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

Percentage Accurate: 99.8% → 99.8%

Time: 7.5s

Alternatives: 11

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.9

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

4. Applied rewrites 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

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

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lower--.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-fma.f64 N/A

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

   17. metadata-eval 99.9

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

6. Applied rewrites 99.9%

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

Alternative 2: 99.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 96.9

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

   5. Applied rewrites 96.9%

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

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

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000} \cdot 1, x, \frac{16316775383}{10000000000}\right) \]

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

         \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) + \left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right) \cdot 1, x, \frac{16316775383}{10000000000}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) \cdot x + \left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right) \cdot 1, x, \frac{16316775383}{10000000000}\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) \cdot x + \frac{-2134856267379707}{1000000000000000} \cdot 1, x, \frac{16316775383}{10000000000}\right) \]

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 99.2

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

   7. Applied rewrites 99.2%

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

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

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{70711}{100000} - \frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{70711}{100000} - \frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{70711}{100000} + \left(\mathsf{neg}\left(\frac{1913510371}{448100000}\right)\right) \cdot \frac{1}{{x}^{2}}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\left(\mathsf{neg}\left(\frac{1913510371}{448100000}\right)\right) \cdot \frac{1}{{x}^{2}} + \frac{70711}{100000}\right)\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{neg}\left(\left(x \cdot \left(\left(\mathsf{neg}\left(\frac{1913510371}{448100000}\right)\right) \cdot \frac{1}{{x}^{2}}\right) + x \cdot \frac{70711}{100000}\right)\right) \]

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

         \[\leadsto \mathsf{neg}\left(\left(x \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

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

         \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(x \cdot \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\frac{1913510371}{448100000} \cdot 1}{{x}^{2}} \cdot x\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\frac{1913510371}{448100000} \cdot 1}{x \cdot x} \cdot x\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

      10. times-frac N/A

          \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\left(\frac{\frac{1913510371}{448100000}}{x} \cdot \frac{1}{x}\right) \cdot x\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\frac{1913510371}{448100000}}{x} \cdot \left(\frac{1}{x} \cdot x\right)\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

      12. lft-mult-inverse N/A

          \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\frac{1913510371}{448100000}}{x} \cdot 1\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(1 \cdot \frac{\frac{1913510371}{448100000}}{x}\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

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

          \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \frac{\frac{1913510371}{448100000}}{x} + x \cdot \frac{70711}{100000}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \frac{\frac{1913510371}{448100000}}{x} + \frac{70711}{100000} \cdot x\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \frac{\frac{1913510371}{448100000}}{x} + \left(\mathsf{neg}\left(\frac{-70711}{100000}\right)\right) \cdot x\right)\right) \]

   7. Applied rewrites 99.9%

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

Alternative 3: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.9

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 99.9

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 99.9

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 4: 99.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.2692862305735844, x, 1.3436228731669864\right), x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{4.2702753202410175}{x} - 0.70711 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (fma -0.70711 x (/ 4.2702753202410175 x))
   (if (<= x 2.5)
     (fma
      (fma (fma -1.2692862305735844 x 1.3436228731669864) x -2.134856267379707)
      x
      1.6316775383)
     (- (/ 4.2702753202410175 x) (* 0.70711 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = fma(-0.70711, x, (4.2702753202410175 / x));
	} else if (x <= 2.5) {
		tmp = fma(fma(fma(-1.2692862305735844, x, 1.3436228731669864), x, -2.134856267379707), x, 1.6316775383);
	} else {
		tmp = (4.2702753202410175 / x) - (0.70711 * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = fma(-0.70711, x, Float64(4.2702753202410175 / x));
	elseif (x <= 2.5)
		tmp = fma(fma(fma(-1.2692862305735844, x, 1.3436228731669864), x, -2.134856267379707), x, 1.6316775383);
	else
		tmp = Float64(Float64(4.2702753202410175 / x) - Float64(0.70711 * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(-0.70711 * x + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.5], N[(N[(N[(-1.2692862305735844 * x + 1.3436228731669864), $MachinePrecision] * x + -2.134856267379707), $MachinePrecision] * x + 1.6316775383), $MachinePrecision], N[(N[(4.2702753202410175 / x), $MachinePrecision] - N[(0.70711 * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{70711}{100000} - \frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{70711}{100000} - \frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{70711}{100000} + \left(\mathsf{neg}\left(\frac{1913510371}{448100000}\right)\right) \cdot \frac{1}{{x}^{2}}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\left(\mathsf{neg}\left(\frac{1913510371}{448100000}\right)\right) \cdot \frac{1}{{x}^{2}} + \frac{70711}{100000}\right)\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{neg}\left(\left(x \cdot \left(\left(\mathsf{neg}\left(\frac{1913510371}{448100000}\right)\right) \cdot \frac{1}{{x}^{2}}\right) + x \cdot \frac{70711}{100000}\right)\right) \]

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

         \[\leadsto \mathsf{neg}\left(\left(x \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

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

         \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(x \cdot \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\frac{1913510371}{448100000} \cdot 1}{{x}^{2}} \cdot x\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\frac{1913510371}{448100000} \cdot 1}{x \cdot x} \cdot x\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

      10. times-frac N/A

          \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\left(\frac{\frac{1913510371}{448100000}}{x} \cdot \frac{1}{x}\right) \cdot x\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\frac{1913510371}{448100000}}{x} \cdot \left(\frac{1}{x} \cdot x\right)\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

      12. lft-mult-inverse N/A

          \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\frac{1913510371}{448100000}}{x} \cdot 1\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(1 \cdot \frac{\frac{1913510371}{448100000}}{x}\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

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

          \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \frac{\frac{1913510371}{448100000}}{x} + x \cdot \frac{70711}{100000}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \frac{\frac{1913510371}{448100000}}{x} + \frac{70711}{100000} \cdot x\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \frac{\frac{1913510371}{448100000}}{x} + \left(\mathsf{neg}\left(\frac{-70711}{100000}\right)\right) \cdot x\right)\right) \]

   7. Applied rewrites 98.4%

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

   if -1.05000000000000004 < x < 2.5

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000} \cdot 1, x, \frac{16316775383}{10000000000}\right) \]

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

         \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) + \left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right) \cdot 1, x, \frac{16316775383}{10000000000}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) \cdot x + \left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right) \cdot 1, x, \frac{16316775383}{10000000000}\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) \cdot x + \frac{-2134856267379707}{1000000000000000} \cdot 1, x, \frac{16316775383}{10000000000}\right) \]

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 99.2

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

   7. Applied rewrites 99.2%

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

   if 2.5 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{70711}{100000} - \frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{70711}{100000} - \frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right) \]

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

         \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{70711}{100000} - \frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \]

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

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\left(\frac{70711}{100000} + \left(\mathsf{neg}\left(\frac{1913510371}{448100000}\right)\right) \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{1913510371}{448100000}\right)\right) \cdot \frac{1}{{x}^{2}} + \frac{70711}{100000}\right)\right)\right) \]

      5. distribute-neg-in N/A

         \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1913510371}{448100000}\right)\right) \cdot \frac{1}{{x}^{2}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right)}\right) \]

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

         \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1913510371}{448100000}\right)\right)\right)\right) \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(\color{blue}{\frac{70711}{100000}}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto x \cdot \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto x \cdot \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} + \frac{-70711}{100000}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right) \cdot x + \color{blue}{\frac{-70711}{100000} \cdot x} \]

      10. metadata-eval N/A

          \[\leadsto \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right) \cdot x + \left(\frac{70711}{100000} \cdot -1\right) \cdot x \]

      11. associate-*r* N/A

          \[\leadsto \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right) \cdot x + \frac{70711}{100000} \cdot \color{blue}{\left(-1 \cdot x\right)} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{\frac{1913510371}{448100000} \cdot 1}{{x}^{2}} \cdot x + \frac{70711}{100000} \cdot \left(-1 \cdot x\right) \]

      13. unpow2 N/A

          \[\leadsto \frac{\frac{1913510371}{448100000} \cdot 1}{x \cdot x} \cdot x + \frac{70711}{100000} \cdot \left(-1 \cdot x\right) \]

      14. times-frac N/A

          \[\leadsto \left(\frac{\frac{1913510371}{448100000}}{x} \cdot \frac{1}{x}\right) \cdot x + \frac{70711}{100000} \cdot \left(-1 \cdot x\right) \]

      15. associate-*l* N/A

          \[\leadsto \frac{\frac{1913510371}{448100000}}{x} \cdot \left(\frac{1}{x} \cdot x\right) + \color{blue}{\frac{70711}{100000}} \cdot \left(-1 \cdot x\right) \]

      16. lft-mult-inverse N/A

          \[\leadsto \frac{\frac{1913510371}{448100000}}{x} \cdot 1 + \frac{70711}{100000} \cdot \left(-1 \cdot x\right) \]

   5. Applied rewrites 98.1%

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

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{1913510371}{448100000}}{x} \cdot 1 + \color{blue}{\frac{-70711}{100000} \cdot x} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1913510371}{448100000}}{x} \cdot 1 + \frac{-70711}{100000} \cdot \color{blue}{x} \]

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

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

      4. lower--.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 98.1

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

   7. Applied rewrites 98.1%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.2692862305735844, x, 1.3436228731669864\right), x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{4.2702753202410175}{x} - 0.70711 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 2.5\right):\\ \;\;\;\;\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.2692862305735844, x, 1.3436228731669864\right), x, -2.134856267379707\right), x, 1.6316775383\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 2.5)))
   (fma -0.70711 x (/ 4.2702753202410175 x))
   (fma
    (fma (fma -1.2692862305735844 x 1.3436228731669864) x -2.134856267379707)
    x
    1.6316775383)))

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 2.5)) {
		tmp = fma(-0.70711, x, (4.2702753202410175 / x));
	} else {
		tmp = fma(fma(fma(-1.2692862305735844, x, 1.3436228731669864), x, -2.134856267379707), x, 1.6316775383);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.05) || !(x <= 2.5))
		tmp = fma(-0.70711, x, Float64(4.2702753202410175 / x));
	else
		tmp = fma(fma(fma(-1.2692862305735844, x, 1.3436228731669864), x, -2.134856267379707), x, 1.6316775383);
	end
	return tmp
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.05], N[Not[LessEqual[x, 2.5]], $MachinePrecision]], N[(-0.70711 * x + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.2692862305735844 * x + 1.3436228731669864), $MachinePrecision] * x + -2.134856267379707), $MachinePrecision] * x + 1.6316775383), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 2.5 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{70711}{100000} - \frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{70711}{100000} - \frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{70711}{100000} + \left(\mathsf{neg}\left(\frac{1913510371}{448100000}\right)\right) \cdot \frac{1}{{x}^{2}}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\left(\mathsf{neg}\left(\frac{1913510371}{448100000}\right)\right) \cdot \frac{1}{{x}^{2}} + \frac{70711}{100000}\right)\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{neg}\left(\left(x \cdot \left(\left(\mathsf{neg}\left(\frac{1913510371}{448100000}\right)\right) \cdot \frac{1}{{x}^{2}}\right) + x \cdot \frac{70711}{100000}\right)\right) \]

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

         \[\leadsto \mathsf{neg}\left(\left(x \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

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

         \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(x \cdot \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\frac{1913510371}{448100000} \cdot 1}{{x}^{2}} \cdot x\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\frac{1913510371}{448100000} \cdot 1}{x \cdot x} \cdot x\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

      10. times-frac N/A

          \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\left(\frac{\frac{1913510371}{448100000}}{x} \cdot \frac{1}{x}\right) \cdot x\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\frac{1913510371}{448100000}}{x} \cdot \left(\frac{1}{x} \cdot x\right)\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

      12. lft-mult-inverse N/A

          \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\frac{1913510371}{448100000}}{x} \cdot 1\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(1 \cdot \frac{\frac{1913510371}{448100000}}{x}\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

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

          \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \frac{\frac{1913510371}{448100000}}{x} + x \cdot \frac{70711}{100000}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \frac{\frac{1913510371}{448100000}}{x} + \frac{70711}{100000} \cdot x\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \frac{\frac{1913510371}{448100000}}{x} + \left(\mathsf{neg}\left(\frac{-70711}{100000}\right)\right) \cdot x\right)\right) \]

   7. Applied rewrites 98.3%

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

   if -1.05000000000000004 < x < 2.5

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000} \cdot 1, x, \frac{16316775383}{10000000000}\right) \]

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

         \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) + \left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right) \cdot 1, x, \frac{16316775383}{10000000000}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) \cdot x + \left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right) \cdot 1, x, \frac{16316775383}{10000000000}\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) \cdot x + \frac{-2134856267379707}{1000000000000000} \cdot 1, x, \frac{16316775383}{10000000000}\right) \]

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 99.2

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

   7. Applied rewrites 99.2%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 2.5\right):\\ \;\;\;\;\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.2692862305735844, x, 1.3436228731669864\right), x, -2.134856267379707\right), x, 1.6316775383\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.55\right):\\ \;\;\;\;\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right), x, 1.6316775383\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 1.55)))
   (fma -0.70711 x (/ 4.2702753202410175 x))
   (fma (fma 1.3436228731669864 x -2.134856267379707) x 1.6316775383)))

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 1.55)) {
		tmp = fma(-0.70711, x, (4.2702753202410175 / x));
	} else {
		tmp = fma(fma(1.3436228731669864, x, -2.134856267379707), x, 1.6316775383);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.05) || !(x <= 1.55))
		tmp = fma(-0.70711, x, Float64(4.2702753202410175 / x));
	else
		tmp = fma(fma(1.3436228731669864, x, -2.134856267379707), x, 1.6316775383);
	end
	return tmp
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.05], N[Not[LessEqual[x, 1.55]], $MachinePrecision]], N[(-0.70711 * x + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision], N[(N[(1.3436228731669864 * x + -2.134856267379707), $MachinePrecision] * x + 1.6316775383), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.55000000000000004 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{70711}{100000} - \frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{70711}{100000} - \frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{70711}{100000} + \left(\mathsf{neg}\left(\frac{1913510371}{448100000}\right)\right) \cdot \frac{1}{{x}^{2}}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\left(\mathsf{neg}\left(\frac{1913510371}{448100000}\right)\right) \cdot \frac{1}{{x}^{2}} + \frac{70711}{100000}\right)\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{neg}\left(\left(x \cdot \left(\left(\mathsf{neg}\left(\frac{1913510371}{448100000}\right)\right) \cdot \frac{1}{{x}^{2}}\right) + x \cdot \frac{70711}{100000}\right)\right) \]

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

         \[\leadsto \mathsf{neg}\left(\left(x \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

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

         \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(x \cdot \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\frac{1913510371}{448100000} \cdot 1}{{x}^{2}} \cdot x\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\frac{1913510371}{448100000} \cdot 1}{x \cdot x} \cdot x\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

      10. times-frac N/A

          \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\left(\frac{\frac{1913510371}{448100000}}{x} \cdot \frac{1}{x}\right) \cdot x\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\frac{1913510371}{448100000}}{x} \cdot \left(\frac{1}{x} \cdot x\right)\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

      12. lft-mult-inverse N/A

          \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\frac{1913510371}{448100000}}{x} \cdot 1\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(1 \cdot \frac{\frac{1913510371}{448100000}}{x}\right)\right) + x \cdot \frac{70711}{100000}\right)\right) \]

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

          \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \frac{\frac{1913510371}{448100000}}{x} + x \cdot \frac{70711}{100000}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \frac{\frac{1913510371}{448100000}}{x} + \frac{70711}{100000} \cdot x\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \frac{\frac{1913510371}{448100000}}{x} + \left(\mathsf{neg}\left(\frac{-70711}{100000}\right)\right) \cdot x\right)\right) \]

   7. Applied rewrites 97.4%

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

   if -1.05000000000000004 < x < 1.55000000000000004

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000} \cdot 1, x, \frac{16316775383}{10000000000}\right) \]

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

         \[\leadsto \mathsf{fma}\left(\frac{134362287316698645903}{100000000000000000000} \cdot x + \left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right) \cdot 1, x, \frac{16316775383}{10000000000}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{134362287316698645903}{100000000000000000000} \cdot x + \frac{-2134856267379707}{1000000000000000} \cdot 1, x, \frac{16316775383}{10000000000}\right) \]

      7. metadata-eval N/A

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

      8. lower-fma.f64 99.6

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

   7. Applied rewrites 99.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.55\right):\\ \;\;\;\;\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right), x, 1.6316775383\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

   \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\color{blue}{\frac{230753}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right) \]
4. Step-by-step derivation

5. Applied rewrites 97.8%

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

6. Taylor expanded in x around 0

   \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000}}{\color{blue}{1 + \frac{99229}{100000} \cdot x}} - x\right) \]
7. Step-by-step derivation

   1. +-commutative N/A

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

   2. lower-fma.f64 97.8

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

8. Applied rewrites 97.8%

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

Alternative 8: 99.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right), x, 1.6316775383\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 1.15)))
   (* -0.70711 x)
   (fma (fma 1.3436228731669864 x -2.134856267379707) x 1.6316775383)))

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 1.15)) {
		tmp = -0.70711 * x;
	} else {
		tmp = fma(fma(1.3436228731669864, x, -2.134856267379707), x, 1.6316775383);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.05) || !(x <= 1.15))
		tmp = Float64(-0.70711 * x);
	else
		tmp = fma(fma(1.3436228731669864, x, -2.134856267379707), x, 1.6316775383);
	end
	return tmp
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.05], N[Not[LessEqual[x, 1.15]], $MachinePrecision]], N[(-0.70711 * x), $MachinePrecision], N[(N[(1.3436228731669864 * x + -2.134856267379707), $MachinePrecision] * x + 1.6316775383), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.1499999999999999 < 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. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 97.0

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

   5. Applied rewrites 97.0%

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

   if -1.05000000000000004 < x < 1.1499999999999999

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000} \cdot 1, x, \frac{16316775383}{10000000000}\right) \]

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

         \[\leadsto \mathsf{fma}\left(\frac{134362287316698645903}{100000000000000000000} \cdot x + \left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right) \cdot 1, x, \frac{16316775383}{10000000000}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{134362287316698645903}{100000000000000000000} \cdot x + \frac{-2134856267379707}{1000000000000000} \cdot 1, x, \frac{16316775383}{10000000000}\right) \]

      7. metadata-eval N/A

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

      8. lower-fma.f64 99.6

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

   7. Applied rewrites 99.6%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right), x, 1.6316775383\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.8% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 1.15)))
   (* -0.70711 x)
   (fma -2.134856267379707 x 1.6316775383)))

C

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

Julia

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

Wolfram

code[x_] := If[Or[LessEqual[x, -1.05], N[Not[LessEqual[x, 1.15]], $MachinePrecision]], N[(-0.70711 * x), $MachinePrecision], N[(-2.134856267379707 * x + 1.6316775383), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.1499999999999999 < 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. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 97.0

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

   5. Applied rewrites 97.0%

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

   if -1.05000000000000004 < x < 1.1499999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.0

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

   5. Applied rewrites 99.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2.134856267379707, x, 1.6316775383\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{else}:\\ \;\;\;\;1.6316775383\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 1.15))) (* -0.70711 x) 1.6316775383))

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 1.15)) {
		tmp = -0.70711 * x;
	} else {
		tmp = 1.6316775383;
	}
	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 <= (-1.05d0)) .or. (.not. (x <= 1.15d0))) then
        tmp = (-0.70711d0) * x
    else
        tmp = 1.6316775383d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 1.15)) {
		tmp = -0.70711 * x;
	} else {
		tmp = 1.6316775383;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.05) or not (x <= 1.15):
		tmp = -0.70711 * x
	else:
		tmp = 1.6316775383
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.05) || !(x <= 1.15))
		tmp = Float64(-0.70711 * x);
	else
		tmp = 1.6316775383;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.05) || ~((x <= 1.15)))
		tmp = -0.70711 * x;
	else
		tmp = 1.6316775383;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.05], N[Not[LessEqual[x, 1.15]], $MachinePrecision]], N[(-0.70711 * x), $MachinePrecision], 1.6316775383]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.1499999999999999 < 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. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 97.0

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

   5. Applied rewrites 97.0%

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

   if -1.05000000000000004 < x < 1.1499999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.1%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{else}:\\ \;\;\;\;1.6316775383\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 51.7% accurate, 44.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.6316775383)

C

double code(double x) {
	return 1.6316775383;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x):
	return 1.6316775383

Julia

function code(x)
	return 1.6316775383
end

MATLAB

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

Wolfram

code[x_] := 1.6316775383

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 53.9%

   \[\leadsto \color{blue}{1.6316775383} \]

6. Final simplification 53.9%

   \[\leadsto 1.6316775383 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025026 
(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)))