Kahan p13 Example 2

Percentage Accurate: 100.0% → 100.0%

Time: 3.2s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

Alternative 2: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;\frac{1 + t\_2}{2 + t\_2} \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1.2 + -1 \cdot \frac{-1 \cdot \frac{0.032 - 0.0768 \cdot \frac{1}{t}}{t} - 0.32}{t}}\\ \end{array} \end{array} \]

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 (fma (- t 2.0) t 1.0) (* t t) 0.5)
     (/
      1.0
      (+
       1.2
       (*
        -1.0
        (/ (- (* -1.0 (/ (- 0.032 (* 0.0768 (/ 1.0 t))) t)) 0.32) 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(fma((t - 2.0), t, 1.0), (t * t), 0.5);
	} else {
		tmp = 1.0 / (1.2 + (-1.0 * (((-1.0 * ((0.032 - (0.0768 * (1.0 / t))) / t)) - 0.32) / 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(fma(Float64(t - 2.0), t, 1.0), Float64(t * t), 0.5);
	else
		tmp = Float64(1.0 / Float64(1.2 + Float64(-1.0 * Float64(Float64(Float64(-1.0 * Float64(Float64(0.032 - Float64(0.0768 * Float64(1.0 / t))) / t)) - 0.32) / 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[(N[(N[(t - 2.0), $MachinePrecision] * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(1.0 / N[(1.2 + N[(-1.0 * N[(N[(N[(-1.0 * N[(N[(0.032 - N[(0.0768 * N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] - 0.32), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   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

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 50.8

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

   4. Applied rewrites 50.8%

      \[\leadsto \color{blue}{0.5 + {t}^{2} \cdot \left(1 + t \cdot \left(t - 2\right)\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 50.8

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 50.8

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 50.8

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

   6. Applied rewrites 50.8%

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

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

   3. Applied rewrites 100.0%

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in t around -inf

      \[\leadsto \frac{1}{\color{blue}{\frac{6}{5} + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{125} - \frac{48}{625} \cdot \frac{1}{t}}{t} - \frac{8}{25}}{t}}} \]
   7. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{6}{5} + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{4}{125} - \frac{48}{625} \cdot \frac{1}{t}}{t} - \frac{8}{25}}{t}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{6}{5} + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{4}{125} - \frac{48}{625} \cdot \frac{1}{t}}{t} - \frac{8}{25}}{t}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{6}{5} + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{125} - \frac{48}{625} \cdot \frac{1}{t}}{t} - \frac{8}{25}}{\color{blue}{t}}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{1}{\frac{6}{5} + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{125} - \frac{48}{625} \cdot \frac{1}{t}}{t} - \frac{8}{25}}{t}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{6}{5} + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{125} - \frac{48}{625} \cdot \frac{1}{t}}{t} - \frac{8}{25}}{t}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{6}{5} + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{125} - \frac{48}{625} \cdot \frac{1}{t}}{t} - \frac{8}{25}}{t}} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{1}{\frac{6}{5} + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{125} - \frac{48}{625} \cdot \frac{1}{t}}{t} - \frac{8}{25}}{t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{6}{5} + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{125} - \frac{48}{625} \cdot \frac{1}{t}}{t} - \frac{8}{25}}{t}} \]

      9. lower-/.f64 52.3

         \[\leadsto \frac{1}{1.2 + -1 \cdot \frac{-1 \cdot \frac{0.032 - 0.0768 \cdot \frac{1}{t}}{t} - 0.32}{t}} \]

   8. Applied rewrites 52.3%

      \[\leadsto \frac{1}{\color{blue}{1.2 + -1 \cdot \frac{-1 \cdot \frac{0.032 - 0.0768 \cdot \frac{1}{t}}{t} - 0.32}{t}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.5% accurate, 0.8× speedup

Math

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

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)
     (fma (fma (- t 2.0) t 1.0) (* t t) 0.5)
     (-
      (/
       (-
        (/ (- (/ 0.04938271604938271 t) -0.037037037037037035) t)
        0.2222222222222222)
       t)
      -0.8333333333333334))))

C

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

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.6)
		tmp = fma(fma(Float64(t - 2.0), t, 1.0), Float64(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[(N[(N[(t - 2.0), $MachinePrecision] * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.04938271604938271 / t), $MachinePrecision] - -0.037037037037037035), $MachinePrecision] / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision] - -0.8333333333333334), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   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

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 50.8

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

   4. Applied rewrites 50.8%

      \[\leadsto \color{blue}{0.5 + {t}^{2} \cdot \left(1 + t \cdot \left(t - 2\right)\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 50.8

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 50.8

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 50.8

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

   6. Applied rewrites 50.8%

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

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in t around -inf

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

      1. 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 51.7

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

   6. Applied rewrites 51.7%

      \[\leadsto \color{blue}{0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}} \]
   7. 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. metadata-eval N/A

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

      5. lower--.f64 51.7

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

   8. Applied rewrites 51.7%

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

Alternative 4: 99.5% accurate, 0.8× speedup

Math

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

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 (fma (- t 2.0) t 1.0) (* t t) 0.5)
     (-
      (/ 0.037037037037037035 (* t t))
      (- (/ 0.2222222222222222 t) 0.8333333333333334)))))

C

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

Julia

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

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 50.8

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

   4. Applied rewrites 50.8%

      \[\leadsto \color{blue}{0.5 + {t}^{2} \cdot \left(1 + t \cdot \left(t - 2\right)\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 50.8

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 50.8

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 50.8

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

   6. Applied rewrites 50.8%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in t around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 52.6

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

   4. Applied rewrites 52.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. lower--.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. mult-flip-rev N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval 52.6

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

   6. Applied rewrites 52.6%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. div-sub N/A

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

      6. metadata-eval N/A

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

      7. associate-+l- N/A

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

      8. lower--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lift-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 52.6

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

   8. Applied rewrites 52.6%

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

Alternative 5: 99.4% accurate, 0.8× speedup

Math

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

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 (* (fma -2.0 t 1.0) t) t 0.5)
     (-
      (/ 0.037037037037037035 (* t t))
      (- (/ 0.2222222222222222 t) 0.8333333333333334)))))

C

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

Julia

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

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 50.8

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

   4. Applied rewrites 50.8%

      \[\leadsto \color{blue}{0.5 + {t}^{2} \cdot \left(1 + t \cdot \left(t - 2\right)\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 50.8

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 50.8

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 50.8

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

   6. Applied rewrites 50.8%

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

   7. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 50.1

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

   9. Applied rewrites 50.1%

      \[\leadsto \mathsf{fma}\left(1 + -2 \cdot t, \color{blue}{t} \cdot t, 0.5\right) \]
   10. Step-by-step derivation

       1. lift-fma.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 50.1

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

       6. lift-+.f64 N/A

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

       7. +-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. lower-fma.f64 50.1

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

   11. Applied rewrites 50.1%

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

   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

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 52.6

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

   4. Applied rewrites 52.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. lower--.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. mult-flip-rev N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval 52.6

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

   6. Applied rewrites 52.6%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. div-sub N/A

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

      6. metadata-eval N/A

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

      7. associate-+l- N/A

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

      8. lower--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lift-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 52.6

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

   8. Applied rewrites 52.6%

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

Alternative 6: 99.4% accurate, 0.8× speedup

Math

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

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)
     (fma (* (fma -2.0 t 1.0) t) t 0.5)
     (-
      (/ (- (/ 0.037037037037037035 t) 0.2222222222222222) t)
      -0.8333333333333334))))

C

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

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.6)
		tmp = fma(Float64(fma(-2.0, t, 1.0) * t), t, 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(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[(N[(N[(-2.0 * t + 1.0), $MachinePrecision] * t), $MachinePrecision] * t + 0.5), $MachinePrecision], N[(N[(N[(N[(0.037037037037037035 / 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

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 50.8

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

   4. Applied rewrites 50.8%

      \[\leadsto \color{blue}{0.5 + {t}^{2} \cdot \left(1 + t \cdot \left(t - 2\right)\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 50.8

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 50.8

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 50.8

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

   6. Applied rewrites 50.8%

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

   7. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 50.1

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

   9. Applied rewrites 50.1%

      \[\leadsto \mathsf{fma}\left(1 + -2 \cdot t, \color{blue}{t} \cdot t, 0.5\right) \]
   10. Step-by-step derivation

       1. lift-fma.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 50.1

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

       6. lift-+.f64 N/A

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

       7. +-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. lower-fma.f64 50.1

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

   11. Applied rewrites 50.1%

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

   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

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 52.6

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

   4. Applied rewrites 52.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. lower--.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. mult-flip-rev N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval 52.6

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

   6. Applied rewrites 52.6%

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

Alternative 7: 99.3% accurate, 0.8× speedup

Math

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

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)
     (fma 1.0 (* t t) 0.5)
     (-
      (/ (- (/ 0.037037037037037035 t) 0.2222222222222222) t)
      -0.8333333333333334))))

C

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

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.6)
		tmp = fma(1.0, Float64(t * t), 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(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[(1.0 * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(N[(0.037037037037037035 / 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

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 50.8

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

   4. Applied rewrites 50.8%

      \[\leadsto \color{blue}{0.5 + {t}^{2} \cdot \left(1 + t \cdot \left(t - 2\right)\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 50.8

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 50.8

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 50.8

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

   6. Applied rewrites 50.8%

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

   7. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 50.1

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

   9. Applied rewrites 50.1%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 51.0%

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

   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

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 52.6

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

   4. Applied rewrites 52.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. lower--.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. mult-flip-rev N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval 52.6

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

   6. Applied rewrites 52.6%

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

Alternative 8: 99.2% accurate, 0.9× speedup

Math

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

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)
     (fma 1.0 (* t t) 0.5)
     (- (/ -0.2222222222222222 t) -0.8333333333333334))))

C

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

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.6)
		tmp = fma(1.0, Float64(t * t), 0.5);
	else
		tmp = Float64(Float64(-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[(1.0 * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(-0.2222222222222222 / 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

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 50.8

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

   4. Applied rewrites 50.8%

      \[\leadsto \color{blue}{0.5 + {t}^{2} \cdot \left(1 + t \cdot \left(t - 2\right)\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 50.8

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 50.8

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 50.8

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

   6. Applied rewrites 50.8%

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

   7. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 50.1

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

   9. Applied rewrites 50.1%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 51.0%

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

   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

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 52.6

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

   4. Applied rewrites 52.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. lower--.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. mult-flip-rev N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval 52.6

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

   6. Applied rewrites 52.6%

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

   7. Taylor expanded in t around inf

      \[\leadsto \frac{\frac{-2}{9}}{t} - \frac{-5}{6} \]
   8. Step-by-step derivation

   9. Applied rewrites 52.0%

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

Alternative 9: 99.1% accurate, 0.9× speedup

Math

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

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
     (- (/ -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;
	} else {
		tmp = (-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
    else
        tmp = ((-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;
	} else {
		tmp = (-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
	else:
		tmp = (-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 = 0.5;
	else
		tmp = Float64(Float64(-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;
	else
		tmp = (-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], 0.5, N[(N[(-0.2222222222222222 / 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

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

   4. Applied rewrites 58.7%

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

   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

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 52.6

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

   4. Applied rewrites 52.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. lower--.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. mult-flip-rev N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval 52.6

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

   6. Applied rewrites 52.6%

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

   7. Taylor expanded in t around inf

      \[\leadsto \frac{\frac{-2}{9}}{t} - \frac{-5}{6} \]
   8. Step-by-step derivation

   9. Applied rewrites 52.0%

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

Alternative 10: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))
    t_2 = t_1 * t_1
    if (((1.0d0 + t_2) / (2.0d0 + t_2)) <= 0.68d0) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.68) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))
	t_2 = t_1 * t_1
	tmp = 0
	if ((1.0 + t_2) / (2.0 + t_2)) <= 0.68:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + 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)) <= 0.68)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	t_2 = t_1 * t_1;
	tmp = 0.0;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.68)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(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], 0.68], 0.5, 0.8333333333333334]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 #s(literal 1 binary64) (*.f64 (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))) (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))))) (+.f64 #s(literal 2 binary64) (*.f64 (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))) (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))))))) < 0.680000000000000049

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 58.7%

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

   if 0.680000000000000049 < (/.f64 (+.f64 #s(literal 1 binary64) (*.f64 (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))) (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))))) (+.f64 #s(literal 2 binary64) (*.f64 (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))) (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))))))

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 59.5%

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

Alternative 11: 58.7% 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 58.7%

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

Reproduce

herbie shell --seed 2025162 
(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))))))))