Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, B

Percentage Accurate: 99.9% → 99.9%

Time: 2.8s

Alternatives: 10

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ 0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  0.70711
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x)))

C

double code(double x) {
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
}

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(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = 0.70711d0 * (((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x)
end function

Java

public static double code(double x) {
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
}

Python

def code(x):
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x)

Julia

function code(x)
	return Float64(0.70711 * Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x))
end

MATLAB

function tmp = code(x)
	tmp = 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
end

Wolfram

code[x_] := N[(0.70711 * N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  0.70711
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x)))

C

double code(double x) {
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
}

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(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = 0.70711d0 * (((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x)
end function

Java

public static double code(double x) {
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
}

Python

def code(x):
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x)

Julia

function code(x)
	return Float64(0.70711 * Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x))
end

MATLAB

function tmp = code(x)
	tmp = 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
end

Wolfram

code[x_] := N[(0.70711 * N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-x, 0.70711, \frac{0.70711 \cdot \mathsf{fma}\left(-0.27061, x, -2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.04481, x, -0.99229\right), x, -1\right)}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (- x)
  0.70711
  (/
   (* 0.70711 (fma -0.27061 x -2.30753))
   (fma (fma -0.04481 x -0.99229) x -1.0))))

C

double code(double x) {
	return fma(-x, 0.70711, ((0.70711 * fma(-0.27061, x, -2.30753)) / fma(fma(-0.04481, x, -0.99229), x, -1.0)));
}

Julia

function code(x)
	return fma(Float64(-x), 0.70711, Float64(Float64(0.70711 * fma(-0.27061, x, -2.30753)) / fma(fma(-0.04481, x, -0.99229), x, -1.0)))
end

Wolfram

code[x_] := N[((-x) * 0.70711 + N[(N[(0.70711 * N[(-0.27061 * x + -2.30753), $MachinePrecision]), $MachinePrecision] / N[(N[(-0.04481 * x + -0.99229), $MachinePrecision] * x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

   2. lift--.f64 N/A

      \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

   3. sub-flip N/A

      \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + \left(\mathsf{neg}\left(x\right)\right)\right)} \]

   4. +-commutative N/A

      \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}\right)} \]

   5. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{70711}{100000} + \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}} \]

   6. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{70711}{100000}, \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}\right)} \]

   7. lower-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, \frac{70711}{100000}, \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \color{blue}{\frac{70711}{100000} \cdot \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}}\right) \]

   9. lift-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \frac{70711}{100000} \cdot \color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}}\right) \]

   10. frac-2neg N/A

       \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \frac{70711}{100000} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)\right)}{\mathsf{neg}\left(\left(1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}}\right) \]

   11. associate-*r/ N/A

       \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \color{blue}{\frac{\frac{70711}{100000} \cdot \left(\mathsf{neg}\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)\right)\right)}{\mathsf{neg}\left(\left(1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}}\right) \]

   12. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \color{blue}{\frac{\frac{70711}{100000} \cdot \left(\mathsf{neg}\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)\right)\right)}{\mathsf{neg}\left(\left(1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}}\right) \]

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-x, 0.70711, \frac{0.70711 \cdot \mathsf{fma}\left(-0.27061, x, -2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.04481, x, -0.99229\right), x, -1\right)}\right)} \]
4. Add Preprocessing

Alternative 2: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(-0.1913510371, x, -1.6316775383\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.04481, x, -0.99229\right), x, -1\right)}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  x
  -0.70711
  (/
   (fma -0.1913510371 x -1.6316775383)
   (fma (fma -0.04481 x -0.99229) x -1.0))))

C

double code(double x) {
	return fma(x, -0.70711, (fma(-0.1913510371, x, -1.6316775383) / fma(fma(-0.04481, x, -0.99229), x, -1.0)));
}

Julia

function code(x)
	return fma(x, -0.70711, Float64(fma(-0.1913510371, x, -1.6316775383) / fma(fma(-0.04481, x, -0.99229), x, -1.0)))
end

Wolfram

code[x_] := N[(x * -0.70711 + N[(N[(-0.1913510371 * x + -1.6316775383), $MachinePrecision] / N[(N[(-0.04481 * x + -0.99229), $MachinePrecision] * x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

   2. lift--.f64 N/A

      \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

   3. sub-flip N/A

      \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + \left(\mathsf{neg}\left(x\right)\right)\right)} \]

   4. +-commutative N/A

      \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}\right)} \]

   5. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{70711}{100000} + \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}} \]

   6. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{70711}{100000}, \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}\right)} \]

   7. lower-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, \frac{70711}{100000}, \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \color{blue}{\frac{70711}{100000} \cdot \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}}\right) \]

   9. lift-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \frac{70711}{100000} \cdot \color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}}\right) \]

   10. frac-2neg N/A

       \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \frac{70711}{100000} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)\right)}{\mathsf{neg}\left(\left(1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}}\right) \]

   11. associate-*r/ N/A

       \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \color{blue}{\frac{\frac{70711}{100000} \cdot \left(\mathsf{neg}\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)\right)\right)}{\mathsf{neg}\left(\left(1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}}\right) \]

   12. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \color{blue}{\frac{\frac{70711}{100000} \cdot \left(\mathsf{neg}\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)\right)\right)}{\mathsf{neg}\left(\left(1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}}\right) \]

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-x, 0.70711, \frac{0.70711 \cdot \mathsf{fma}\left(-0.27061, x, -2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.04481, x, -0.99229\right), x, -1\right)}\right)} \]
4. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{70711}{100000} + \frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}} \]

   2. add-flip N/A

      \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{70711}{100000} - \left(\mathsf{neg}\left(\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right)\right)} \]

   3. sub-flip N/A

      \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{70711}{100000} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right)\right)\right)\right)} \]

   4. lift-neg.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \frac{70711}{100000} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{70711}{100000}\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right)\right)\right)\right) \]

   7. lift-/.f64 N/A

      \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}}\right)\right)\right)\right) \]

   8. distribute-neg-frac N/A

      \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{\mathsf{neg}\left(\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}}\right)\right) \]

   9. distribute-frac-neg2 N/A

      \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right) + \color{blue}{\frac{\mathsf{neg}\left(\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)\right)}{\mathsf{neg}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)\right)}} \]

   10. frac-2neg N/A

       \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right) + \color{blue}{\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}} \]

   11. lift-/.f64 N/A

       \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right) + \color{blue}{\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}} \]

   12. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{neg}\left(\frac{70711}{100000}\right), \frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right)} \]

   13. metadata-eval 99.9

       \[\leadsto \mathsf{fma}\left(x, \color{blue}{-0.70711}, \frac{0.70711 \cdot \mathsf{fma}\left(-0.27061, x, -2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.04481, x, -0.99229\right), x, -1\right)}\right) \]

5. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(-0.1913510371, x, -1.6316775383\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.04481, x, -0.99229\right), x, -1\right)}\right)} \]
6. Add Preprocessing

Alternative 3: 99.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{6.039053782637804 - \frac{82.23527511657367}{x}}{x} - x\right) \cdot 0.70711\\ \mathbf{if}\;x \leq -4.8:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.3:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.2692862305735844, x, 1.3436228731669864\right), x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (*
          (- (/ (- 6.039053782637804 (/ 82.23527511657367 x)) x) x)
          0.70711)))
   (if (<= x -4.8)
     t_0
     (if (<= x 2.3)
       (fma
        (fma
         (fma -1.2692862305735844 x 1.3436228731669864)
         x
         -2.134856267379707)
        x
        1.6316775383)
       t_0))))

C

double code(double x) {
	double t_0 = (((6.039053782637804 - (82.23527511657367 / x)) / x) - x) * 0.70711;
	double tmp;
	if (x <= -4.8) {
		tmp = t_0;
	} else if (x <= 2.3) {
		tmp = fma(fma(fma(-1.2692862305735844, x, 1.3436228731669864), x, -2.134856267379707), x, 1.6316775383);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(Float64(Float64(6.039053782637804 - Float64(82.23527511657367 / x)) / x) - x) * 0.70711)
	tmp = 0.0
	if (x <= -4.8)
		tmp = t_0;
	elseif (x <= 2.3)
		tmp = fma(fma(fma(-1.2692862305735844, x, 1.3436228731669864), x, -2.134856267379707), x, 1.6316775383);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(N[(6.039053782637804 - N[(82.23527511657367 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - x), $MachinePrecision] * 0.70711), $MachinePrecision]}, If[LessEqual[x, -4.8], t$95$0, If[LessEqual[x, 2.3], N[(N[(N[(-1.2692862305735844 * x + 1.3436228731669864), $MachinePrecision] * x + -2.134856267379707), $MachinePrecision] * x + 1.6316775383), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.79999999999999982 or 2.2999999999999998 < x

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

   2. Taylor expanded in x around inf

      \[\leadsto \frac{70711}{100000} \cdot \left(\color{blue}{\frac{\frac{27061}{4481} - \frac{1651231776}{20079361} \cdot \frac{1}{x}}{x}} - x\right) \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{27061}{4481} - \frac{1651231776}{20079361} \cdot \frac{1}{x}}{\color{blue}{x}} - x\right) \]

      2. lower--.f64 N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{27061}{4481} - \frac{1651231776}{20079361} \cdot \frac{1}{x}}{x} - x\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{27061}{4481} - \frac{1651231776}{20079361} \cdot \frac{1}{x}}{x} - x\right) \]

      4. lower-/.f64 50.8

         \[\leadsto 0.70711 \cdot \left(\frac{6.039053782637804 - 82.23527511657367 \cdot \frac{1}{x}}{x} - x\right) \]

   4. Applied rewrites 50.8%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\frac{6.039053782637804 - 82.23527511657367 \cdot \frac{1}{x}}{x}} - x\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{70711}{100000} \cdot \left(\frac{\frac{27061}{4481} - \frac{1651231776}{20079361} \cdot \frac{1}{x}}{x} - x\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{27061}{4481} - \frac{1651231776}{20079361} \cdot \frac{1}{x}}{x} - x\right) \cdot \frac{70711}{100000}} \]

      3. lower-*.f64 50.8

         \[\leadsto \color{blue}{\left(\frac{6.039053782637804 - 82.23527511657367 \cdot \frac{1}{x}}{x} - x\right) \cdot 0.70711} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\frac{\frac{27061}{4481} - \frac{1651231776}{20079361} \cdot \frac{1}{x}}{x} - x\right) \cdot \frac{70711}{100000} \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\frac{\frac{27061}{4481} - \frac{1651231776}{20079361} \cdot \frac{1}{x}}{x} - x\right) \cdot \frac{70711}{100000} \]

      6. mult-flip-rev N/A

         \[\leadsto \left(\frac{\frac{27061}{4481} - \frac{\frac{1651231776}{20079361}}{x}}{x} - x\right) \cdot \frac{70711}{100000} \]

      7. lower-/.f64 50.8

         \[\leadsto \left(\frac{6.039053782637804 - \frac{82.23527511657367}{x}}{x} - x\right) \cdot 0.70711 \]

   6. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\left(\frac{6.039053782637804 - \frac{82.23527511657367}{x}}{x} - x\right) \cdot 0.70711} \]

   if -4.79999999999999982 < x < 2.2999999999999998

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{16316775383}{10000000000} + x \cdot \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right)} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{16316775383}{10000000000} + \color{blue}{x \cdot \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{16316775383}{10000000000} + x \cdot \color{blue}{\left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{16316775383}{10000000000} + x \cdot \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \color{blue}{\frac{2134856267379707}{1000000000000000}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{16316775383}{10000000000} + x \cdot \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{16316775383}{10000000000} + x \cdot \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right) \]

      6. lower-*.f64 52.0

         \[\leadsto 1.6316775383 + x \cdot \left(x \cdot \left(1.3436228731669864 + -1.2692862305735844 \cdot x\right) - 2.134856267379707\right) \]

   4. Applied rewrites 52.0%

      \[\leadsto \color{blue}{1.6316775383 + x \cdot \left(x \cdot \left(1.3436228731669864 + -1.2692862305735844 \cdot x\right) - 2.134856267379707\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{16316775383}{10000000000} + \color{blue}{x \cdot \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right) + \color{blue}{\frac{16316775383}{10000000000}} \]

      3. lift-*.f64 N/A

         \[\leadsto x \cdot \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right) + \frac{16316775383}{10000000000} \]

      4. *-commutative N/A

         \[\leadsto \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right) \cdot x + \frac{16316775383}{10000000000} \]

      5. lower-fma.f64 52.0

         \[\leadsto \mathsf{fma}\left(x \cdot \left(1.3436228731669864 + -1.2692862305735844 \cdot x\right) - 2.134856267379707, \color{blue}{x}, 1.6316775383\right) \]

      6. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}, x, \frac{16316775383}{10000000000}\right) \]

      7. sub-flip N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) + \left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right), x, \frac{16316775383}{10000000000}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) + \left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right), x, \frac{16316775383}{10000000000}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) \cdot x + \left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right), x, \frac{16316775383}{10000000000}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x, x, \mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right), x, \frac{16316775383}{10000000000}\right) \]

      11. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x, x, \mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right), x, \frac{16316775383}{10000000000}\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x + \frac{134362287316698645903}{100000000000000000000}, x, \mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right), x, \frac{16316775383}{10000000000}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x + \frac{134362287316698645903}{100000000000000000000}, x, \mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right), x, \frac{16316775383}{10000000000}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-12692862305735843227608787}{10000000000000000000000000}, x, \frac{134362287316698645903}{100000000000000000000}\right), x, \mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right), x, \frac{16316775383}{10000000000}\right) \]

      15. metadata-eval 52.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.2692862305735844, x, 1.3436228731669864\right), x, -2.134856267379707\right), x, 1.6316775383\right) \]

   6. Applied rewrites 52.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.2692862305735844, x, 1.3436228731669864\right), x, -2.134856267379707\right), x, 1.6316775383\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right)\\ \mathbf{if}\;t\_0 \leq -40:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.2692862305735844, x, 1.3436228731669864\right), x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (*
          0.70711
          (-
           (/
            (+ 2.30753 (* x 0.27061))
            (+ 1.0 (* x (+ 0.99229 (* x 0.04481)))))
           x))))
   (if (<= t_0 -40.0)
     (* 0.70711 (- (/ 6.039053782637804 x) x))
     (if (<= t_0 2.0)
       (fma
        (fma
         (fma -1.2692862305735844 x 1.3436228731669864)
         x
         -2.134856267379707)
        x
        1.6316775383)
       (fma x -0.70711 (/ 4.2702753202410175 x))))))

C

double code(double x) {
	double t_0 = 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
	double tmp;
	if (t_0 <= -40.0) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else if (t_0 <= 2.0) {
		tmp = fma(fma(fma(-1.2692862305735844, x, 1.3436228731669864), x, -2.134856267379707), x, 1.6316775383);
	} else {
		tmp = fma(x, -0.70711, (4.2702753202410175 / x));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.70711 * Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x))
	tmp = 0.0
	if (t_0 <= -40.0)
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	elseif (t_0 <= 2.0)
		tmp = fma(fma(fma(-1.2692862305735844, x, 1.3436228731669864), x, -2.134856267379707), x, 1.6316775383);
	else
		tmp = fma(x, -0.70711, Float64(4.2702753202410175 / x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.70711 * N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -40.0], N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(N[(-1.2692862305735844 * x + 1.3436228731669864), $MachinePrecision] * x + -2.134856267379707), $MachinePrecision] * x + 1.6316775383), $MachinePrecision], N[(x * -0.70711 + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 70711/100000 binary64) (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x)) < -40

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

   2. Taylor expanded in x around inf

      \[\leadsto \frac{70711}{100000} \cdot \left(\color{blue}{\frac{\frac{27061}{4481}}{x}} - x\right) \]
   3. Step-by-step derivation

      1. lower-/.f64 51.9

         \[\leadsto 0.70711 \cdot \left(\frac{6.039053782637804}{\color{blue}{x}} - x\right) \]

   4. Applied rewrites 51.9%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\frac{6.039053782637804}{x}} - x\right) \]

   if -40 < (*.f64 #s(literal 70711/100000 binary64) (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x)) < 2

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{16316775383}{10000000000} + x \cdot \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right)} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{16316775383}{10000000000} + \color{blue}{x \cdot \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{16316775383}{10000000000} + x \cdot \color{blue}{\left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{16316775383}{10000000000} + x \cdot \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \color{blue}{\frac{2134856267379707}{1000000000000000}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{16316775383}{10000000000} + x \cdot \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{16316775383}{10000000000} + x \cdot \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right) \]

      6. lower-*.f64 52.0

         \[\leadsto 1.6316775383 + x \cdot \left(x \cdot \left(1.3436228731669864 + -1.2692862305735844 \cdot x\right) - 2.134856267379707\right) \]

   4. Applied rewrites 52.0%

      \[\leadsto \color{blue}{1.6316775383 + x \cdot \left(x \cdot \left(1.3436228731669864 + -1.2692862305735844 \cdot x\right) - 2.134856267379707\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{16316775383}{10000000000} + \color{blue}{x \cdot \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right) + \color{blue}{\frac{16316775383}{10000000000}} \]

      3. lift-*.f64 N/A

         \[\leadsto x \cdot \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right) + \frac{16316775383}{10000000000} \]

      4. *-commutative N/A

         \[\leadsto \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right) \cdot x + \frac{16316775383}{10000000000} \]

      5. lower-fma.f64 52.0

         \[\leadsto \mathsf{fma}\left(x \cdot \left(1.3436228731669864 + -1.2692862305735844 \cdot x\right) - 2.134856267379707, \color{blue}{x}, 1.6316775383\right) \]

      6. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}, x, \frac{16316775383}{10000000000}\right) \]

      7. sub-flip N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) + \left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right), x, \frac{16316775383}{10000000000}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) + \left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right), x, \frac{16316775383}{10000000000}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) \cdot x + \left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right), x, \frac{16316775383}{10000000000}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x, x, \mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right), x, \frac{16316775383}{10000000000}\right) \]

      11. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x, x, \mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right), x, \frac{16316775383}{10000000000}\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x + \frac{134362287316698645903}{100000000000000000000}, x, \mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right), x, \frac{16316775383}{10000000000}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x + \frac{134362287316698645903}{100000000000000000000}, x, \mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right), x, \frac{16316775383}{10000000000}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-12692862305735843227608787}{10000000000000000000000000}, x, \frac{134362287316698645903}{100000000000000000000}\right), x, \mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right), x, \frac{16316775383}{10000000000}\right) \]

      15. metadata-eval 52.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.2692862305735844, x, 1.3436228731669864\right), x, -2.134856267379707\right), x, 1.6316775383\right) \]

   6. Applied rewrites 52.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.2692862305735844, x, 1.3436228731669864\right), x, -2.134856267379707\right), x, 1.6316775383\right)} \]

   if 2 < (*.f64 #s(literal 70711/100000 binary64) (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x))

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

      3. sub-flip N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}\right)} \]

      5. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{70711}{100000} + \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{70711}{100000}, \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, \frac{70711}{100000}, \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \color{blue}{\frac{70711}{100000} \cdot \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \frac{70711}{100000} \cdot \color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}}\right) \]

      10. frac-2neg N/A

          \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \frac{70711}{100000} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)\right)}{\mathsf{neg}\left(\left(1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}}\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \color{blue}{\frac{\frac{70711}{100000} \cdot \left(\mathsf{neg}\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)\right)\right)}{\mathsf{neg}\left(\left(1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \color{blue}{\frac{\frac{70711}{100000} \cdot \left(\mathsf{neg}\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)\right)\right)}{\mathsf{neg}\left(\left(1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}}\right) \]

   3. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, 0.70711, \frac{0.70711 \cdot \mathsf{fma}\left(-0.27061, x, -2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.04481, x, -0.99229\right), x, -1\right)}\right)} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{70711}{100000} + \frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{70711}{100000} - \left(\mathsf{neg}\left(\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right)\right)} \]

      3. sub-flip N/A

         \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{70711}{100000} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right)\right)\right)\right)} \]

      4. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \frac{70711}{100000} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{70711}{100000}\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right)\right)\right)\right) \]

      7. lift-/.f64 N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}}\right)\right)\right)\right) \]

      8. distribute-neg-frac N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{\mathsf{neg}\left(\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}}\right)\right) \]

      9. distribute-frac-neg2 N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right) + \color{blue}{\frac{\mathsf{neg}\left(\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)\right)}{\mathsf{neg}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)\right)}} \]

      10. frac-2neg N/A

          \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right) + \color{blue}{\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}} \]

      11. lift-/.f64 N/A

          \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right) + \color{blue}{\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}} \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{neg}\left(\frac{70711}{100000}\right), \frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right)} \]

      13. metadata-eval 99.9

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{-0.70711}, \frac{0.70711 \cdot \mathsf{fma}\left(-0.27061, x, -2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.04481, x, -0.99229\right), x, -1\right)}\right) \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(-0.1913510371, x, -1.6316775383\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.04481, x, -0.99229\right), x, -1\right)}\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \color{blue}{\frac{\frac{1913510371}{448100000}}{x}}\right) \]
   7. Step-by-step derivation

      1. lower-/.f64 51.9

         \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175}{\color{blue}{x}}\right) \]

   8. Applied rewrites 51.9%

      \[\leadsto \mathsf{fma}\left(x, -0.70711, \color{blue}{\frac{4.2702753202410175}{x}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right)\\ \mathbf{if}\;t\_0 \leq -40:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (*
          0.70711
          (-
           (/
            (+ 2.30753 (* x 0.27061))
            (+ 1.0 (* x (+ 0.99229 (* x 0.04481)))))
           x))))
   (if (<= t_0 -40.0)
     (* 0.70711 (- (/ 6.039053782637804 x) x))
     (if (<= t_0 2.0)
       (fma (fma 1.3436228731669864 x -2.134856267379707) x 1.6316775383)
       (fma x -0.70711 (/ 4.2702753202410175 x))))))

C

double code(double x) {
	double t_0 = 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
	double tmp;
	if (t_0 <= -40.0) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else if (t_0 <= 2.0) {
		tmp = fma(fma(1.3436228731669864, x, -2.134856267379707), x, 1.6316775383);
	} else {
		tmp = fma(x, -0.70711, (4.2702753202410175 / x));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.70711 * Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x))
	tmp = 0.0
	if (t_0 <= -40.0)
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	elseif (t_0 <= 2.0)
		tmp = fma(fma(1.3436228731669864, x, -2.134856267379707), x, 1.6316775383);
	else
		tmp = fma(x, -0.70711, Float64(4.2702753202410175 / x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.70711 * N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -40.0], N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(1.3436228731669864 * x + -2.134856267379707), $MachinePrecision] * x + 1.6316775383), $MachinePrecision], N[(x * -0.70711 + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 70711/100000 binary64) (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x)) < -40

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

   2. Taylor expanded in x around inf

      \[\leadsto \frac{70711}{100000} \cdot \left(\color{blue}{\frac{\frac{27061}{4481}}{x}} - x\right) \]
   3. Step-by-step derivation

      1. lower-/.f64 51.9

         \[\leadsto 0.70711 \cdot \left(\frac{6.039053782637804}{\color{blue}{x}} - x\right) \]

   4. Applied rewrites 51.9%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\frac{6.039053782637804}{x}} - x\right) \]

   if -40 < (*.f64 #s(literal 70711/100000 binary64) (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x)) < 2

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

      3. sub-flip N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}\right)} \]

      5. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{70711}{100000} + \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{70711}{100000}, \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, \frac{70711}{100000}, \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \color{blue}{\frac{70711}{100000} \cdot \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \frac{70711}{100000} \cdot \color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}}\right) \]

      10. frac-2neg N/A

          \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \frac{70711}{100000} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)\right)}{\mathsf{neg}\left(\left(1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}}\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \color{blue}{\frac{\frac{70711}{100000} \cdot \left(\mathsf{neg}\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)\right)\right)}{\mathsf{neg}\left(\left(1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \color{blue}{\frac{\frac{70711}{100000} \cdot \left(\mathsf{neg}\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)\right)\right)}{\mathsf{neg}\left(\left(1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}}\right) \]

   3. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, 0.70711, \frac{0.70711 \cdot \mathsf{fma}\left(-0.27061, x, -2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.04481, x, -0.99229\right), x, -1\right)}\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{16316775383}{10000000000} + x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right)} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{16316775383}{10000000000} + \color{blue}{x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{16316775383}{10000000000} + x \cdot \color{blue}{\left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{16316775383}{10000000000} + x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \color{blue}{\frac{2134856267379707}{1000000000000000}}\right) \]

      4. lower-*.f64 50.7

         \[\leadsto 1.6316775383 + x \cdot \left(1.3436228731669864 \cdot x - 2.134856267379707\right) \]

   6. Applied rewrites 50.7%

      \[\leadsto \color{blue}{1.6316775383 + x \cdot \left(1.3436228731669864 \cdot x - 2.134856267379707\right)} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{16316775383}{10000000000} + \color{blue}{x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right) + \color{blue}{\frac{16316775383}{10000000000}} \]

      3. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right) + \frac{16316775383}{10000000000} \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right) \cdot x + \frac{16316775383}{10000000000} \]

      5. lower-fma.f64 50.7

         \[\leadsto \mathsf{fma}\left(1.3436228731669864 \cdot x - 2.134856267379707, \color{blue}{x}, 1.6316775383\right) \]

      6. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}, x, \frac{16316775383}{10000000000}\right) \]

      7. sub-flip N/A

         \[\leadsto \mathsf{fma}\left(\frac{134362287316698645903}{100000000000000000000} \cdot x + \left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right), x, \frac{16316775383}{10000000000}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{134362287316698645903}{100000000000000000000} \cdot x + \left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right), x, \frac{16316775383}{10000000000}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{134362287316698645903}{100000000000000000000}, x, \mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right), x, \frac{16316775383}{10000000000}\right) \]

      10. metadata-eval 50.7

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right), x, 1.6316775383\right) \]

   8. Applied rewrites 50.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right), x, 1.6316775383\right)} \]

   if 2 < (*.f64 #s(literal 70711/100000 binary64) (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x))

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

      3. sub-flip N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}\right)} \]

      5. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{70711}{100000} + \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{70711}{100000}, \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, \frac{70711}{100000}, \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \color{blue}{\frac{70711}{100000} \cdot \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \frac{70711}{100000} \cdot \color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}}\right) \]

      10. frac-2neg N/A

          \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \frac{70711}{100000} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)\right)}{\mathsf{neg}\left(\left(1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}}\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \color{blue}{\frac{\frac{70711}{100000} \cdot \left(\mathsf{neg}\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)\right)\right)}{\mathsf{neg}\left(\left(1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \color{blue}{\frac{\frac{70711}{100000} \cdot \left(\mathsf{neg}\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)\right)\right)}{\mathsf{neg}\left(\left(1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}}\right) \]

   3. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, 0.70711, \frac{0.70711 \cdot \mathsf{fma}\left(-0.27061, x, -2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.04481, x, -0.99229\right), x, -1\right)}\right)} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{70711}{100000} + \frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{70711}{100000} - \left(\mathsf{neg}\left(\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right)\right)} \]

      3. sub-flip N/A

         \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{70711}{100000} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right)\right)\right)\right)} \]

      4. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \frac{70711}{100000} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{70711}{100000}\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right)\right)\right)\right) \]

      7. lift-/.f64 N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}}\right)\right)\right)\right) \]

      8. distribute-neg-frac N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{\mathsf{neg}\left(\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}}\right)\right) \]

      9. distribute-frac-neg2 N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right) + \color{blue}{\frac{\mathsf{neg}\left(\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)\right)}{\mathsf{neg}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)\right)}} \]

      10. frac-2neg N/A

          \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right) + \color{blue}{\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}} \]

      11. lift-/.f64 N/A

          \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right) + \color{blue}{\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}} \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{neg}\left(\frac{70711}{100000}\right), \frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right)} \]

      13. metadata-eval 99.9

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{-0.70711}, \frac{0.70711 \cdot \mathsf{fma}\left(-0.27061, x, -2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.04481, x, -0.99229\right), x, -1\right)}\right) \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(-0.1913510371, x, -1.6316775383\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.04481, x, -0.99229\right), x, -1\right)}\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \color{blue}{\frac{\frac{1913510371}{448100000}}{x}}\right) \]
   7. Step-by-step derivation

      1. lower-/.f64 51.9

         \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175}{\color{blue}{x}}\right) \]

   8. Applied rewrites 51.9%

      \[\leadsto \mathsf{fma}\left(x, -0.70711, \color{blue}{\frac{4.2702753202410175}{x}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right)\\ \mathbf{if}\;t\_0 \leq -40:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-2.134856267379707, x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (*
          0.70711
          (-
           (/
            (+ 2.30753 (* x 0.27061))
            (+ 1.0 (* x (+ 0.99229 (* x 0.04481)))))
           x))))
   (if (<= t_0 -40.0)
     (* 0.70711 (- (/ 6.039053782637804 x) x))
     (if (<= t_0 2.0)
       (fma -2.134856267379707 x 1.6316775383)
       (fma x -0.70711 (/ 4.2702753202410175 x))))))

C

double code(double x) {
	double t_0 = 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
	double tmp;
	if (t_0 <= -40.0) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else if (t_0 <= 2.0) {
		tmp = fma(-2.134856267379707, x, 1.6316775383);
	} else {
		tmp = fma(x, -0.70711, (4.2702753202410175 / x));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.70711 * Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x))
	tmp = 0.0
	if (t_0 <= -40.0)
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	elseif (t_0 <= 2.0)
		tmp = fma(-2.134856267379707, x, 1.6316775383);
	else
		tmp = fma(x, -0.70711, Float64(4.2702753202410175 / x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.70711 * N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -40.0], N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(-2.134856267379707 * x + 1.6316775383), $MachinePrecision], N[(x * -0.70711 + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 70711/100000 binary64) (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x)) < -40

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

   2. Taylor expanded in x around inf

      \[\leadsto \frac{70711}{100000} \cdot \left(\color{blue}{\frac{\frac{27061}{4481}}{x}} - x\right) \]
   3. Step-by-step derivation

      1. lower-/.f64 51.9

         \[\leadsto 0.70711 \cdot \left(\frac{6.039053782637804}{\color{blue}{x}} - x\right) \]

   4. Applied rewrites 51.9%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\frac{6.039053782637804}{x}} - x\right) \]

   if -40 < (*.f64 #s(literal 70711/100000 binary64) (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x)) < 2

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

      3. sub-flip N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}\right)} \]

      5. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{70711}{100000} + \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{70711}{100000}, \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, \frac{70711}{100000}, \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \color{blue}{\frac{70711}{100000} \cdot \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \frac{70711}{100000} \cdot \color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}}\right) \]

      10. frac-2neg N/A

          \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \frac{70711}{100000} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)\right)}{\mathsf{neg}\left(\left(1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}}\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \color{blue}{\frac{\frac{70711}{100000} \cdot \left(\mathsf{neg}\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)\right)\right)}{\mathsf{neg}\left(\left(1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \color{blue}{\frac{\frac{70711}{100000} \cdot \left(\mathsf{neg}\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)\right)\right)}{\mathsf{neg}\left(\left(1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}}\right) \]

   3. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, 0.70711, \frac{0.70711 \cdot \mathsf{fma}\left(-0.27061, x, -2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.04481, x, -0.99229\right), x, -1\right)}\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{16316775383}{10000000000} + x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right)} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{16316775383}{10000000000} + \color{blue}{x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{16316775383}{10000000000} + x \cdot \color{blue}{\left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{16316775383}{10000000000} + x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \color{blue}{\frac{2134856267379707}{1000000000000000}}\right) \]

      4. lower-*.f64 50.7

         \[\leadsto 1.6316775383 + x \cdot \left(1.3436228731669864 \cdot x - 2.134856267379707\right) \]

   6. Applied rewrites 50.7%

      \[\leadsto \color{blue}{1.6316775383 + x \cdot \left(1.3436228731669864 \cdot x - 2.134856267379707\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{16316775383}{10000000000} + x \cdot \frac{-2134856267379707}{1000000000000000} \]
   8. Step-by-step derivation

   9. Applied rewrites 57.6%

      \[\leadsto 1.6316775383 + x \cdot -2.134856267379707 \]
   10. Step-by-step derivation

       1. lift-+.f64 N/A

          \[\leadsto \frac{16316775383}{10000000000} + \color{blue}{x \cdot \frac{-2134856267379707}{1000000000000000}} \]

       2. +-commutative N/A

          \[\leadsto x \cdot \frac{-2134856267379707}{1000000000000000} + \color{blue}{\frac{16316775383}{10000000000}} \]

       3. lift-*.f64 N/A

          \[\leadsto x \cdot \frac{-2134856267379707}{1000000000000000} + \frac{16316775383}{10000000000} \]

       4. *-commutative N/A

          \[\leadsto \frac{-2134856267379707}{1000000000000000} \cdot x + \frac{16316775383}{10000000000} \]

       5. lower-fma.f64 57.6

          \[\leadsto \mathsf{fma}\left(-2.134856267379707, \color{blue}{x}, 1.6316775383\right) \]

   11. Applied rewrites 57.6%

       \[\leadsto \mathsf{fma}\left(-2.134856267379707, \color{blue}{x}, 1.6316775383\right) \]

   if 2 < (*.f64 #s(literal 70711/100000 binary64) (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x))

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

      3. sub-flip N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}\right)} \]

      5. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{70711}{100000} + \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{70711}{100000}, \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, \frac{70711}{100000}, \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \color{blue}{\frac{70711}{100000} \cdot \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \frac{70711}{100000} \cdot \color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}}\right) \]

      10. frac-2neg N/A

          \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \frac{70711}{100000} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)\right)}{\mathsf{neg}\left(\left(1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}}\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \color{blue}{\frac{\frac{70711}{100000} \cdot \left(\mathsf{neg}\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)\right)\right)}{\mathsf{neg}\left(\left(1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \color{blue}{\frac{\frac{70711}{100000} \cdot \left(\mathsf{neg}\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)\right)\right)}{\mathsf{neg}\left(\left(1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}}\right) \]

   3. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, 0.70711, \frac{0.70711 \cdot \mathsf{fma}\left(-0.27061, x, -2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.04481, x, -0.99229\right), x, -1\right)}\right)} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{70711}{100000} + \frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{70711}{100000} - \left(\mathsf{neg}\left(\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right)\right)} \]

      3. sub-flip N/A

         \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{70711}{100000} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right)\right)\right)\right)} \]

      4. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \frac{70711}{100000} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{70711}{100000}\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right)\right)\right)\right) \]

      7. lift-/.f64 N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}}\right)\right)\right)\right) \]

      8. distribute-neg-frac N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{\mathsf{neg}\left(\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}}\right)\right) \]

      9. distribute-frac-neg2 N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right) + \color{blue}{\frac{\mathsf{neg}\left(\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)\right)}{\mathsf{neg}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)\right)}} \]

      10. frac-2neg N/A

          \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right) + \color{blue}{\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}} \]

      11. lift-/.f64 N/A

          \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right) + \color{blue}{\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}} \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{neg}\left(\frac{70711}{100000}\right), \frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right)} \]

      13. metadata-eval 99.9

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{-0.70711}, \frac{0.70711 \cdot \mathsf{fma}\left(-0.27061, x, -2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.04481, x, -0.99229\right), x, -1\right)}\right) \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(-0.1913510371, x, -1.6316775383\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.04481, x, -0.99229\right), x, -1\right)}\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \color{blue}{\frac{\frac{1913510371}{448100000}}{x}}\right) \]
   7. Step-by-step derivation

      1. lower-/.f64 51.9

         \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175}{\color{blue}{x}}\right) \]

   8. Applied rewrites 51.9%

      \[\leadsto \mathsf{fma}\left(x, -0.70711, \color{blue}{\frac{4.2702753202410175}{x}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right)\\ t_1 := \mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175}{x}\right)\\ \mathbf{if}\;t\_0 \leq -40:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-2.134856267379707, x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (*
          0.70711
          (-
           (/
            (+ 2.30753 (* x 0.27061))
            (+ 1.0 (* x (+ 0.99229 (* x 0.04481)))))
           x)))
        (t_1 (fma x -0.70711 (/ 4.2702753202410175 x))))
   (if (<= t_0 -40.0)
     t_1
     (if (<= t_0 2.0) (fma -2.134856267379707 x 1.6316775383) t_1))))

C

double code(double x) {
	double t_0 = 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
	double t_1 = fma(x, -0.70711, (4.2702753202410175 / x));
	double tmp;
	if (t_0 <= -40.0) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = fma(-2.134856267379707, x, 1.6316775383);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.70711 * Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x))
	t_1 = fma(x, -0.70711, Float64(4.2702753202410175 / x))
	tmp = 0.0
	if (t_0 <= -40.0)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = fma(-2.134856267379707, x, 1.6316775383);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.70711 * N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * -0.70711 + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -40.0], t$95$1, If[LessEqual[t$95$0, 2.0], N[(-2.134856267379707 * x + 1.6316775383), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 70711/100000 binary64) (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x)) < -40 or 2 < (*.f64 #s(literal 70711/100000 binary64) (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x))

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

      3. sub-flip N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}\right)} \]

      5. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{70711}{100000} + \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{70711}{100000}, \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, \frac{70711}{100000}, \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \color{blue}{\frac{70711}{100000} \cdot \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \frac{70711}{100000} \cdot \color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}}\right) \]

      10. frac-2neg N/A

          \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \frac{70711}{100000} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)\right)}{\mathsf{neg}\left(\left(1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}}\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \color{blue}{\frac{\frac{70711}{100000} \cdot \left(\mathsf{neg}\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)\right)\right)}{\mathsf{neg}\left(\left(1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \color{blue}{\frac{\frac{70711}{100000} \cdot \left(\mathsf{neg}\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)\right)\right)}{\mathsf{neg}\left(\left(1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}}\right) \]

   3. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, 0.70711, \frac{0.70711 \cdot \mathsf{fma}\left(-0.27061, x, -2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.04481, x, -0.99229\right), x, -1\right)}\right)} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{70711}{100000} + \frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{70711}{100000} - \left(\mathsf{neg}\left(\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right)\right)} \]

      3. sub-flip N/A

         \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{70711}{100000} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right)\right)\right)\right)} \]

      4. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \frac{70711}{100000} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{70711}{100000}\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right)\right)\right)\right) \]

      7. lift-/.f64 N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}}\right)\right)\right)\right) \]

      8. distribute-neg-frac N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{\mathsf{neg}\left(\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}}\right)\right) \]

      9. distribute-frac-neg2 N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right) + \color{blue}{\frac{\mathsf{neg}\left(\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)\right)}{\mathsf{neg}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)\right)}} \]

      10. frac-2neg N/A

          \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right) + \color{blue}{\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}} \]

      11. lift-/.f64 N/A

          \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right) + \color{blue}{\frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}} \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{neg}\left(\frac{70711}{100000}\right), \frac{\frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{-27061}{100000}, x, \frac{-230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right)} \]

      13. metadata-eval 99.9

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{-0.70711}, \frac{0.70711 \cdot \mathsf{fma}\left(-0.27061, x, -2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.04481, x, -0.99229\right), x, -1\right)}\right) \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(-0.1913510371, x, -1.6316775383\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.04481, x, -0.99229\right), x, -1\right)}\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \color{blue}{\frac{\frac{1913510371}{448100000}}{x}}\right) \]
   7. Step-by-step derivation

      1. lower-/.f64 51.9

         \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175}{\color{blue}{x}}\right) \]

   8. Applied rewrites 51.9%

      \[\leadsto \mathsf{fma}\left(x, -0.70711, \color{blue}{\frac{4.2702753202410175}{x}}\right) \]

   if -40 < (*.f64 #s(literal 70711/100000 binary64) (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x)) < 2

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

      3. sub-flip N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}\right)} \]

      5. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{70711}{100000} + \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{70711}{100000}, \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, \frac{70711}{100000}, \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \color{blue}{\frac{70711}{100000} \cdot \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \frac{70711}{100000} \cdot \color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}}\right) \]

      10. frac-2neg N/A

          \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \frac{70711}{100000} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)\right)}{\mathsf{neg}\left(\left(1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}}\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \color{blue}{\frac{\frac{70711}{100000} \cdot \left(\mathsf{neg}\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)\right)\right)}{\mathsf{neg}\left(\left(1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \color{blue}{\frac{\frac{70711}{100000} \cdot \left(\mathsf{neg}\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)\right)\right)}{\mathsf{neg}\left(\left(1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}}\right) \]

   3. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, 0.70711, \frac{0.70711 \cdot \mathsf{fma}\left(-0.27061, x, -2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.04481, x, -0.99229\right), x, -1\right)}\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{16316775383}{10000000000} + x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right)} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{16316775383}{10000000000} + \color{blue}{x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{16316775383}{10000000000} + x \cdot \color{blue}{\left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{16316775383}{10000000000} + x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \color{blue}{\frac{2134856267379707}{1000000000000000}}\right) \]

      4. lower-*.f64 50.7

         \[\leadsto 1.6316775383 + x \cdot \left(1.3436228731669864 \cdot x - 2.134856267379707\right) \]

   6. Applied rewrites 50.7%

      \[\leadsto \color{blue}{1.6316775383 + x \cdot \left(1.3436228731669864 \cdot x - 2.134856267379707\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{16316775383}{10000000000} + x \cdot \frac{-2134856267379707}{1000000000000000} \]
   8. Step-by-step derivation

   9. Applied rewrites 57.6%

      \[\leadsto 1.6316775383 + x \cdot -2.134856267379707 \]
   10. Step-by-step derivation

       1. lift-+.f64 N/A

          \[\leadsto \frac{16316775383}{10000000000} + \color{blue}{x \cdot \frac{-2134856267379707}{1000000000000000}} \]

       2. +-commutative N/A

          \[\leadsto x \cdot \frac{-2134856267379707}{1000000000000000} + \color{blue}{\frac{16316775383}{10000000000}} \]

       3. lift-*.f64 N/A

          \[\leadsto x \cdot \frac{-2134856267379707}{1000000000000000} + \frac{16316775383}{10000000000} \]

       4. *-commutative N/A

          \[\leadsto \frac{-2134856267379707}{1000000000000000} \cdot x + \frac{16316775383}{10000000000} \]

       5. lower-fma.f64 57.6

          \[\leadsto \mathsf{fma}\left(-2.134856267379707, \color{blue}{x}, 1.6316775383\right) \]

   11. Applied rewrites 57.6%

       \[\leadsto \mathsf{fma}\left(-2.134856267379707, \color{blue}{x}, 1.6316775383\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 98.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;\mathsf{fma}\left(-2.134856267379707, x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;-0.70711 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (* -0.70711 x)
   (if (<= x 1.15) (fma -2.134856267379707 x 1.6316775383) (* -0.70711 x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = -0.70711 * x;
	} else if (x <= 1.15) {
		tmp = fma(-2.134856267379707, x, 1.6316775383);
	} else {
		tmp = -0.70711 * x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(-0.70711 * x);
	elseif (x <= 1.15)
		tmp = fma(-2.134856267379707, x, 1.6316775383);
	else
		tmp = Float64(-0.70711 * x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(-0.70711 * x), $MachinePrecision], If[LessEqual[x, 1.15], N[(-2.134856267379707 * x + 1.6316775383), $MachinePrecision], N[(-0.70711 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.1499999999999999 < x

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-70711}{100000} \cdot x} \]
   3. Step-by-step derivation

      1. lower-*.f64 52.0

         \[\leadsto -0.70711 \cdot \color{blue}{x} \]

   4. Applied rewrites 52.0%

      \[\leadsto \color{blue}{-0.70711 \cdot x} \]

   if -1.05000000000000004 < x < 1.1499999999999999

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

      3. sub-flip N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}\right)} \]

      5. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{70711}{100000} + \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{70711}{100000}, \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, \frac{70711}{100000}, \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \color{blue}{\frac{70711}{100000} \cdot \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \frac{70711}{100000} \cdot \color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}}\right) \]

      10. frac-2neg N/A

          \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \frac{70711}{100000} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)\right)}{\mathsf{neg}\left(\left(1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}}\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \color{blue}{\frac{\frac{70711}{100000} \cdot \left(\mathsf{neg}\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)\right)\right)}{\mathsf{neg}\left(\left(1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(-x, \frac{70711}{100000}, \color{blue}{\frac{\frac{70711}{100000} \cdot \left(\mathsf{neg}\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)\right)\right)}{\mathsf{neg}\left(\left(1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}}\right) \]

   3. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, 0.70711, \frac{0.70711 \cdot \mathsf{fma}\left(-0.27061, x, -2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.04481, x, -0.99229\right), x, -1\right)}\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{16316775383}{10000000000} + x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right)} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{16316775383}{10000000000} + \color{blue}{x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{16316775383}{10000000000} + x \cdot \color{blue}{\left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{16316775383}{10000000000} + x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \color{blue}{\frac{2134856267379707}{1000000000000000}}\right) \]

      4. lower-*.f64 50.7

         \[\leadsto 1.6316775383 + x \cdot \left(1.3436228731669864 \cdot x - 2.134856267379707\right) \]

   6. Applied rewrites 50.7%

      \[\leadsto \color{blue}{1.6316775383 + x \cdot \left(1.3436228731669864 \cdot x - 2.134856267379707\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{16316775383}{10000000000} + x \cdot \frac{-2134856267379707}{1000000000000000} \]
   8. Step-by-step derivation

   9. Applied rewrites 57.6%

      \[\leadsto 1.6316775383 + x \cdot -2.134856267379707 \]
   10. Step-by-step derivation

       1. lift-+.f64 N/A

          \[\leadsto \frac{16316775383}{10000000000} + \color{blue}{x \cdot \frac{-2134856267379707}{1000000000000000}} \]

       2. +-commutative N/A

          \[\leadsto x \cdot \frac{-2134856267379707}{1000000000000000} + \color{blue}{\frac{16316775383}{10000000000}} \]

       3. lift-*.f64 N/A

          \[\leadsto x \cdot \frac{-2134856267379707}{1000000000000000} + \frac{16316775383}{10000000000} \]

       4. *-commutative N/A

          \[\leadsto \frac{-2134856267379707}{1000000000000000} \cdot x + \frac{16316775383}{10000000000} \]

       5. lower-fma.f64 57.6

          \[\leadsto \mathsf{fma}\left(-2.134856267379707, \color{blue}{x}, 1.6316775383\right) \]

   11. Applied rewrites 57.6%

       \[\leadsto \mathsf{fma}\left(-2.134856267379707, \color{blue}{x}, 1.6316775383\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 98.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4:\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;-0.70711 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.4) (* -0.70711 x) (if (<= x 1.2) 1.6316775383 (* -0.70711 x))))

C

double code(double x) {
	double tmp;
	if (x <= -3.4) {
		tmp = -0.70711 * x;
	} else if (x <= 1.2) {
		tmp = 1.6316775383;
	} else {
		tmp = -0.70711 * x;
	}
	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(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.4d0)) then
        tmp = (-0.70711d0) * x
    else if (x <= 1.2d0) then
        tmp = 1.6316775383d0
    else
        tmp = (-0.70711d0) * x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.4) {
		tmp = -0.70711 * x;
	} else if (x <= 1.2) {
		tmp = 1.6316775383;
	} else {
		tmp = -0.70711 * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.4:
		tmp = -0.70711 * x
	elif x <= 1.2:
		tmp = 1.6316775383
	else:
		tmp = -0.70711 * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.4)
		tmp = Float64(-0.70711 * x);
	elseif (x <= 1.2)
		tmp = 1.6316775383;
	else
		tmp = Float64(-0.70711 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.4)
		tmp = -0.70711 * x;
	elseif (x <= 1.2)
		tmp = 1.6316775383;
	else
		tmp = -0.70711 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.4], N[(-0.70711 * x), $MachinePrecision], If[LessEqual[x, 1.2], 1.6316775383, N[(-0.70711 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.39999999999999991 or 1.19999999999999996 < x

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-70711}{100000} \cdot x} \]
   3. Step-by-step derivation

      1. lower-*.f64 52.0

         \[\leadsto -0.70711 \cdot \color{blue}{x} \]

   4. Applied rewrites 52.0%

      \[\leadsto \color{blue}{-0.70711 \cdot x} \]

   if -3.39999999999999991 < x < 1.19999999999999996

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{16316775383}{10000000000}} \]
   3. Step-by-step derivation

   4. Applied rewrites 49.7%

      \[\leadsto \color{blue}{1.6316775383} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 49.7% accurate, 27.0× speedup

Math

\[\begin{array}{l} \\ 1.6316775383 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.6316775383)

C

double code(double x) {
	return 1.6316775383;
}

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(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = 1.6316775383d0
end function

Java

public static double code(double x) {
	return 1.6316775383;
}

Python

def code(x):
	return 1.6316775383

Julia

function code(x)
	return 1.6316775383
end

MATLAB

function tmp = code(x)
	tmp = 1.6316775383;
end

Wolfram

code[x_] := 1.6316775383

Derivation

1. Initial program 99.9%

   \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{16316775383}{10000000000}} \]
3. Step-by-step derivation

4. Applied rewrites 49.7%

   \[\leadsto \color{blue}{1.6316775383} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025155 
(FPCore (x)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, B"
  :precision binary64
  (* 0.70711 (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x)))