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

Percentage Accurate: 99.8% → 99.9%

Time: 4.9s

Alternatives: 8

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-flip N/A

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

   4. distribute-rgt-in N/A

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

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

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

   6. lower--.f64 N/A

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

3. Applied rewrites 99.8%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. +-commutative N/A

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

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

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

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

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

   7. lower-fma.f64 N/A

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

   8. metadata-eval 99.8%

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

   9. lift-*.f64 N/A

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

   10. lift-fma.f64 N/A

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

   11. distribute-lft-in N/A

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

   12. *-commutative N/A

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

   13. associate-*r* N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f64 N/A

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

   17. metadata-eval N/A

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

   18. metadata-eval 99.9%

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

5. Applied rewrites 99.9%

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

Alternative 2: 98.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175}{x}\right)\\ \mathbf{if}\;x \leq -2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 470000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma x -0.70711 (/ 4.2702753202410175 x))))
   (if (<= x -2.0)
     t_0
     (if (<= x 470000.0)
       (fma (fma 1.3436228731669864 x -2.134856267379707) x 1.6316775383)
       t_0))))

C

double code(double x) {
	double t_0 = fma(x, -0.70711, (4.2702753202410175 / x));
	double tmp;
	if (x <= -2.0) {
		tmp = t_0;
	} else if (x <= 470000.0) {
		tmp = fma(fma(1.3436228731669864, x, -2.134856267379707), x, 1.6316775383);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(x, -0.70711, Float64(4.2702753202410175 / x))
	tmp = 0.0
	if (x <= -2.0)
		tmp = t_0;
	elseif (x <= 470000.0)
		tmp = fma(fma(1.3436228731669864, x, -2.134856267379707), x, 1.6316775383);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * -0.70711 + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.0], t$95$0, If[LessEqual[x, 470000.0], N[(N[(1.3436228731669864 * x + -2.134856267379707), $MachinePrecision] * x + 1.6316775383), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2 or 4.7e5 < x

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. distribute-rgt-in N/A

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

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

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

      6. lower--.f64 N/A

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

   3. Applied rewrites 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval 99.8%

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. distribute-lft-in N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval 99.9%

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 51.7%

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

   8. Applied rewrites 51.7%

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

   if -2 < x < 4.7e5

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. distribute-rgt-in N/A

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

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

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

      6. lower--.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 50.8%

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

   6. Applied rewrites 50.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-in N/A

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

      12. sub-flip N/A

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

      13. lift--.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 50.8%

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

      16. lift--.f64 N/A

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

      17. sub-flip N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 N/A

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

      20. metadata-eval 50.8%

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

   8. Applied rewrites 50.8%

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

Alternative 3: 98.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175}{x}\right)\\ \mathbf{if}\;x \leq -2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 180:\\ \;\;\;\;\mathsf{fma}\left(-2.134856267379707, x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma x -0.70711 (/ 4.2702753202410175 x))))
   (if (<= x -2.0)
     t_0
     (if (<= x 180.0) (fma -2.134856267379707 x 1.6316775383) t_0))))

C

double code(double x) {
	double t_0 = fma(x, -0.70711, (4.2702753202410175 / x));
	double tmp;
	if (x <= -2.0) {
		tmp = t_0;
	} else if (x <= 180.0) {
		tmp = fma(-2.134856267379707, x, 1.6316775383);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(x, -0.70711, Float64(4.2702753202410175 / x))
	tmp = 0.0
	if (x <= -2.0)
		tmp = t_0;
	elseif (x <= 180.0)
		tmp = fma(-2.134856267379707, x, 1.6316775383);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * -0.70711 + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.0], t$95$0, If[LessEqual[x, 180.0], N[(-2.134856267379707 * x + 1.6316775383), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2 or 180 < x

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. distribute-rgt-in N/A

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

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

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

      6. lower--.f64 N/A

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

   3. Applied rewrites 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval 99.8%

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. distribute-lft-in N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval 99.9%

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 51.7%

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

   8. Applied rewrites 51.7%

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

   if -2 < x < 180

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. distribute-rgt-in N/A

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

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

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

      6. lower--.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 50.8%

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

   6. Applied rewrites 50.8%

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

   7. Taylor expanded in x around 0

      \[\leadsto 1.6316775383 + x \cdot \frac{-2134856267379707}{1000000000000000} \]
   8. Step-by-step derivation

   9. Applied rewrites 57.7%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 57.7%

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

   11. Applied rewrites 57.7%

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

Alternative 4: 98.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-flip N/A

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

   4. distribute-rgt-in N/A

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

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

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

   6. lower--.f64 N/A

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 98.3%

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

7. Taylor expanded in x around 0

   \[\leadsto \frac{-1.6316775383}{\mathsf{fma}\left(\color{blue}{\frac{-99229}{100000}}, x, -1\right)} - x \cdot 0.70711 \]
8. Step-by-step derivation

9. Applied rewrites 98.3%

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

Alternative 5: 98.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-flip N/A

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

   4. distribute-rgt-in N/A

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

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

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

   6. lower--.f64 N/A

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 98.3%

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

7. Taylor expanded in x around 0

   \[\leadsto \frac{-1.6316775383}{\mathsf{fma}\left(\color{blue}{\frac{-99229}{100000}}, x, -1\right)} - x \cdot 0.70711 \]
8. Step-by-step derivation

9. Applied rewrites 98.3%

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

    1. lift--.f64 N/A

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

    2. lift-*.f64 N/A

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

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

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

    4. +-commutative N/A

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

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

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

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

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

    7. metadata-eval N/A

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

    8. lower-fma.f64 98.3%

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

11. Applied rewrites 98.3%

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

Alternative 6: 98.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -0.98:\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{elif}\;x \leq 180:\\ \;\;\;\;\mathsf{fma}\left(-2.134856267379707, x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;-0.70711 \cdot x\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.98)
   (* -0.70711 x)
   (if (<= x 180.0) (fma -2.134856267379707 x 1.6316775383) (* -0.70711 x))))

C

double code(double x) {
	double tmp;
	if (x <= -0.98) {
		tmp = -0.70711 * x;
	} else if (x <= 180.0) {
		tmp = fma(-2.134856267379707, x, 1.6316775383);
	} else {
		tmp = -0.70711 * x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.98)
		tmp = Float64(-0.70711 * x);
	elseif (x <= 180.0)
		tmp = fma(-2.134856267379707, x, 1.6316775383);
	else
		tmp = Float64(-0.70711 * x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -0.98], N[(-0.70711 * x), $MachinePrecision], If[LessEqual[x, 180.0], N[(-2.134856267379707 * x + 1.6316775383), $MachinePrecision], N[(-0.70711 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.97999999999999998 or 180 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 51.9%

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

   4. Applied rewrites 51.9%

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

   if -0.97999999999999998 < x < 180

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. distribute-rgt-in N/A

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

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

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

      6. lower--.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 50.8%

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

   6. Applied rewrites 50.8%

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

   7. Taylor expanded in x around 0

      \[\leadsto 1.6316775383 + x \cdot \frac{-2134856267379707}{1000000000000000} \]
   8. Step-by-step derivation

   9. Applied rewrites 57.7%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 57.7%

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

   11. Applied rewrites 57.7%

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

Alternative 7: 98.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{elif}\;x \leq 470000:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;-0.70711 \cdot x\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.0)
   (* -0.70711 x)
   (if (<= x 470000.0) 1.6316775383 (* -0.70711 x))))

C

double code(double x) {
	double tmp;
	if (x <= -2.0) {
		tmp = -0.70711 * x;
	} else if (x <= 470000.0) {
		tmp = 1.6316775383;
	} else {
		tmp = -0.70711 * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2.0d0)) then
        tmp = (-0.70711d0) * x
    else if (x <= 470000.0d0) then
        tmp = 1.6316775383d0
    else
        tmp = (-0.70711d0) * x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.0) {
		tmp = -0.70711 * x;
	} else if (x <= 470000.0) {
		tmp = 1.6316775383;
	} else {
		tmp = -0.70711 * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.0:
		tmp = -0.70711 * x
	elif x <= 470000.0:
		tmp = 1.6316775383
	else:
		tmp = -0.70711 * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.0)
		tmp = Float64(-0.70711 * x);
	elseif (x <= 470000.0)
		tmp = 1.6316775383;
	else
		tmp = Float64(-0.70711 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.0)
		tmp = -0.70711 * x;
	elseif (x <= 470000.0)
		tmp = 1.6316775383;
	else
		tmp = -0.70711 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.0], N[(-0.70711 * x), $MachinePrecision], If[LessEqual[x, 470000.0], 1.6316775383, N[(-0.70711 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2 or 4.7e5 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 51.9%

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

   4. Applied rewrites 51.9%

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

   if -2 < x < 4.7e5

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 49.9%

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

Alternative 8: 49.9% accurate, 27.0× speedup

Math

\[1.6316775383 \]

FPCore

(FPCore (x) :precision binary64 1.6316775383)

C

double code(double x) {
	return 1.6316775383;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x):
	return 1.6316775383

Julia

function code(x)
	return 1.6316775383
end

MATLAB

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

Wolfram

code[x_] := 1.6316775383

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 49.9%

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

Reproduce

herbie shell --seed 2025187 
(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)))