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

Percentage Accurate: 99.9% → 99.9%

Time: 3.0s

Alternatives: 9

Speedup: 1.1×

Specification

Math

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

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

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

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, 1.1× speedup

Math

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

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

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. distribute-rgt-in N/A

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

   5. associate-+r+ N/A

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

   6. *-commutative N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. sqr-neg-rev N/A

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

   12. lower-*.f32 N/A

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

   13. lower-unsound-*.f32 N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-unsound-*.f32 N/A

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

   16. lower-*.f32 N/A

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

   17. sqr-neg-rev N/A

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

   18. lower-*.f64 N/A

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

   19. +-commutative N/A

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

   20. *-commutative N/A

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

   21. lower-fma.f64 99.9%

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

3. Applied rewrites 99.9%

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

5. Applied rewrites 99.9%

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

6. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{\frac{-1913510371}{10000000000} \cdot x - \frac{16316775383}{10000000000}}}{\mathsf{fma}\left(\mathsf{fma}\left(-0.04481, x, -0.99229\right), x, -1\right)} - x \cdot 0.70711 \]
7. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \frac{\frac{-1913510371}{10000000000} \cdot x - \color{blue}{\frac{16316775383}{10000000000}}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)} - x \cdot \frac{70711}{100000} \]

   2. lower-*.f64 99.9%

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

8. Applied rewrites 99.9%

   \[\leadsto \frac{\color{blue}{-0.1913510371 \cdot x - 1.6316775383}}{\mathsf{fma}\left(\mathsf{fma}\left(-0.04481, x, -0.99229\right), x, -1\right)} - x \cdot 0.70711 \]
9. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-1913510371}{10000000000} \cdot x - \frac{16316775383}{10000000000}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)} - x \cdot \frac{70711}{100000}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\frac{-1913510371}{10000000000} \cdot x - \frac{16316775383}{10000000000}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)} - \color{blue}{x \cdot \frac{70711}{100000}} \]

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

      \[\leadsto \color{blue}{\frac{\frac{-1913510371}{10000000000} \cdot x - \frac{16316775383}{10000000000}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)} + \left(\mathsf{neg}\left(x\right)\right) \cdot \frac{70711}{100000}} \]

   4. +-commutative N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{70711}{100000} + \frac{\frac{-1913510371}{10000000000} \cdot x - \frac{16316775383}{10000000000}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}} \]

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

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{70711}{100000}\right)\right)} + \frac{\frac{-1913510371}{10000000000} \cdot x - \frac{16316775383}{10000000000}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)} \]

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

      \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right)} + \frac{\frac{-1913510371}{10000000000} \cdot x - \frac{16316775383}{10000000000}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{neg}\left(\frac{70711}{100000}\right), \frac{\frac{-1913510371}{10000000000} \cdot x - \frac{16316775383}{10000000000}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right)} \]

   8. metadata-eval 99.9%

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

   9. lift--.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\frac{-1913510371}{10000000000} \cdot x - \color{blue}{\frac{16316775383}{10000000000}}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right) \]

   10. sub-flip N/A

       \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\frac{-1913510371}{10000000000} \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{16316775383}{10000000000}\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right) \]

   11. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\frac{-1913510371}{10000000000} \cdot x + \left(\mathsf{neg}\left(\color{blue}{\frac{16316775383}{10000000000}}\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right) \]

   12. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\mathsf{fma}\left(\frac{-1913510371}{10000000000}, \color{blue}{x}, \mathsf{neg}\left(\frac{16316775383}{10000000000}\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right) \]

   13. metadata-eval 99.9%

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

10. Applied rewrites 99.9%

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

Alternative 2: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := 0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right)\\ t_1 := \left(\frac{6.039053782637804 - \frac{82.23527511657367}{x}}{x} - x\right) \cdot 0.70711\\ \mathbf{if}\;t\_0 \leq -200000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.2692862305735844, x, 1.3436228731669864\right), x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

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
         (*
          (- (/ (- 6.039053782637804 (/ 82.23527511657367 x)) x) x)
          0.70711)))
   (if (<= t_0 -200000.0)
     t_1
     (if (<= t_0 2.0)
       (fma
        (fma
         (fma -1.2692862305735844 x 1.3436228731669864)
         x
         -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 = (((6.039053782637804 - (82.23527511657367 / x)) / x) - x) * 0.70711;
	double tmp;
	if (t_0 <= -200000.0) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = fma(fma(fma(-1.2692862305735844, x, 1.3436228731669864), x, -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 = Float64(Float64(Float64(Float64(6.039053782637804 - Float64(82.23527511657367 / x)) / x) - x) * 0.70711)
	tmp = 0.0
	if (t_0 <= -200000.0)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = fma(fma(fma(-1.2692862305735844, x, 1.3436228731669864), x, -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[(N[(N[(N[(6.039053782637804 - N[(82.23527511657367 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - x), $MachinePrecision] * 0.70711), $MachinePrecision]}, If[LessEqual[t$95$0, -200000.0], t$95$1, If[LessEqual[t$95$0, 2.0], N[(N[(N[(-1.2692862305735844 * x + 1.3436228731669864), $MachinePrecision] * x + -2.134856267379707), $MachinePrecision] * 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)) < -2e5 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

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. sqr-neg-rev N/A

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

      12. lower-*.f32 N/A

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

      13. lower-unsound-*.f32 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-unsound-*.f32 N/A

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

      16. lower-*.f32 N/A

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

      17. sqr-neg-rev N/A

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

      18. lower-*.f64 N/A

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

      19. +-commutative N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 99.9%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around inf

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\frac{\frac{27061}{4481} - \frac{1651231776}{20079361} \cdot \frac{1}{x}}{x}} - x\right) \]
   5. 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 51.0%

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

   6. Applied rewrites 51.0%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\frac{6.039053782637804 - 82.23527511657367 \cdot \frac{1}{x}}{x}} - x\right) \]
   7. 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 51.0%

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

      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 51.0%

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

   8. Applied rewrites 51.0%

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

   if -2e5 < (*.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 51.9%

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

   4. Applied rewrites 51.9%

      \[\leadsto \color{blue}{1.6316775383 + x \cdot \left(x \cdot \left(1.3436228731669864 + -1.2692862305735844 \cdot x\right) - 2.134856267379707\right)} \]
   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 51.9%

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

      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 51.9%

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

   6. Applied rewrites 51.9%

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

Alternative 3: 99.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := 0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{if}\;x \leq -2.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 760:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.2692862305735844, x, 1.3436228731669864\right), x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 0.70711 (- (/ 6.039053782637804 x) x))))
   (if (<= x -2.5)
     t_0
     (if (<= x 760.0)
       (fma
        (fma
         (fma -1.2692862305735844 x 1.3436228731669864)
         x
         -2.134856267379707)
        x
        1.6316775383)
       t_0))))

C

double code(double x) {
	double t_0 = 0.70711 * ((6.039053782637804 / x) - x);
	double tmp;
	if (x <= -2.5) {
		tmp = t_0;
	} else if (x <= 760.0) {
		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(0.70711 * Float64(Float64(6.039053782637804 / x) - x))
	tmp = 0.0
	if (x <= -2.5)
		tmp = t_0;
	elseif (x <= 760.0)
		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[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.5], t$95$0, If[LessEqual[x, 760.0], 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 < -2.5 or 760 < 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

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\frac{\frac{27061}{4481}}{x}} - x\right) \]
   3. Step-by-step derivation

      1. lower-/.f64 52.1%

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

   4. Applied rewrites 52.1%

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

   if -2.5 < x < 760

   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 51.9%

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

   4. Applied rewrites 51.9%

      \[\leadsto \color{blue}{1.6316775383 + x \cdot \left(x \cdot \left(1.3436228731669864 + -1.2692862305735844 \cdot x\right) - 2.134856267379707\right)} \]
   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 51.9%

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

      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 51.9%

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

   6. Applied rewrites 51.9%

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

Alternative 4: 99.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := 0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 0.70711 (- (/ 6.039053782637804 x) x))))
   (if (<= x -2.6)
     t_0
     (if (<= x 1.45)
       (fma (fma 1.3436228731669864 x -2.134856267379707) x 1.6316775383)
       t_0))))

C

double code(double x) {
	double t_0 = 0.70711 * ((6.039053782637804 / x) - x);
	double tmp;
	if (x <= -2.6) {
		tmp = t_0;
	} else if (x <= 1.45) {
		tmp = fma(fma(1.3436228731669864, x, -2.134856267379707), x, 1.6316775383);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x))
	tmp = 0.0
	if (x <= -2.6)
		tmp = t_0;
	elseif (x <= 1.45)
		tmp = fma(fma(1.3436228731669864, x, -2.134856267379707), x, 1.6316775383);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.6], t$95$0, If[LessEqual[x, 1.45], N[(N[(1.3436228731669864 * x + -2.134856267379707), $MachinePrecision] * x + 1.6316775383), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.60000000000000009 or 1.44999999999999996 < 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

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\frac{\frac{27061}{4481}}{x}} - x\right) \]
   3. Step-by-step derivation

      1. lower-/.f64 52.1%

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

   4. Applied rewrites 52.1%

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

   if -2.60000000000000009 < x < 1.44999999999999996

   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

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. sqr-neg-rev N/A

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

      12. lower-*.f32 N/A

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

      13. lower-unsound-*.f32 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-unsound-*.f32 N/A

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

      16. lower-*.f32 N/A

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

      17. sqr-neg-rev N/A

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

      18. lower-*.f64 N/A

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

      19. +-commutative N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 99.9%

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

   3. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9%

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{16316775383}{10000000000} + x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right)} \]
   7. 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 50.6%

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

   8. Applied rewrites 50.6%

      \[\leadsto \color{blue}{1.6316775383 + x \cdot \left(1.3436228731669864 \cdot x - 2.134856267379707\right)} \]
   9. 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 50.6%

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

      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. metadata-eval N/A

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

      10. lower-fma.f64 50.6%

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

   10. Applied rewrites 50.6%

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

Alternative 5: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := 0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{if}\;x \leq -2.7:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 760:\\ \;\;\;\;\mathsf{fma}\left(x, -2.134856267379707, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 0.70711 (- (/ 6.039053782637804 x) x))))
   (if (<= x -2.7)
     t_0
     (if (<= x 760.0) (fma x -2.134856267379707 1.6316775383) t_0))))

C

double code(double x) {
	double t_0 = 0.70711 * ((6.039053782637804 / x) - x);
	double tmp;
	if (x <= -2.7) {
		tmp = t_0;
	} else if (x <= 760.0) {
		tmp = fma(x, -2.134856267379707, 1.6316775383);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x))
	tmp = 0.0
	if (x <= -2.7)
		tmp = t_0;
	elseif (x <= 760.0)
		tmp = fma(x, -2.134856267379707, 1.6316775383);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.7], t$95$0, If[LessEqual[x, 760.0], N[(x * -2.134856267379707 + 1.6316775383), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.7000000000000002 or 760 < 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

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\frac{\frac{27061}{4481}}{x}} - x\right) \]
   3. Step-by-step derivation

      1. lower-/.f64 52.1%

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

   4. Applied rewrites 52.1%

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

   if -2.7000000000000002 < x < 760

   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

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. sqr-neg-rev N/A

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

      12. lower-*.f32 N/A

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

      13. lower-unsound-*.f32 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-unsound-*.f32 N/A

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

      16. lower-*.f32 N/A

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

      17. sqr-neg-rev N/A

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

      18. lower-*.f64 N/A

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

      19. +-commutative N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 99.9%

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

   3. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9%

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 57.5%

         \[\leadsto 1.6316775383 + -2.134856267379707 \cdot \color{blue}{x} \]

   8. Applied rewrites 57.5%

      \[\leadsto \color{blue}{1.6316775383 + -2.134856267379707 \cdot x} \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 57.5%

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

   10. Applied rewrites 57.5%

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

Alternative 6: 98.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{elif}\;x \leq 0.76:\\ \;\;\;\;\mathsf{fma}\left(x, -2.134856267379707, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;-0.70711 \cdot x\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (* -0.70711 x)
   (if (<= x 0.76) (fma x -2.134856267379707 1.6316775383) (* -0.70711 x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = -0.70711 * x;
	} else if (x <= 0.76) {
		tmp = fma(x, -2.134856267379707, 1.6316775383);
	} else {
		tmp = -0.70711 * x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(-0.70711 * x);
	elseif (x <= 0.76)
		tmp = fma(x, -2.134856267379707, 1.6316775383);
	else
		tmp = Float64(-0.70711 * x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], N[(-0.70711 * x), $MachinePrecision], If[LessEqual[x, 0.76], N[(x * -2.134856267379707 + 1.6316775383), $MachinePrecision], N[(-0.70711 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.76000000000000001 < 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 52.2%

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

   4. Applied rewrites 52.2%

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

   if -1 < x < 0.76000000000000001

   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

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. sqr-neg-rev N/A

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

      12. lower-*.f32 N/A

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

      13. lower-unsound-*.f32 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-unsound-*.f32 N/A

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

      16. lower-*.f32 N/A

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

      17. sqr-neg-rev N/A

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

      18. lower-*.f64 N/A

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

      19. +-commutative N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 99.9%

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

   3. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9%

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 57.5%

         \[\leadsto 1.6316775383 + -2.134856267379707 \cdot \color{blue}{x} \]

   8. Applied rewrites 57.5%

      \[\leadsto \color{blue}{1.6316775383 + -2.134856267379707 \cdot x} \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 57.5%

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

   10. Applied rewrites 57.5%

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

Alternative 7: 98.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. lower-*.f64 99.9%

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 99.9%

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 99.9%

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f64 99.9%

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

3. Applied rewrites 99.9%

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

4. Taylor expanded in x around 0

   \[\leadsto \left(\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\color{blue}{\frac{99229}{100000}}, x, 1\right)} - x\right) \cdot 0.70711 \]
5. Step-by-step derivation

6. Applied rewrites 98.5%

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

Alternative 8: 98.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -64000:\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{elif}\;x \leq 1.65:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;-0.70711 \cdot x\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -64000.0)
   (* -0.70711 x)
   (if (<= x 1.65) 1.6316775383 (* -0.70711 x))))

C

double code(double x) {
	double tmp;
	if (x <= -64000.0) {
		tmp = -0.70711 * x;
	} else if (x <= 1.65) {
		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 <= (-64000.0d0)) then
        tmp = (-0.70711d0) * x
    else if (x <= 1.65d0) 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 <= -64000.0) {
		tmp = -0.70711 * x;
	} else if (x <= 1.65) {
		tmp = 1.6316775383;
	} else {
		tmp = -0.70711 * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -64000.0:
		tmp = -0.70711 * x
	elif x <= 1.65:
		tmp = 1.6316775383
	else:
		tmp = -0.70711 * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -64000.0)
		tmp = Float64(-0.70711 * x);
	elseif (x <= 1.65)
		tmp = 1.6316775383;
	else
		tmp = Float64(-0.70711 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -64000.0)
		tmp = -0.70711 * x;
	elseif (x <= 1.65)
		tmp = 1.6316775383;
	else
		tmp = -0.70711 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -64000.0], N[(-0.70711 * x), $MachinePrecision], If[LessEqual[x, 1.65], 1.6316775383, N[(-0.70711 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -64000 or 1.6499999999999999 < 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 52.2%

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

   4. Applied rewrites 52.2%

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

   if -64000 < x < 1.6499999999999999

   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 49.6%

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

Alternative 9: 49.6% accurate, 27.0× speedup

Math

\[1.6316775383 \]

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 49.6%

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

Reproduce

herbie shell --seed 2025183 
(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)))