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

Percentage Accurate: 99.8% → 99.9%

Time: 3.5s

Alternatives: 9

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

   5. distribute-rgt-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-neg.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-/.f64 N/A

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

   10. frac-2neg N/A

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

   11. associate-*r/ N/A

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

   12. lower-/.f64 N/A

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

3. Applied rewrites 99.9%

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

Alternative 2: 99.9% accurate, 1.1× speedup

Math

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

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. lift--.f64 N/A

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

   3. sub-flip N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-neg.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-/.f64 N/A

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

   10. frac-2neg N/A

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

   11. associate-*r/ N/A

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

   12. lower-/.f64 N/A

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

3. Applied rewrites 99.9%

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

   1. lift-fma.f64 N/A

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

   2. lift-neg.f64 N/A

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

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

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

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

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

   5. lower-fma.f64 N/A

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

   6. metadata-eval 99.9

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

   7. lift-*.f64 N/A

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

   8. lift-fma.f64 N/A

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

   9. distribute-rgt-in N/A

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

   10. *-commutative N/A

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

   11. metadata-eval N/A

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

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

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

   13. *-commutative N/A

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

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

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

   15. metadata-eval N/A

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

   16. *-commutative N/A

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

   17. associate-*r* N/A

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

   18. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\color{blue}{\frac{-1913510371}{10000000000}} \cdot x + \frac{-230753}{100000} \cdot \frac{70711}{100000}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right) \]

   19. metadata-eval N/A

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

   20. lower-fma.f64 N/A

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

   21. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\mathsf{fma}\left(\color{blue}{\frac{-1913510371}{10000000000}}, x, \frac{-230753}{100000} \cdot \frac{70711}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right) \]

   22. metadata-eval 99.9

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

5. Applied rewrites 99.9%

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

Alternative 3: 99.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -4.8:\\ \;\;\;\;\mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175 - 58.14938538768042 \cdot \frac{1}{x}}{x}\right)\\ \mathbf{elif}\;x \leq 0.66:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.2692862305735844, x, 1.3436228731669864\right), x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175}{x}\right)\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.8)
   (fma
    x
    -0.70711
    (/ (- 4.2702753202410175 (* 58.14938538768042 (/ 1.0 x))) x))
   (if (<= x 0.66)
     (fma
      (fma (fma -1.2692862305735844 x 1.3436228731669864) x -2.134856267379707)
      x
      1.6316775383)
     (fma x -0.70711 (/ 4.2702753202410175 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -4.8) {
		tmp = fma(x, -0.70711, ((4.2702753202410175 - (58.14938538768042 * (1.0 / x))) / x));
	} else if (x <= 0.66) {
		tmp = fma(fma(fma(-1.2692862305735844, x, 1.3436228731669864), x, -2.134856267379707), x, 1.6316775383);
	} else {
		tmp = fma(x, -0.70711, (4.2702753202410175 / x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.8)
		tmp = fma(x, -0.70711, Float64(Float64(4.2702753202410175 - Float64(58.14938538768042 * Float64(1.0 / x))) / x));
	elseif (x <= 0.66)
		tmp = fma(fma(fma(-1.2692862305735844, x, 1.3436228731669864), x, -2.134856267379707), x, 1.6316775383);
	else
		tmp = fma(x, -0.70711, Float64(4.2702753202410175 / x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -4.8], N[(x * -0.70711 + N[(N[(4.2702753202410175 - N[(58.14938538768042 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.66], N[(N[(N[(-1.2692862305735844 * x + 1.3436228731669864), $MachinePrecision] * x + -2.134856267379707), $MachinePrecision] * x + 1.6316775383), $MachinePrecision], N[(x * -0.70711 + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.79999999999999982

   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. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 99.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-neg.f64 N/A

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

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

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

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

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

      5. lower-fma.f64 N/A

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

      6. metadata-eval 99.9

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. metadata-eval N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\color{blue}{\frac{-1913510371}{10000000000}} \cdot x + \frac{-230753}{100000} \cdot \frac{70711}{100000}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right) \]

      19. metadata-eval N/A

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

      20. lower-fma.f64 N/A

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

      21. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\mathsf{fma}\left(\color{blue}{\frac{-1913510371}{10000000000}}, x, \frac{-230753}{100000} \cdot \frac{70711}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right) \]

      22. metadata-eval 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \color{blue}{\frac{\frac{1913510371}{448100000} - \frac{3648757816023}{62748003125} \cdot \frac{1}{x}}{x}}\right) \]
   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 50.5

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

   8. Applied rewrites 50.5%

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

   if -4.79999999999999982 < x < 0.660000000000000031

   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. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{16316775383}{10000000000} + x \cdot \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right)} \]
   5. 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 52.3

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

   6. Applied rewrites 52.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 52.3

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

      6. lift--.f64 N/A

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

      7. sub-flip N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 52.3

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 52.3

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

   8. Applied rewrites 52.3%

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

   if 0.660000000000000031 < 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. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 99.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-neg.f64 N/A

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

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

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

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

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

      5. lower-fma.f64 N/A

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

      6. metadata-eval 99.9

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. metadata-eval N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\color{blue}{\frac{-1913510371}{10000000000}} \cdot x + \frac{-230753}{100000} \cdot \frac{70711}{100000}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right) \]

      19. metadata-eval N/A

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

      20. lower-fma.f64 N/A

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

      21. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\mathsf{fma}\left(\color{blue}{\frac{-1913510371}{10000000000}}, x, \frac{-230753}{100000} \cdot \frac{70711}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right) \]

      22. metadata-eval 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 51.6

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

   8. Applied rewrites 51.6%

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

Alternative 4: 99.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175}{x}\right)\\ \mathbf{if}\;x \leq -2.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.66:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.2692862305735844, x, 1.3436228731669864\right), x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma x -0.70711 (/ 4.2702753202410175 x))))
   (if (<= x -2.5)
     t_0
     (if (<= x 0.66)
       (fma
        (fma
         (fma -1.2692862305735844 x 1.3436228731669864)
         x
         -2.134856267379707)
        x
        1.6316775383)
       t_0))))

C

double code(double x) {
	double t_0 = fma(x, -0.70711, (4.2702753202410175 / x));
	double tmp;
	if (x <= -2.5) {
		tmp = t_0;
	} else if (x <= 0.66) {
		tmp = fma(fma(fma(-1.2692862305735844, x, 1.3436228731669864), x, -2.134856267379707), x, 1.6316775383);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(x, -0.70711, Float64(4.2702753202410175 / x))
	tmp = 0.0
	if (x <= -2.5)
		tmp = t_0;
	elseif (x <= 0.66)
		tmp = fma(fma(fma(-1.2692862305735844, x, 1.3436228731669864), x, -2.134856267379707), x, 1.6316775383);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * -0.70711 + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.5], t$95$0, If[LessEqual[x, 0.66], N[(N[(N[(-1.2692862305735844 * x + 1.3436228731669864), $MachinePrecision] * x + -2.134856267379707), $MachinePrecision] * x + 1.6316775383), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5 or 0.660000000000000031 < 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. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 99.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-neg.f64 N/A

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

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

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

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

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

      5. lower-fma.f64 N/A

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

      6. metadata-eval 99.9

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. metadata-eval N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\color{blue}{\frac{-1913510371}{10000000000}} \cdot x + \frac{-230753}{100000} \cdot \frac{70711}{100000}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right) \]

      19. metadata-eval N/A

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

      20. lower-fma.f64 N/A

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

      21. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\mathsf{fma}\left(\color{blue}{\frac{-1913510371}{10000000000}}, x, \frac{-230753}{100000} \cdot \frac{70711}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right) \]

      22. metadata-eval 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 51.6

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

   8. Applied rewrites 51.6%

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

   if -2.5 < x < 0.660000000000000031

   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. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{16316775383}{10000000000} + x \cdot \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right)} \]
   5. 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 52.3

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

   6. Applied rewrites 52.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 52.3

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

      6. lift--.f64 N/A

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

      7. sub-flip N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 52.3

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 52.3

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

   8. Applied rewrites 52.3%

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

Alternative 5: 98.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175}{x}\right)\\ \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3900000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma x -0.70711 (/ 4.2702753202410175 x))))
   (if (<= x -2.6)
     t_0
     (if (<= x 3900000.0)
       (fma (fma 1.3436228731669864 x -2.134856267379707) x 1.6316775383)
       t_0))))

C

double code(double x) {
	double t_0 = fma(x, -0.70711, (4.2702753202410175 / x));
	double tmp;
	if (x <= -2.6) {
		tmp = t_0;
	} else if (x <= 3900000.0) {
		tmp = fma(fma(1.3436228731669864, x, -2.134856267379707), x, 1.6316775383);
	} 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 <= -2.6)
		tmp = t_0;
	elseif (x <= 3900000.0)
		tmp = fma(fma(1.3436228731669864, x, -2.134856267379707), x, 1.6316775383);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * -0.70711 + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.6], t$95$0, If[LessEqual[x, 3900000.0], N[(N[(1.3436228731669864 * x + -2.134856267379707), $MachinePrecision] * x + 1.6316775383), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.60000000000000009 or 3.9e6 < 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. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 99.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-neg.f64 N/A

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

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

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

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

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

      5. lower-fma.f64 N/A

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

      6. metadata-eval 99.9

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. metadata-eval N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\color{blue}{\frac{-1913510371}{10000000000}} \cdot x + \frac{-230753}{100000} \cdot \frac{70711}{100000}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right) \]

      19. metadata-eval N/A

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

      20. lower-fma.f64 N/A

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

      21. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\mathsf{fma}\left(\color{blue}{\frac{-1913510371}{10000000000}}, x, \frac{-230753}{100000} \cdot \frac{70711}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right) \]

      22. metadata-eval 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 51.6

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

   8. Applied rewrites 51.6%

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

   if -2.60000000000000009 < x < 3.9e6

   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. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 50.8

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

   6. Applied rewrites 50.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 50.8

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

      6. lift--.f64 N/A

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

      7. sub-flip N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval 50.8

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

   8. Applied rewrites 50.8%

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

Alternative 6: 98.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175}{x}\right)\\ \mathbf{if}\;x \leq -2.7:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.58:\\ \;\;\;\;\mathsf{fma}\left(-2.134856267379707, x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma x -0.70711 (/ 4.2702753202410175 x))))
   (if (<= x -2.7)
     t_0
     (if (<= x 0.58) (fma -2.134856267379707 x 1.6316775383) t_0))))

C

double code(double x) {
	double t_0 = fma(x, -0.70711, (4.2702753202410175 / x));
	double tmp;
	if (x <= -2.7) {
		tmp = t_0;
	} else if (x <= 0.58) {
		tmp = fma(-2.134856267379707, x, 1.6316775383);
	} 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 <= -2.7)
		tmp = t_0;
	elseif (x <= 0.58)
		tmp = fma(-2.134856267379707, x, 1.6316775383);
	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, -2.7], t$95$0, If[LessEqual[x, 0.58], N[(-2.134856267379707 * x + 1.6316775383), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.7000000000000002 or 0.57999999999999996 < 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. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 99.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-neg.f64 N/A

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

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

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

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

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

      5. lower-fma.f64 N/A

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

      6. metadata-eval 99.9

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. metadata-eval N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\color{blue}{\frac{-1913510371}{10000000000}} \cdot x + \frac{-230753}{100000} \cdot \frac{70711}{100000}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right) \]

      19. metadata-eval N/A

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

      20. lower-fma.f64 N/A

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

      21. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\mathsf{fma}\left(\color{blue}{\frac{-1913510371}{10000000000}}, x, \frac{-230753}{100000} \cdot \frac{70711}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right) \]

      22. metadata-eval 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 51.6

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

   8. Applied rewrites 51.6%

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

   if -2.7000000000000002 < x < 0.57999999999999996

   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. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 50.8

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

   6. Applied rewrites 50.8%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 57.7%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 57.7

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

   11. Applied rewrites 57.7%

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

Alternative 7: 98.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -0.98:\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{elif}\;x \leq 0.75:\\ \;\;\;\;\mathsf{fma}\left(-2.134856267379707, x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;-0.70711 \cdot x\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.98)
   (* -0.70711 x)
   (if (<= x 0.75) (fma -2.134856267379707 x 1.6316775383) (* -0.70711 x))))

C

double code(double x) {
	double tmp;
	if (x <= -0.98) {
		tmp = -0.70711 * x;
	} else if (x <= 0.75) {
		tmp = fma(-2.134856267379707, x, 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 <= 0.75)
		tmp = fma(-2.134856267379707, x, 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, 0.75], N[(-2.134856267379707 * x + 1.6316775383), $MachinePrecision], N[(-0.70711 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.97999999999999998 or 0.75 < 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 51.8

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

   4. Applied rewrites 51.8%

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

   if -0.97999999999999998 < x < 0.75

   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. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 50.8

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

   6. Applied rewrites 50.8%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 57.7%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 57.7

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

   11. Applied rewrites 57.7%

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

Alternative 8: 97.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -1500:\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{elif}\;x \leq 3900000:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;-0.70711 \cdot x\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1500.0)
   (* -0.70711 x)
   (if (<= x 3900000.0) 1.6316775383 (* -0.70711 x))))

C

double code(double x) {
	double tmp;
	if (x <= -1500.0) {
		tmp = -0.70711 * x;
	} else if (x <= 3900000.0) {
		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 <= (-1500.0d0)) then
        tmp = (-0.70711d0) * x
    else if (x <= 3900000.0d0) 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 <= -1500.0) {
		tmp = -0.70711 * x;
	} else if (x <= 3900000.0) {
		tmp = 1.6316775383;
	} else {
		tmp = -0.70711 * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1500.0:
		tmp = -0.70711 * x
	elif x <= 3900000.0:
		tmp = 1.6316775383
	else:
		tmp = -0.70711 * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1500.0)
		tmp = Float64(-0.70711 * x);
	elseif (x <= 3900000.0)
		tmp = 1.6316775383;
	else
		tmp = Float64(-0.70711 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1500.0)
		tmp = -0.70711 * x;
	elseif (x <= 3900000.0)
		tmp = 1.6316775383;
	else
		tmp = -0.70711 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1500.0], N[(-0.70711 * x), $MachinePrecision], If[LessEqual[x, 3900000.0], 1.6316775383, N[(-0.70711 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1500 or 3.9e6 < 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 51.8

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

   4. Applied rewrites 51.8%

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

   if -1500 < x < 3.9e6

   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 49.8%

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

Alternative 9: 49.8% accurate, 27.0× speedup

Math

\[1.6316775383 \]

FPCore

(FPCore (x) :precision binary64 1.6316775383)

C

double code(double x) {
	return 1.6316775383;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x):
	return 1.6316775383

Julia

function code(x)
	return 1.6316775383
end

MATLAB

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

Wolfram

code[x_] := 1.6316775383

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 49.8%

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

Reproduce

herbie shell --seed 2025172 
(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)))