Kahan p13 Example 2

Percentage Accurate: 100.0% → 100.0%

Time: 4.2s

Alternatives: 11

Speedup: 1.6×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    t_1 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))
    t_2 = t_1 * t_1
    code = (1.0d0 + t_2) / (2.0d0 + t_2)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    t_1 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))
    t_2 = t_1 * t_1
    code = (1.0d0 + t_2) / (2.0d0 + t_2)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lower-/.f64 N/A

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

3. Applied rewrites 100.0%

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

   1. lift--.f64 N/A

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

   2. sub-negate-rev N/A

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

   3. lift-/.f64 N/A

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

   4. sub-to-fraction N/A

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

   5. distribute-neg-frac2 N/A

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

   6. lower-/.f64 N/A

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

   7. metadata-eval N/A

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

   8. add-flip-rev N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lift--.f64 N/A

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

   12. sub-negate-rev N/A

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

   13. lower--.f64 100.0

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

5. Applied rewrites 100.0%

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

   1. lift--.f64 N/A

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

   2. sub-negate-rev N/A

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

   3. lift-/.f64 N/A

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

   4. sub-to-fraction N/A

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

   5. distribute-neg-frac2 N/A

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

   6. lower-/.f64 N/A

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

   7. metadata-eval N/A

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

   8. add-flip-rev N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lift--.f64 N/A

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

   12. sub-negate-rev N/A

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

   13. lower--.f64 100.0

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in t around 0

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

   1. lower-*.f64 100.0

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

10. Applied rewrites 100.0%

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

11. Taylor expanded in t around 0

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

    1. lower-*.f64 100.0

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

13. Applied rewrites 100.0%

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

15. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{t - -1}\\ t_2 := t\_1 - 2\\ t_3 := 2 - t\_1\\ \frac{\mathsf{fma}\left(t\_2, t\_3, -1\right)}{\mathsf{fma}\left(t\_2, t\_3, -2\right)} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ 2.0 (- t -1.0))) (t_2 (- t_1 2.0)) (t_3 (- 2.0 t_1)))
   (/ (fma t_2 t_3 -1.0) (fma t_2 t_3 -2.0))))

C

double code(double t) {
	double t_1 = 2.0 / (t - -1.0);
	double t_2 = t_1 - 2.0;
	double t_3 = 2.0 - t_1;
	return fma(t_2, t_3, -1.0) / fma(t_2, t_3, -2.0);
}

Julia

function code(t)
	t_1 = Float64(2.0 / Float64(t - -1.0))
	t_2 = Float64(t_1 - 2.0)
	t_3 = Float64(2.0 - t_1)
	return Float64(fma(t_2, t_3, -1.0) / fma(t_2, t_3, -2.0))
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 / N[(t - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 - t$95$1), $MachinePrecision]}, N[(N[(t$95$2 * t$95$3 + -1.0), $MachinePrecision] / N[(t$95$2 * t$95$3 + -2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 100.0%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lower-/.f64 N/A

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

3. Applied rewrites 100.0%

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

Alternative 3: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))
    t_2 = t_1 * t_1
    if (((1.0d0 + t_2) / (2.0d0 + t_2)) <= 0.6d0) then
        tmp = 0.5d0 + ((t ** 2.0d0) * (1.0d0 + ((-2.0d0) * t)))
    else
        tmp = (((((0.04938271604938271d0 / t) - (-0.037037037037037035d0)) / t) - 0.2222222222222222d0) / t) - (-0.8333333333333334d0)
    end if
    code = tmp
end function

Java

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

Python

def code(t):
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))
	t_2 = t_1 * t_1
	tmp = 0
	if ((1.0 + t_2) / (2.0 + t_2)) <= 0.6:
		tmp = 0.5 + (math.pow(t, 2.0) * (1.0 + (-2.0 * t)))
	else:
		tmp = (((((0.04938271604938271 / t) - -0.037037037037037035) / t) - 0.2222222222222222) / t) - -0.8333333333333334
	return tmp

Julia

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

MATLAB

function tmp_2 = code(t)
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	t_2 = t_1 * t_1;
	tmp = 0.0;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.6)
		tmp = 0.5 + ((t ^ 2.0) * (1.0 + (-2.0 * t)));
	else
		tmp = (((((0.04938271604938271 / t) - -0.037037037037037035) / t) - 0.2222222222222222) / t) - -0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 50.9

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

   4. Applied rewrites 50.9%

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

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

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

   3. Applied rewrites 100.0%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-/.f64 N/A

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

      4. sub-to-fraction N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. add-flip-rev N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-/.f64 N/A

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

      4. sub-to-fraction N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. add-flip-rev N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower--.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in t around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 50.9

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

   10. Applied rewrites 50.9%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. add-flip N/A

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

       4. lower--.f64 N/A

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

   12. Applied rewrites 50.9%

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

Alternative 4: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;\frac{1 + t\_2}{2 + t\_2} \leq 0.6:\\ \;\;\;\;\frac{-0.25}{\mathsf{fma}\left(t, t, -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.04938271604938271}{t} - -0.037037037037037035}{t} - 0.2222222222222222}{t} - -0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))) (t_2 (* t_1 t_1)))
   (if (<= (/ (+ 1.0 t_2) (+ 2.0 t_2)) 0.6)
     (/ -0.25 (fma t t -0.5))
     (-
      (/
       (-
        (/ (- (/ 0.04938271604938271 t) -0.037037037037037035) t)
        0.2222222222222222)
       t)
      -0.8333333333333334))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.6) {
		tmp = -0.25 / fma(t, t, -0.5);
	} else {
		tmp = (((((0.04938271604938271 / t) - -0.037037037037037035) / t) - 0.2222222222222222) / t) - -0.8333333333333334;
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.6)
		tmp = Float64(-0.25 / fma(t, t, -0.5));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.04938271604938271 / t) - -0.037037037037037035) / t) - 0.2222222222222222) / t) - -0.8333333333333334);
	end
	return tmp
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], 0.6], N[(-0.25 / N[(t * t + -0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.04938271604938271 / t), $MachinePrecision] - -0.037037037037037035), $MachinePrecision] / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision] - -0.8333333333333334), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-pow.f64 51.8

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

   4. Applied rewrites 51.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lift-pow.f64 N/A

         \[\leadsto \frac{{t}^{2} \cdot {t}^{2} - \frac{1}{2} \cdot \frac{1}{2}}{{t}^{2} - \frac{1}{2}} \]

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow3 N/A

          \[\leadsto \frac{{t}^{3} \cdot t - \frac{1}{2} \cdot \frac{1}{2}}{{t}^{2} - \frac{1}{2}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{{t}^{3} \cdot t - \frac{1}{2} \cdot \frac{1}{2}}{{\color{blue}{t}}^{2} - \frac{1}{2}} \]

      13. unpow3 N/A

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

      14. unpow2 N/A

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

      15. lift-pow.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-pow.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 50.9

          \[\leadsto \frac{\left(\left(t \cdot t\right) \cdot t\right) \cdot t - 0.25}{{t}^{2} - \color{blue}{0.5}} \]

      22. lift-pow.f64 N/A

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

      23. unpow2 N/A

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

      24. lower-*.f64 50.9

          \[\leadsto \frac{\left(\left(t \cdot t\right) \cdot t\right) \cdot t - 0.25}{t \cdot t - 0.5} \]

   6. Applied rewrites 50.9%

      \[\leadsto \frac{\left(\left(t \cdot t\right) \cdot t\right) \cdot t - 0.25}{\color{blue}{t \cdot t - 0.5}} \]

   7. Taylor expanded in t around 0

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

   9. Applied rewrites 51.0%

      \[\leadsto \frac{-0.25}{\color{blue}{t \cdot t} - 0.5} \]

   11. Applied rewrites 51.0%

       \[\leadsto \color{blue}{\frac{-0.25}{\mathsf{fma}\left(t, t, -0.5\right)}} \]

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

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

   3. Applied rewrites 100.0%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-/.f64 N/A

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

      4. sub-to-fraction N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. add-flip-rev N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-/.f64 N/A

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

      4. sub-to-fraction N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. add-flip-rev N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower--.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in t around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 50.9

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

   10. Applied rewrites 50.9%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. add-flip N/A

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

       4. lower--.f64 N/A

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

   12. Applied rewrites 50.9%

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

Alternative 5: 99.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;\frac{1 + t\_2}{2 + t\_2} \leq 0.6:\\ \;\;\;\;\frac{-0.25}{\mathsf{fma}\left(t, t, -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{0.037037037037037035 \cdot \frac{1}{t} - 0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))) (t_2 (* t_1 t_1)))
   (if (<= (/ (+ 1.0 t_2) (+ 2.0 t_2)) 0.6)
     (/ -0.25 (fma t t -0.5))
     (+
      0.8333333333333334
      (/ (- (* 0.037037037037037035 (/ 1.0 t)) 0.2222222222222222) t)))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.6) {
		tmp = -0.25 / fma(t, t, -0.5);
	} else {
		tmp = 0.8333333333333334 + (((0.037037037037037035 * (1.0 / t)) - 0.2222222222222222) / t);
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.6)
		tmp = Float64(-0.25 / fma(t, t, -0.5));
	else
		tmp = Float64(0.8333333333333334 + Float64(Float64(Float64(0.037037037037037035 * Float64(1.0 / t)) - 0.2222222222222222) / t));
	end
	return tmp
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], 0.6], N[(-0.25 / N[(t * t + -0.5), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 + N[(N[(N[(0.037037037037037035 * N[(1.0 / t), $MachinePrecision]), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-pow.f64 51.8

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

   4. Applied rewrites 51.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lift-pow.f64 N/A

         \[\leadsto \frac{{t}^{2} \cdot {t}^{2} - \frac{1}{2} \cdot \frac{1}{2}}{{t}^{2} - \frac{1}{2}} \]

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow3 N/A

          \[\leadsto \frac{{t}^{3} \cdot t - \frac{1}{2} \cdot \frac{1}{2}}{{t}^{2} - \frac{1}{2}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{{t}^{3} \cdot t - \frac{1}{2} \cdot \frac{1}{2}}{{\color{blue}{t}}^{2} - \frac{1}{2}} \]

      13. unpow3 N/A

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

      14. unpow2 N/A

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

      15. lift-pow.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-pow.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 50.9

          \[\leadsto \frac{\left(\left(t \cdot t\right) \cdot t\right) \cdot t - 0.25}{{t}^{2} - \color{blue}{0.5}} \]

      22. lift-pow.f64 N/A

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

      23. unpow2 N/A

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

      24. lower-*.f64 50.9

          \[\leadsto \frac{\left(\left(t \cdot t\right) \cdot t\right) \cdot t - 0.25}{t \cdot t - 0.5} \]

   6. Applied rewrites 50.9%

      \[\leadsto \frac{\left(\left(t \cdot t\right) \cdot t\right) \cdot t - 0.25}{\color{blue}{t \cdot t - 0.5}} \]

   7. Taylor expanded in t around 0

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

   9. Applied rewrites 51.0%

      \[\leadsto \frac{-0.25}{\color{blue}{t \cdot t} - 0.5} \]

   11. Applied rewrites 51.0%

       \[\leadsto \color{blue}{\frac{-0.25}{\mathsf{fma}\left(t, t, -0.5\right)}} \]

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

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

   3. Applied rewrites 100.0%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-/.f64 N/A

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

      4. sub-to-fraction N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. add-flip-rev N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-/.f64 N/A

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

      4. sub-to-fraction N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. add-flip-rev N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower--.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in t around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 50.9

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

   10. Applied rewrites 50.9%

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

   11. Taylor expanded in t around inf

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

       1. lower-/.f64 N/A

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

       2. lower--.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 51.7

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

   13. Applied rewrites 51.7%

       \[\leadsto 0.8333333333333334 + \frac{0.037037037037037035 \cdot \frac{1}{t} - 0.2222222222222222}{\color{blue}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;\frac{1 + t\_2}{2 + t\_2} \leq 0.6:\\ \;\;\;\;\frac{-0.25}{\mathsf{fma}\left(t, t, -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{0.26666666666666666}{t}\right) \cdot 0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))) (t_2 (* t_1 t_1)))
   (if (<= (/ (+ 1.0 t_2) (+ 2.0 t_2)) 0.6)
     (/ -0.25 (fma t t -0.5))
     (* (- 1.0 (/ 0.26666666666666666 t)) 0.8333333333333334))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.6) {
		tmp = -0.25 / fma(t, t, -0.5);
	} else {
		tmp = (1.0 - (0.26666666666666666 / t)) * 0.8333333333333334;
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.6)
		tmp = Float64(-0.25 / fma(t, t, -0.5));
	else
		tmp = Float64(Float64(1.0 - Float64(0.26666666666666666 / t)) * 0.8333333333333334);
	end
	return tmp
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], 0.6], N[(-0.25 / N[(t * t + -0.5), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(0.26666666666666666 / t), $MachinePrecision]), $MachinePrecision] * 0.8333333333333334), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-pow.f64 51.8

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

   4. Applied rewrites 51.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lift-pow.f64 N/A

         \[\leadsto \frac{{t}^{2} \cdot {t}^{2} - \frac{1}{2} \cdot \frac{1}{2}}{{t}^{2} - \frac{1}{2}} \]

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow3 N/A

          \[\leadsto \frac{{t}^{3} \cdot t - \frac{1}{2} \cdot \frac{1}{2}}{{t}^{2} - \frac{1}{2}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{{t}^{3} \cdot t - \frac{1}{2} \cdot \frac{1}{2}}{{\color{blue}{t}}^{2} - \frac{1}{2}} \]

      13. unpow3 N/A

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

      14. unpow2 N/A

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

      15. lift-pow.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-pow.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 50.9

          \[\leadsto \frac{\left(\left(t \cdot t\right) \cdot t\right) \cdot t - 0.25}{{t}^{2} - \color{blue}{0.5}} \]

      22. lift-pow.f64 N/A

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

      23. unpow2 N/A

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

      24. lower-*.f64 50.9

          \[\leadsto \frac{\left(\left(t \cdot t\right) \cdot t\right) \cdot t - 0.25}{t \cdot t - 0.5} \]

   6. Applied rewrites 50.9%

      \[\leadsto \frac{\left(\left(t \cdot t\right) \cdot t\right) \cdot t - 0.25}{\color{blue}{t \cdot t - 0.5}} \]

   7. Taylor expanded in t around 0

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

   9. Applied rewrites 51.0%

      \[\leadsto \frac{-0.25}{\color{blue}{t \cdot t} - 0.5} \]

   11. Applied rewrites 51.0%

       \[\leadsto \color{blue}{\frac{-0.25}{\mathsf{fma}\left(t, t, -0.5\right)}} \]

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

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 51.1

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

   4. Applied rewrites 51.1%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mult-flip N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip-rev N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval 51.1

          \[\leadsto \left(1 - \frac{0.2222222222222222}{t} \cdot 1.2\right) \cdot 0.8333333333333334 \]

   6. Applied rewrites 51.1%

      \[\leadsto \left(1 - \frac{0.2222222222222222}{t} \cdot 1.2\right) \cdot \color{blue}{0.8333333333333334} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval 51.1

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

   8. Applied rewrites 51.1%

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

Alternative 7: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;\frac{1 + t\_2}{2 + t\_2} \leq 0.6:\\ \;\;\;\;\frac{-0.25}{\mathsf{fma}\left(t, t, -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))) (t_2 (* t_1 t_1)))
   (if (<= (/ (+ 1.0 t_2) (+ 2.0 t_2)) 0.6)
     (/ -0.25 (fma t t -0.5))
     (- 0.8333333333333334 (/ 0.2222222222222222 t)))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.6) {
		tmp = -0.25 / fma(t, t, -0.5);
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.6)
		tmp = Float64(-0.25 / fma(t, t, -0.5));
	else
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	end
	return tmp
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], 0.6], N[(-0.25 / N[(t * t + -0.5), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-pow.f64 51.8

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

   4. Applied rewrites 51.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lift-pow.f64 N/A

         \[\leadsto \frac{{t}^{2} \cdot {t}^{2} - \frac{1}{2} \cdot \frac{1}{2}}{{t}^{2} - \frac{1}{2}} \]

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow3 N/A

          \[\leadsto \frac{{t}^{3} \cdot t - \frac{1}{2} \cdot \frac{1}{2}}{{t}^{2} - \frac{1}{2}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{{t}^{3} \cdot t - \frac{1}{2} \cdot \frac{1}{2}}{{\color{blue}{t}}^{2} - \frac{1}{2}} \]

      13. unpow3 N/A

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

      14. unpow2 N/A

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

      15. lift-pow.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-pow.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 50.9

          \[\leadsto \frac{\left(\left(t \cdot t\right) \cdot t\right) \cdot t - 0.25}{{t}^{2} - \color{blue}{0.5}} \]

      22. lift-pow.f64 N/A

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

      23. unpow2 N/A

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

      24. lower-*.f64 50.9

          \[\leadsto \frac{\left(\left(t \cdot t\right) \cdot t\right) \cdot t - 0.25}{t \cdot t - 0.5} \]

   6. Applied rewrites 50.9%

      \[\leadsto \frac{\left(\left(t \cdot t\right) \cdot t\right) \cdot t - 0.25}{\color{blue}{t \cdot t - 0.5}} \]

   7. Taylor expanded in t around 0

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

   9. Applied rewrites 51.0%

      \[\leadsto \frac{-0.25}{\color{blue}{t \cdot t} - 0.5} \]

   11. Applied rewrites 51.0%

       \[\leadsto \color{blue}{\frac{-0.25}{\mathsf{fma}\left(t, t, -0.5\right)}} \]

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

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 51.1

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

   4. Applied rewrites 51.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lower-/.f64 51.1

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

   6. Applied rewrites 51.1%

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

Alternative 8: 99.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-pow.f64 51.8

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

   4. Applied rewrites 51.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 51.8

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

   6. Applied rewrites 51.8%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 51.1

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

   4. Applied rewrites 51.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lower-/.f64 51.1

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

   6. Applied rewrites 51.1%

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

Alternative 9: 98.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-pow.f64 51.8

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

   4. Applied rewrites 51.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 51.8

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

   6. Applied rewrites 51.8%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 58.7%

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

Alternative 10: 98.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 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 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.66) {
		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 - ((2.0d0 / t) / (1.0d0 + (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 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.66) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], 0.66], 0.5, 0.8333333333333334]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 59.3%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 58.7%

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

Alternative 11: 59.3% accurate, 77.5× speedup

Math

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

FPCore

(FPCore (t) :precision binary64 0.5)

C

double code(double t) {
	return 0.5;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t)
use fmin_fmax_functions
    real(8), intent (in) :: t
    code = 0.5d0
end function

Java

public static double code(double t) {
	return 0.5;
}

Python

def code(t):
	return 0.5

Julia

function code(t)
	return 0.5
end

MATLAB

function tmp = code(t)
	tmp = 0.5;
end

Wolfram

code[t_] := 0.5

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in t around 0

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

4. Applied rewrites 59.3%

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

Reproduce

herbie shell --seed 2025149 
(FPCore (t)
  :name "Kahan p13 Example 2"
  :precision binary64
  (/ (+ 1.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))) (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))))