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

Percentage Accurate: 99.8% → 99.9%

Time: 3.2s

Alternatives: 12

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: 99.8% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
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 99.8%

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

   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 99.8%

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

   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 99.8%

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

   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 99.8%

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

3. Applied rewrites 99.8%

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

Alternative 2: 99.8% 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 99.8%

   \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
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 99.8%

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

   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 99.8%

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

   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 99.8%

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

   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 99.8%

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

3. Applied rewrites 99.8%

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

5. Applied rewrites 99.9%

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

Alternative 3: 99.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -2.5:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;x \leq 600:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot \left(1.3436228731669864 + -1.2692862305735844 \cdot x\right) - 2.134856267379707\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175 - \frac{58.14938538768042}{x}}{x}\right)\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= x -2.5)
  (* 0.70711 (- (/ 6.039053782637804 x) x))
  (if (<= x 600.0)
    (+
     1.6316775383
     (*
      x
      (-
       (* x (+ 1.3436228731669864 (* -1.2692862305735844 x)))
       2.134856267379707)))
    (fma
     x
     -0.70711
     (/ (- 4.2702753202410175 (/ 58.14938538768042 x)) x)))))

C

double code(double x) {
	double tmp;
	if (x <= -2.5) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else if (x <= 600.0) {
		tmp = 1.6316775383 + (x * ((x * (1.3436228731669864 + (-1.2692862305735844 * x))) - 2.134856267379707));
	} else {
		tmp = fma(x, -0.70711, ((4.2702753202410175 - (58.14938538768042 / x)) / x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.5)
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	elseif (x <= 600.0)
		tmp = Float64(1.6316775383 + Float64(x * Float64(Float64(x * Float64(1.3436228731669864 + Float64(-1.2692862305735844 * x))) - 2.134856267379707)));
	else
		tmp = fma(x, -0.70711, Float64(Float64(4.2702753202410175 - Float64(58.14938538768042 / x)) / x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -2.5], N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 600.0], N[(1.6316775383 + N[(x * N[(N[(x * N[(1.3436228731669864 + N[(-1.2692862305735844 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.134856267379707), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * -0.70711 + N[(N[(4.2702753202410175 - N[(58.14938538768042 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.5

   1. Initial program 99.8%

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

   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 51.0%

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

   4. Applied rewrites 51.0%

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

   if -2.5 < x < 600

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 52.9%

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

   4. Applied rewrites 52.9%

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

   if 600 < x

   1. Initial program 99.8%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   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 99.8%

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

      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 99.8%

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

      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 99.8%

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

      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 99.8%

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

   3. Applied rewrites 99.8%

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(x, -0.70711, \color{blue}{\frac{\frac{1913510371}{448100000} - \frac{3648757816023}{62748003125} \cdot \frac{1}{x}}{x}}\right) \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\frac{1913510371}{448100000} - \frac{3648757816023}{62748003125} \cdot \frac{1}{x}}{\color{blue}{x}}\right) \]

      2. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\frac{1913510371}{448100000} - \frac{3648757816023}{62748003125} \cdot \frac{1}{x}}{x}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\frac{1913510371}{448100000} - \frac{3648757816023}{62748003125} \cdot \frac{1}{x}}{x}\right) \]

      4. lower-/.f64 49.9%

         \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175 - 58.14938538768042 \cdot \frac{1}{x}}{x}\right) \]

   8. Applied rewrites 49.9%

      \[\leadsto \mathsf{fma}\left(x, -0.70711, \color{blue}{\frac{4.2702753202410175 - 58.14938538768042 \cdot \frac{1}{x}}{x}}\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\frac{1913510371}{448100000} - \frac{3648757816023}{62748003125} \cdot \frac{1}{x}}{x}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\frac{1913510371}{448100000} - \frac{3648757816023}{62748003125} \cdot \frac{1}{x}}{x}\right) \]

      3. mult-flip-rev N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\frac{1913510371}{448100000} - \frac{\frac{3648757816023}{62748003125}}{x}}{x}\right) \]

      4. lower-/.f64 49.9%

         \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175 - \frac{58.14938538768042}{x}}{x}\right) \]

   10. Applied rewrites 49.9%

       \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175 - \frac{58.14938538768042}{x}}{x}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;x \leq 75000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175 - \frac{58.14938538768042}{x}}{x}\right)\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= x -2.6)
  (* 0.70711 (- (/ 6.039053782637804 x) x))
  (if (<= x 75000.0)
    (fma (fma 1.3436228731669864 x -2.134856267379707) x 1.6316775383)
    (fma
     x
     -0.70711
     (/ (- 4.2702753202410175 (/ 58.14938538768042 x)) x)))))

C

double code(double x) {
	double tmp;
	if (x <= -2.6) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else if (x <= 75000.0) {
		tmp = fma(fma(1.3436228731669864, x, -2.134856267379707), x, 1.6316775383);
	} else {
		tmp = fma(x, -0.70711, ((4.2702753202410175 - (58.14938538768042 / x)) / x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.6)
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	elseif (x <= 75000.0)
		tmp = fma(fma(1.3436228731669864, x, -2.134856267379707), x, 1.6316775383);
	else
		tmp = fma(x, -0.70711, Float64(Float64(4.2702753202410175 - Float64(58.14938538768042 / x)) / x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -2.6], N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 75000.0], N[(N[(1.3436228731669864 * x + -2.134856267379707), $MachinePrecision] * x + 1.6316775383), $MachinePrecision], N[(x * -0.70711 + N[(N[(4.2702753202410175 - N[(58.14938538768042 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.6000000000000001

   1. Initial program 99.8%

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

   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 51.0%

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

   4. Applied rewrites 51.0%

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

   if -2.6000000000000001 < x < 75000

   1. Initial program 99.8%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   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 99.8%

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

      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 99.8%

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

      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 99.8%

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

      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 99.8%

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

   3. Applied rewrites 99.8%

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 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 51.5%

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

   8. Applied rewrites 51.5%

      \[\leadsto \color{blue}{1.6316775383 + x \cdot \left(1.3436228731669864 \cdot x - 2.134856267379707\right)} \]
   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 51.5%

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

      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 51.5%

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

   10. Applied rewrites 51.5%

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

   if 75000 < x

   1. Initial program 99.8%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   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 99.8%

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

      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 99.8%

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

      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 99.8%

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

      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 99.8%

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

   3. Applied rewrites 99.8%

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(x, -0.70711, \color{blue}{\frac{\frac{1913510371}{448100000} - \frac{3648757816023}{62748003125} \cdot \frac{1}{x}}{x}}\right) \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\frac{1913510371}{448100000} - \frac{3648757816023}{62748003125} \cdot \frac{1}{x}}{\color{blue}{x}}\right) \]

      2. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\frac{1913510371}{448100000} - \frac{3648757816023}{62748003125} \cdot \frac{1}{x}}{x}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\frac{1913510371}{448100000} - \frac{3648757816023}{62748003125} \cdot \frac{1}{x}}{x}\right) \]

      4. lower-/.f64 49.9%

         \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175 - 58.14938538768042 \cdot \frac{1}{x}}{x}\right) \]

   8. Applied rewrites 49.9%

      \[\leadsto \mathsf{fma}\left(x, -0.70711, \color{blue}{\frac{4.2702753202410175 - 58.14938538768042 \cdot \frac{1}{x}}{x}}\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\frac{1913510371}{448100000} - \frac{3648757816023}{62748003125} \cdot \frac{1}{x}}{x}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\frac{1913510371}{448100000} - \frac{3648757816023}{62748003125} \cdot \frac{1}{x}}{x}\right) \]

      3. mult-flip-rev N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\frac{1913510371}{448100000} - \frac{\frac{3648757816023}{62748003125}}{x}}{x}\right) \]

      4. lower-/.f64 49.9%

         \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175 - \frac{58.14938538768042}{x}}{x}\right) \]

   10. Applied rewrites 49.9%

       \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175 - \frac{58.14938538768042}{x}}{x}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;x \leq 75000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{4.2702753202410175}{x} - x \cdot 0.70711\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= x -2.6)
  (* 0.70711 (- (/ 6.039053782637804 x) x))
  (if (<= x 75000.0)
    (fma (fma 1.3436228731669864 x -2.134856267379707) x 1.6316775383)
    (- (/ 4.2702753202410175 x) (* x 0.70711)))))

C

double code(double x) {
	double tmp;
	if (x <= -2.6) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else if (x <= 75000.0) {
		tmp = fma(fma(1.3436228731669864, x, -2.134856267379707), x, 1.6316775383);
	} else {
		tmp = (4.2702753202410175 / x) - (x * 0.70711);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.6)
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	elseif (x <= 75000.0)
		tmp = fma(fma(1.3436228731669864, x, -2.134856267379707), x, 1.6316775383);
	else
		tmp = Float64(Float64(4.2702753202410175 / x) - Float64(x * 0.70711));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -2.6], N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 75000.0], N[(N[(1.3436228731669864 * x + -2.134856267379707), $MachinePrecision] * x + 1.6316775383), $MachinePrecision], N[(N[(4.2702753202410175 / x), $MachinePrecision] - N[(x * 0.70711), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.6000000000000001

   1. Initial program 99.8%

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

   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 51.0%

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

   4. Applied rewrites 51.0%

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

   if -2.6000000000000001 < x < 75000

   1. Initial program 99.8%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   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 99.8%

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

      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 99.8%

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

      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 99.8%

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

      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 99.8%

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

   3. Applied rewrites 99.8%

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 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 51.5%

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

   8. Applied rewrites 51.5%

      \[\leadsto \color{blue}{1.6316775383 + x \cdot \left(1.3436228731669864 \cdot x - 2.134856267379707\right)} \]
   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 51.5%

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

      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 51.5%

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

   10. Applied rewrites 51.5%

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

   if 75000 < x

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. distribute-rgt-in N/A

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

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

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

      6. lower--.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1913510371}{448100000}}{x}} - x \cdot 0.70711 \]
   5. Step-by-step derivation

      1. lower-/.f64 51.0%

         \[\leadsto \frac{4.2702753202410175}{\color{blue}{x}} - x \cdot 0.70711 \]

   6. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\frac{4.2702753202410175}{x}} - x \cdot 0.70711 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -2.8:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;x \leq 980:\\ \;\;\;\;\mathsf{fma}\left(-3.0191289437, x, 2.30753\right) \cdot 0.70711\\ \mathbf{else}:\\ \;\;\;\;\frac{4.2702753202410175}{x} - x \cdot 0.70711\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= x -2.8)
  (* 0.70711 (- (/ 6.039053782637804 x) x))
  (if (<= x 980.0)
    (* (fma -3.0191289437 x 2.30753) 0.70711)
    (- (/ 4.2702753202410175 x) (* x 0.70711)))))

C

double code(double x) {
	double tmp;
	if (x <= -2.8) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else if (x <= 980.0) {
		tmp = fma(-3.0191289437, x, 2.30753) * 0.70711;
	} else {
		tmp = (4.2702753202410175 / x) - (x * 0.70711);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.8)
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	elseif (x <= 980.0)
		tmp = Float64(fma(-3.0191289437, x, 2.30753) * 0.70711);
	else
		tmp = Float64(Float64(4.2702753202410175 / x) - Float64(x * 0.70711));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -2.8], N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 980.0], N[(N[(-3.0191289437 * x + 2.30753), $MachinePrecision] * 0.70711), $MachinePrecision], N[(N[(4.2702753202410175 / x), $MachinePrecision] - N[(x * 0.70711), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.7999999999999998

   1. Initial program 99.8%

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

   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 51.0%

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

   4. Applied rewrites 51.0%

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

   if -2.7999999999999998 < x < 980

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

      \[\leadsto 0.70711 \cdot \color{blue}{\left(\frac{230753}{100000} + \frac{-30191289437}{10000000000} \cdot x\right)} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\frac{230753}{100000} + \color{blue}{\frac{-30191289437}{10000000000} \cdot x}\right) \]

      2. lower-*.f64 58.4%

         \[\leadsto 0.70711 \cdot \left(2.30753 + -3.0191289437 \cdot \color{blue}{x}\right) \]

   4. Applied rewrites 58.4%

      \[\leadsto 0.70711 \cdot \color{blue}{\left(2.30753 + -3.0191289437 \cdot x\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{70711}{100000} \cdot \left(\frac{230753}{100000} + \frac{-30191289437}{10000000000} \cdot x\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{230753}{100000} + \frac{-30191289437}{10000000000} \cdot x\right) \cdot \frac{70711}{100000}} \]

      3. lower-*.f64 58.4%

         \[\leadsto \color{blue}{\left(2.30753 + -3.0191289437 \cdot x\right) \cdot 0.70711} \]

      4. lift-+.f64 N/A

         \[\leadsto \left(\frac{230753}{100000} + \color{blue}{\frac{-30191289437}{10000000000} \cdot x}\right) \cdot \frac{70711}{100000} \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{-30191289437}{10000000000} \cdot x + \color{blue}{\frac{230753}{100000}}\right) \cdot \frac{70711}{100000} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{-30191289437}{10000000000} \cdot x + \frac{230753}{100000}\right) \cdot \frac{70711}{100000} \]

      7. lower-fma.f64 58.4%

         \[\leadsto \mathsf{fma}\left(-3.0191289437, \color{blue}{x}, 2.30753\right) \cdot 0.70711 \]

   6. Applied rewrites 58.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-3.0191289437, x, 2.30753\right) \cdot 0.70711} \]

   if 980 < x

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. distribute-rgt-in N/A

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

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

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

      6. lower--.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1913510371}{448100000}}{x}} - x \cdot 0.70711 \]
   5. Step-by-step derivation

      1. lower-/.f64 51.0%

         \[\leadsto \frac{4.2702753202410175}{\color{blue}{x}} - x \cdot 0.70711 \]

   6. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\frac{4.2702753202410175}{x}} - x \cdot 0.70711 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 98.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -2.8:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;x \leq 980:\\ \;\;\;\;\mathsf{fma}\left(-3.0191289437, x, 2.30753\right) \cdot 0.70711\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175}{x}\right)\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= x -2.8)
  (* 0.70711 (- (/ 6.039053782637804 x) x))
  (if (<= x 980.0)
    (* (fma -3.0191289437 x 2.30753) 0.70711)
    (fma x -0.70711 (/ 4.2702753202410175 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -2.8) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else if (x <= 980.0) {
		tmp = fma(-3.0191289437, x, 2.30753) * 0.70711;
	} else {
		tmp = fma(x, -0.70711, (4.2702753202410175 / x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.8)
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	elseif (x <= 980.0)
		tmp = Float64(fma(-3.0191289437, x, 2.30753) * 0.70711);
	else
		tmp = fma(x, -0.70711, Float64(4.2702753202410175 / x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -2.8], N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 980.0], N[(N[(-3.0191289437 * x + 2.30753), $MachinePrecision] * 0.70711), $MachinePrecision], N[(x * -0.70711 + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.7999999999999998

   1. Initial program 99.8%

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

   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 51.0%

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

   4. Applied rewrites 51.0%

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

   if -2.7999999999999998 < x < 980

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

      \[\leadsto 0.70711 \cdot \color{blue}{\left(\frac{230753}{100000} + \frac{-30191289437}{10000000000} \cdot x\right)} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\frac{230753}{100000} + \color{blue}{\frac{-30191289437}{10000000000} \cdot x}\right) \]

      2. lower-*.f64 58.4%

         \[\leadsto 0.70711 \cdot \left(2.30753 + -3.0191289437 \cdot \color{blue}{x}\right) \]

   4. Applied rewrites 58.4%

      \[\leadsto 0.70711 \cdot \color{blue}{\left(2.30753 + -3.0191289437 \cdot x\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{70711}{100000} \cdot \left(\frac{230753}{100000} + \frac{-30191289437}{10000000000} \cdot x\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{230753}{100000} + \frac{-30191289437}{10000000000} \cdot x\right) \cdot \frac{70711}{100000}} \]

      3. lower-*.f64 58.4%

         \[\leadsto \color{blue}{\left(2.30753 + -3.0191289437 \cdot x\right) \cdot 0.70711} \]

      4. lift-+.f64 N/A

         \[\leadsto \left(\frac{230753}{100000} + \color{blue}{\frac{-30191289437}{10000000000} \cdot x}\right) \cdot \frac{70711}{100000} \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{-30191289437}{10000000000} \cdot x + \color{blue}{\frac{230753}{100000}}\right) \cdot \frac{70711}{100000} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{-30191289437}{10000000000} \cdot x + \frac{230753}{100000}\right) \cdot \frac{70711}{100000} \]

      7. lower-fma.f64 58.4%

         \[\leadsto \mathsf{fma}\left(-3.0191289437, \color{blue}{x}, 2.30753\right) \cdot 0.70711 \]

   6. Applied rewrites 58.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-3.0191289437, x, 2.30753\right) \cdot 0.70711} \]

   if 980 < x

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. distribute-rgt-in N/A

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

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

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

      6. lower--.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1913510371}{448100000}}{x}} - x \cdot 0.70711 \]
   5. Step-by-step derivation

      1. lower-/.f64 51.0%

         \[\leadsto \frac{4.2702753202410175}{\color{blue}{x}} - x \cdot 0.70711 \]

   6. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\frac{4.2702753202410175}{x}} - x \cdot 0.70711 \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1913510371}{448100000}}{x} - x \cdot \frac{70711}{100000}} \]

      2. sub-flip N/A

         \[\leadsto \color{blue}{\frac{\frac{1913510371}{448100000}}{x} + \left(\mathsf{neg}\left(x \cdot \frac{70711}{100000}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1913510371}{448100000}}{x} + \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{70711}{100000}}\right)\right) \]

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

         \[\leadsto \frac{\frac{1913510371}{448100000}}{x} + \color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\frac{1913510371}{448100000}}{x} + x \cdot \color{blue}{\frac{-70711}{100000}} \]

      6. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{-70711}{100000} + \frac{\frac{1913510371}{448100000}}{x}} \]

      7. lower-fma.f64 51.0%

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

   8. Applied rewrites 51.0%

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

Alternative 8: 98.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175}{x}\right)\\ \mathbf{if}\;x \leq -2.8:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 980:\\ \;\;\;\;\mathsf{fma}\left(-3.0191289437, x, 2.30753\right) \cdot 0.70711\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (fma x -0.70711 (/ 4.2702753202410175 x))))
  (if (<= x -2.8)
    t_0
    (if (<= x 980.0) (* (fma -3.0191289437 x 2.30753) 0.70711) t_0))))

C

double code(double x) {
	double t_0 = fma(x, -0.70711, (4.2702753202410175 / x));
	double tmp;
	if (x <= -2.8) {
		tmp = t_0;
	} else if (x <= 980.0) {
		tmp = fma(-3.0191289437, x, 2.30753) * 0.70711;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(x, -0.70711, Float64(4.2702753202410175 / x))
	tmp = 0.0
	if (x <= -2.8)
		tmp = t_0;
	elseif (x <= 980.0)
		tmp = Float64(fma(-3.0191289437, x, 2.30753) * 0.70711);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * -0.70711 + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.8], t$95$0, If[LessEqual[x, 980.0], N[(N[(-3.0191289437 * x + 2.30753), $MachinePrecision] * 0.70711), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.7999999999999998 or 980 < x

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. distribute-rgt-in N/A

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

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

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

      6. lower--.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1913510371}{448100000}}{x}} - x \cdot 0.70711 \]
   5. Step-by-step derivation

      1. lower-/.f64 51.0%

         \[\leadsto \frac{4.2702753202410175}{\color{blue}{x}} - x \cdot 0.70711 \]

   6. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\frac{4.2702753202410175}{x}} - x \cdot 0.70711 \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1913510371}{448100000}}{x} - x \cdot \frac{70711}{100000}} \]

      2. sub-flip N/A

         \[\leadsto \color{blue}{\frac{\frac{1913510371}{448100000}}{x} + \left(\mathsf{neg}\left(x \cdot \frac{70711}{100000}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1913510371}{448100000}}{x} + \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{70711}{100000}}\right)\right) \]

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

         \[\leadsto \frac{\frac{1913510371}{448100000}}{x} + \color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\frac{1913510371}{448100000}}{x} + x \cdot \color{blue}{\frac{-70711}{100000}} \]

      6. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{-70711}{100000} + \frac{\frac{1913510371}{448100000}}{x}} \]

      7. lower-fma.f64 51.0%

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

   8. Applied rewrites 51.0%

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

   if -2.7999999999999998 < x < 980

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

      \[\leadsto 0.70711 \cdot \color{blue}{\left(\frac{230753}{100000} + \frac{-30191289437}{10000000000} \cdot x\right)} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\frac{230753}{100000} + \color{blue}{\frac{-30191289437}{10000000000} \cdot x}\right) \]

      2. lower-*.f64 58.4%

         \[\leadsto 0.70711 \cdot \left(2.30753 + -3.0191289437 \cdot \color{blue}{x}\right) \]

   4. Applied rewrites 58.4%

      \[\leadsto 0.70711 \cdot \color{blue}{\left(2.30753 + -3.0191289437 \cdot x\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{70711}{100000} \cdot \left(\frac{230753}{100000} + \frac{-30191289437}{10000000000} \cdot x\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{230753}{100000} + \frac{-30191289437}{10000000000} \cdot x\right) \cdot \frac{70711}{100000}} \]

      3. lower-*.f64 58.4%

         \[\leadsto \color{blue}{\left(2.30753 + -3.0191289437 \cdot x\right) \cdot 0.70711} \]

      4. lift-+.f64 N/A

         \[\leadsto \left(\frac{230753}{100000} + \color{blue}{\frac{-30191289437}{10000000000} \cdot x}\right) \cdot \frac{70711}{100000} \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{-30191289437}{10000000000} \cdot x + \color{blue}{\frac{230753}{100000}}\right) \cdot \frac{70711}{100000} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{-30191289437}{10000000000} \cdot x + \frac{230753}{100000}\right) \cdot \frac{70711}{100000} \]

      7. lower-fma.f64 58.4%

         \[\leadsto \mathsf{fma}\left(-3.0191289437, \color{blue}{x}, 2.30753\right) \cdot 0.70711 \]

   6. Applied rewrites 58.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-3.0191289437, x, 2.30753\right) \cdot 0.70711} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 98.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -0.98:\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{elif}\;x \leq 980:\\ \;\;\;\;\mathsf{fma}\left(-3.0191289437, x, 2.30753\right) \cdot 0.70711\\ \mathbf{else}:\\ \;\;\;\;-0.70711 \cdot x\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= x -0.98)
  (* -0.70711 x)
  (if (<= x 980.0)
    (* (fma -3.0191289437 x 2.30753) 0.70711)
    (* -0.70711 x))))

C

double code(double x) {
	double tmp;
	if (x <= -0.98) {
		tmp = -0.70711 * x;
	} else if (x <= 980.0) {
		tmp = fma(-3.0191289437, x, 2.30753) * 0.70711;
	} else {
		tmp = -0.70711 * x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.98)
		tmp = Float64(-0.70711 * x);
	elseif (x <= 980.0)
		tmp = Float64(fma(-3.0191289437, x, 2.30753) * 0.70711);
	else
		tmp = Float64(-0.70711 * x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -0.98], N[(-0.70711 * x), $MachinePrecision], If[LessEqual[x, 980.0], N[(N[(-3.0191289437 * x + 2.30753), $MachinePrecision] * 0.70711), $MachinePrecision], N[(-0.70711 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.97999999999999998 or 980 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 51.2%

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

   4. Applied rewrites 51.2%

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

   if -0.97999999999999998 < x < 980

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

      \[\leadsto 0.70711 \cdot \color{blue}{\left(\frac{230753}{100000} + \frac{-30191289437}{10000000000} \cdot x\right)} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\frac{230753}{100000} + \color{blue}{\frac{-30191289437}{10000000000} \cdot x}\right) \]

      2. lower-*.f64 58.4%

         \[\leadsto 0.70711 \cdot \left(2.30753 + -3.0191289437 \cdot \color{blue}{x}\right) \]

   4. Applied rewrites 58.4%

      \[\leadsto 0.70711 \cdot \color{blue}{\left(2.30753 + -3.0191289437 \cdot x\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{70711}{100000} \cdot \left(\frac{230753}{100000} + \frac{-30191289437}{10000000000} \cdot x\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{230753}{100000} + \frac{-30191289437}{10000000000} \cdot x\right) \cdot \frac{70711}{100000}} \]

      3. lower-*.f64 58.4%

         \[\leadsto \color{blue}{\left(2.30753 + -3.0191289437 \cdot x\right) \cdot 0.70711} \]

      4. lift-+.f64 N/A

         \[\leadsto \left(\frac{230753}{100000} + \color{blue}{\frac{-30191289437}{10000000000} \cdot x}\right) \cdot \frac{70711}{100000} \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{-30191289437}{10000000000} \cdot x + \color{blue}{\frac{230753}{100000}}\right) \cdot \frac{70711}{100000} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{-30191289437}{10000000000} \cdot x + \frac{230753}{100000}\right) \cdot \frac{70711}{100000} \]

      7. lower-fma.f64 58.4%

         \[\leadsto \mathsf{fma}\left(-3.0191289437, \color{blue}{x}, 2.30753\right) \cdot 0.70711 \]

   6. Applied rewrites 58.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-3.0191289437, x, 2.30753\right) \cdot 0.70711} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 98.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -0.98:\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{elif}\;x \leq 980:\\ \;\;\;\;\mathsf{fma}\left(x, -2.134856267379707, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;-0.70711 \cdot x\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= x -0.98)
  (* -0.70711 x)
  (if (<= x 980.0)
    (fma x -2.134856267379707 1.6316775383)
    (* -0.70711 x))))

C

double code(double x) {
	double tmp;
	if (x <= -0.98) {
		tmp = -0.70711 * x;
	} else if (x <= 980.0) {
		tmp = fma(x, -2.134856267379707, 1.6316775383);
	} else {
		tmp = -0.70711 * x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.98)
		tmp = Float64(-0.70711 * x);
	elseif (x <= 980.0)
		tmp = fma(x, -2.134856267379707, 1.6316775383);
	else
		tmp = Float64(-0.70711 * x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -0.98], N[(-0.70711 * x), $MachinePrecision], If[LessEqual[x, 980.0], N[(x * -2.134856267379707 + 1.6316775383), $MachinePrecision], N[(-0.70711 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.97999999999999998 or 980 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 51.2%

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

   4. Applied rewrites 51.2%

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

   if -0.97999999999999998 < x < 980

   1. Initial program 99.8%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   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 99.8%

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

      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 99.8%

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

      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 99.8%

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

      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 99.8%

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

   3. Applied rewrites 99.8%

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 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 58.4%

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

   8. Applied rewrites 58.4%

      \[\leadsto \color{blue}{1.6316775383 + -2.134856267379707 \cdot x} \]
   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 58.4%

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

   10. Applied rewrites 58.4%

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

Alternative 11: 97.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -1750000:\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{elif}\;x \leq 75000:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;-0.70711 \cdot x\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= x -1750000.0)
  (* -0.70711 x)
  (if (<= x 75000.0) 1.6316775383 (* -0.70711 x))))

C

double code(double x) {
	double tmp;
	if (x <= -1750000.0) {
		tmp = -0.70711 * x;
	} else if (x <= 75000.0) {
		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 <= (-1750000.0d0)) then
        tmp = (-0.70711d0) * x
    else if (x <= 75000.0d0) 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 <= -1750000.0) {
		tmp = -0.70711 * x;
	} else if (x <= 75000.0) {
		tmp = 1.6316775383;
	} else {
		tmp = -0.70711 * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1750000.0:
		tmp = -0.70711 * x
	elif x <= 75000.0:
		tmp = 1.6316775383
	else:
		tmp = -0.70711 * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1750000.0)
		tmp = Float64(-0.70711 * x);
	elseif (x <= 75000.0)
		tmp = 1.6316775383;
	else
		tmp = Float64(-0.70711 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1750000.0)
		tmp = -0.70711 * x;
	elseif (x <= 75000.0)
		tmp = 1.6316775383;
	else
		tmp = -0.70711 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1750000.0], N[(-0.70711 * x), $MachinePrecision], If[LessEqual[x, 75000.0], 1.6316775383, N[(-0.70711 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.75e6 or 75000 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 51.2%

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

   4. Applied rewrites 51.2%

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

   if -1.75e6 < x < 75000

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 50.6%

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

Alternative 12: 50.6% accurate, 27.2× 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 99.8%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 50.6%

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

Reproduce

herbie shell --seed 2025212 
(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)))