}\left(4, \frac{\left(-t\right) \cdot t}{\left(-1 - t\right) \cdot \left(t - -1\right)}, 1\right)}{\mathsf{fma}\left(t\_3, t\_3 \cdot 4, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{\frac{0.04938271604938271}{t} - -0.037037037037037035}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* 2.0 t) (+ 1.0 t))) (t_2 (* t_1 t_1)) (t_3 (/ t (- t -1.0))))
   (if (<= (/ (+ 1.0 t_2) (+ 2.0 t_2)) 0.8)
     (/
      (fma 4.0 (/ (* (- t) t) (* (- -1.0 t) (- t -1.0))) 1.0)
      (fma t_3 (* t_3 4.0) 2.0))
     (-
      0.8333333333333334
      (/
       (-
        0.2222222222222222
        (/ (- (/ 0.04938271604938271 t) -0.037037037037037035) t))
       t)))))

C

double code(double t) {
	double t_1 = (2.0 * t) / (1.0 + t);
	double t_2 = t_1 * t_1;
	double t_3 = t / (t - -1.0);
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.8) {
		tmp = fma(4.0, ((-t * t) / ((-1.0 - t) * (t - -1.0))), 1.0) / fma(t_3, (t_3 * 4.0), 2.0);
	} else {
		tmp = 0.8333333333333334 - ((0.2222222222222222 - (((0.04938271604938271 / t) - -0.037037037037037035) / t)) / 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)
	t_3 = Float64(t / Float64(t - -1.0))
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.8)
		tmp = Float64(fma(4.0, Float64(Float64(Float64(-t) * t) / Float64(Float64(-1.0 - t) * Float64(t - -1.0))), 1.0) / fma(t_3, Float64(t_3 * 4.0), 2.0));
	else
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 - Float64(Float64(Float64(0.04938271604938271 / t) - -0.037037037037037035) / t)) / t));
	end
	return tmp
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t / N[(t - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], 0.8], N[(N[(4.0 * N[(N[((-t) * t), $MachinePrecision] / N[(N[(-1.0 - t), $MachinePrecision] * N[(t - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$3 * N[(t$95$3 * 4.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 - N[(N[(0.2222222222222222 - N[(N[(N[(0.04938271604938271 / t), $MachinePrecision] - -0.037037037037037035), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 #s(literal 1 binary64) (*.f64 (/.f64 (*.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))))) < 0.80000000000000004

   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. Add Preprocessing
   3. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. swap-sqr N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   6. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

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

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

      14. metadata-eval N/A

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. lower-+.f64 N/A

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

      18. lower-neg.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in t around 0

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

       1. mul-1-neg N/A

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

       2. lft-mult-inverse N/A

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

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

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

       4. *-lft-identity N/A

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

       5. metadata-eval N/A

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

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

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

       7. metadata-eval N/A

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

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

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

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

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

       10. distribute-rgt-in N/A

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

       11. +-commutative N/A

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

       12. distribute-rgt-in N/A

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

       13. metadata-eval N/A

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

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

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

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

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

       16. lft-mult-inverse N/A

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

       17. metadata-eval N/A

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

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

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

       19. mul-1-neg N/A

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

       20. remove-double-neg N/A

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

       21. lower--.f64 100.0

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

   11. Applied rewrites 100.0%

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

   if 0.80000000000000004 < (/.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 99.2%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 98.9%

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

Alternative 3: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 \cdot t}{1 + t}\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;\frac{1 + t\_2}{2 + t\_2} \leq 0.6:\\ \;\;\;\;\frac{\mathsf{fma}\left(4, \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, t, 3\right), t, -2\right), t, 1\right) \cdot t\right) \cdot t, 1\right)}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{\frac{0.04938271604938271}{t} - -0.037037037037037035}{t}}{t}\\ \end{array} \end{array} \]

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 4.0 (* (* (fma (fma (fma -4.0 t 3.0) t -2.0) t 1.0) t) t) 1.0)
      (+ 2.0 (* (* (fma (fma (fma -16.0 t 12.0) t -8.0) t 4.0) t) t)))
     (-
      0.8333333333333334
      (/
       (-
        0.2222222222222222
        (/ (- (/ 0.04938271604938271 t) -0.037037037037037035) t))
       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(4.0, ((fma(fma(fma(-4.0, t, 3.0), t, -2.0), t, 1.0) * t) * t), 1.0) / (2.0 + ((fma(fma(fma(-16.0, t, 12.0), t, -8.0), t, 4.0) * t) * t));
	} else {
		tmp = 0.8333333333333334 - ((0.2222222222222222 - (((0.04938271604938271 / t) - -0.037037037037037035) / t)) / 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 = Float64(fma(4.0, Float64(Float64(fma(fma(fma(-4.0, t, 3.0), t, -2.0), t, 1.0) * t) * t), 1.0) / Float64(2.0 + Float64(Float64(fma(fma(fma(-16.0, t, 12.0), t, -8.0), t, 4.0) * t) * t)));
	else
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 - Float64(Float64(Float64(0.04938271604938271 / t) - -0.037037037037037035) / t)) / t));
	end
	return tmp
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(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[(4.0 * N[(N[(N[(N[(N[(-4.0 * t + 3.0), $MachinePrecision] * t + -2.0), $MachinePrecision] * t + 1.0), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision] + 1.0), $MachinePrecision] / N[(2.0 + N[(N[(N[(N[(N[(-16.0 * t + 12.0), $MachinePrecision] * t + -8.0), $MachinePrecision] * t + 4.0), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 - N[(N[(0.2222222222222222 - N[(N[(N[(0.04938271604938271 / t), $MachinePrecision] - -0.037037037037037035), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 #s(literal 1 binary64) (*.f64 (/.f64 (*.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. Add Preprocessing
   3. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. swap-sqr N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. metadata-eval N/A

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

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

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in t around 0

      \[\leadsto \frac{\mathsf{fma}\left(4, \color{blue}{{t}^{2} \cdot \left(1 + t \cdot \left(t \cdot \left(3 + -4 \cdot t\right) - 2\right)\right)}, 1\right)}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(4, \color{blue}{\left(1 + t \cdot \left(t \cdot \left(3 + -4 \cdot t\right) - 2\right)\right) \cdot {t}^{2}}, 1\right)}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t} \]

      2. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(4, \left(1 + t \cdot \left(t \cdot \left(3 + -4 \cdot t\right) - 2\right)\right) \cdot \color{blue}{\left(t \cdot t\right)}, 1\right)}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(4, \color{blue}{\left(\left(1 + t \cdot \left(t \cdot \left(3 + -4 \cdot t\right) - 2\right)\right) \cdot t\right) \cdot t}, 1\right)}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(4, \color{blue}{\left(\left(1 + t \cdot \left(t \cdot \left(3 + -4 \cdot t\right) - 2\right)\right) \cdot t\right) \cdot t}, 1\right)}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(4, \color{blue}{\left(\left(1 + t \cdot \left(t \cdot \left(3 + -4 \cdot t\right) - 2\right)\right) \cdot t\right)} \cdot t, 1\right)}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(4, \left(\color{blue}{\left(t \cdot \left(t \cdot \left(3 + -4 \cdot t\right) - 2\right) + 1\right)} \cdot t\right) \cdot t, 1\right)}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(4, \left(\left(\color{blue}{\left(t \cdot \left(3 + -4 \cdot t\right) - 2\right) \cdot t} + 1\right) \cdot t\right) \cdot t, 1\right)}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(4, \left(\color{blue}{\mathsf{fma}\left(t \cdot \left(3 + -4 \cdot t\right) - 2, t, 1\right)} \cdot t\right) \cdot t, 1\right)}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(4, \left(\mathsf{fma}\left(t \cdot \left(3 + -4 \cdot t\right) - \color{blue}{2 \cdot 1}, t, 1\right) \cdot t\right) \cdot t, 1\right)}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(4, \left(\mathsf{fma}\left(\color{blue}{t \cdot \left(3 + -4 \cdot t\right) + \left(\mathsf{neg}\left(2\right)\right) \cdot 1}, t, 1\right) \cdot t\right) \cdot t, 1\right)}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t} \]

      11. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(4, \left(\mathsf{fma}\left(\color{blue}{\left(3 + -4 \cdot t\right) \cdot t} + \left(\mathsf{neg}\left(2\right)\right) \cdot 1, t, 1\right) \cdot t\right) \cdot t, 1\right)}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(4, \left(\mathsf{fma}\left(\left(3 + -4 \cdot t\right) \cdot t + \color{blue}{-2} \cdot 1, t, 1\right) \cdot t\right) \cdot t, 1\right)}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(4, \left(\mathsf{fma}\left(\left(3 + -4 \cdot t\right) \cdot t + \color{blue}{-2}, t, 1\right) \cdot t\right) \cdot t, 1\right)}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(4, \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(3 + -4 \cdot t, t, -2\right)}, t, 1\right) \cdot t\right) \cdot t, 1\right)}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t} \]

      15. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(4, \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-4 \cdot t + 3}, t, -2\right), t, 1\right) \cdot t\right) \cdot t, 1\right)}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t} \]

      16. lower-fma.f64 99.6

          \[\leadsto \frac{\mathsf{fma}\left(4, \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-4, t, 3\right)}, t, -2\right), t, 1\right) \cdot t\right) \cdot t, 1\right)}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t} \]

   10. Applied rewrites 99.6%

       \[\leadsto \frac{\mathsf{fma}\left(4, \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, t, 3\right), t, -2\right), t, 1\right) \cdot t\right) \cdot t}, 1\right)}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t} \]

   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 99.2%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 98.4%

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

Alternative 4: 99.5% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* 2.0 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
        (/ (- (/ 0.04938271604938271 t) -0.037037037037037035) t))
       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 - (((0.04938271604938271 / t) - -0.037037037037037035) / t)) / 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(Float64(0.2222222222222222 - Float64(Float64(Float64(0.04938271604938271 / t) - -0.037037037037037035) / t)) / t));
	end
	return tmp
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(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[(N[(0.2222222222222222 - N[(N[(N[(0.04938271604938271 / t), $MachinePrecision] - -0.037037037037037035), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 #s(literal 1 binary64) (*.f64 (/.f64 (*.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. Add Preprocessing
   3. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. swap-sqr N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.5

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

   7. 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 99.2%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 98.4%

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

Alternative 5: 99.4% accurate, 0.8× speedup

Math

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

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 (/ 0.037037037037037035 t)) 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 - (0.037037037037037035 / t)) / 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(Float64(0.2222222222222222 - Float64(0.037037037037037035 / t)) / 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[(N[(0.2222222222222222 - N[(0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 #s(literal 1 binary64) (*.f64 (/.f64 (*.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. Add Preprocessing
   3. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. swap-sqr N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.5

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

   7. 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 99.2%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. div-sub N/A

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

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

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

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

      15. *-commutative N/A

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

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

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

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

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

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

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

      19. metadata-eval N/A

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

      20. *-rgt-identity N/A

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

   5. Applied rewrites 98.2%

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

Alternative 6: 99.3% 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. Add Preprocessing
   3. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. swap-sqr N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.5

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

   7. 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 99.2%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 98.0

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

   5. Applied rewrites 98.0%

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

Alternative 7: 99.2% accurate, 0.8× speedup

Math

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

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. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 99.4

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

   5. Applied rewrites 99.4%

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

   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 99.2%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 98.0

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

   5. Applied rewrites 98.0%

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

Alternative 8: 98.7% 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. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 99.4

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

   5. Applied rewrites 99.4%

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

   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 99.2%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 96.7%

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

Alternative 9: 98.5% accurate, 0.9× speedup

Math

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

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. Add Preprocessing

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 99.1%

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

   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 99.2%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 96.7%

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

Alternative 10: 59.5% 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 99.6%

   \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

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

5. Applied rewrites 60.6%

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

Reproduce

herbie shell --seed 2025010 
(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))))))