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

Percentage Accurate: 99.9% → 99.9%

Time: 10.2s

Alternatives: 14

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

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

real(8) function code(x)
    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, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(-0.70711 * x + 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. Add Preprocessing

4. Applied rewrites 99.8%

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

5. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

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

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

   3. *-commutative N/A

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

   4. metadata-eval N/A

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

   5. rgt-mult-inverse N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. unsub-neg N/A

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

   9. *-commutative N/A

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

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

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

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

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

   14. mul-1-neg N/A

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

   15. *-commutative N/A

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

   16. associate-*r* N/A

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

   17. mul-1-neg N/A

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

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

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

   19. rgt-mult-inverse N/A

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

   20. metadata-eval N/A

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

   21. metadata-eval 99.8

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

7. Applied rewrites 99.8%

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

Alternative 2: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x\\ \mathbf{if}\;t\_0 \leq 0.5:\\ \;\;\;\;\left(\frac{6.039053782637804}{x} - x\right) \cdot 0.70711\\ \mathbf{elif}\;t\_0 \leq 5:\\ \;\;\;\;1.6316775383 + \mathsf{fma}\left(\mathsf{fma}\left(-1.2692862305735844, x, 1.3436228731669864\right), x, -2.134856267379707\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (-
          (/ (+ (* 0.27061 x) 2.30753) (+ (* (+ (* 0.04481 x) 0.99229) x) 1.0))
          x)))
   (if (<= t_0 0.5)
     (* (- (/ 6.039053782637804 x) x) 0.70711)
     (if (<= t_0 5.0)
       (+
        1.6316775383
        (*
         (fma
          (fma -1.2692862305735844 x 1.3436228731669864)
          x
          -2.134856267379707)
         x))
       (fma -0.70711 x (/ 4.2702753202410175 x))))))

C

double code(double x) {
	double t_0 = (((0.27061 * x) + 2.30753) / ((((0.04481 * x) + 0.99229) * x) + 1.0)) - x;
	double tmp;
	if (t_0 <= 0.5) {
		tmp = ((6.039053782637804 / x) - x) * 0.70711;
	} else if (t_0 <= 5.0) {
		tmp = 1.6316775383 + (fma(fma(-1.2692862305735844, x, 1.3436228731669864), x, -2.134856267379707) * x);
	} else {
		tmp = fma(-0.70711, x, (4.2702753202410175 / x));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(Float64(Float64(0.27061 * x) + 2.30753) / Float64(Float64(Float64(Float64(0.04481 * x) + 0.99229) * x) + 1.0)) - x)
	tmp = 0.0
	if (t_0 <= 0.5)
		tmp = Float64(Float64(Float64(6.039053782637804 / x) - x) * 0.70711);
	elseif (t_0 <= 5.0)
		tmp = Float64(1.6316775383 + Float64(fma(fma(-1.2692862305735844, x, 1.3436228731669864), x, -2.134856267379707) * x));
	else
		tmp = fma(-0.70711, x, Float64(4.2702753202410175 / x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(N[(0.27061 * x), $MachinePrecision] + 2.30753), $MachinePrecision] / N[(N[(N[(N[(0.04481 * x), $MachinePrecision] + 0.99229), $MachinePrecision] * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[t$95$0, 0.5], N[(N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision] * 0.70711), $MachinePrecision], If[LessEqual[t$95$0, 5.0], N[(1.6316775383 + N[(N[(N[(-1.2692862305735844 * x + 1.3436228731669864), $MachinePrecision] * x + -2.134856267379707), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(-0.70711 * x + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.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) < 0.5

   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 97.7

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

   5. Applied rewrites 97.7%

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

   if 0.5 < (-.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) < 5

   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

   4. Applied rewrites 99.9%

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

   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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 97.7

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

   7. Applied rewrites 97.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.2692862305735844, x, 1.3436228731669864\right), x, -2.134856267379707\right), x, 1.6316775383\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 97.7%

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

   if 5 < (-.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 99.8%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. remove-double-neg N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. neg-mul-1 N/A

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

      14. remove-double-neg N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

   5. Applied rewrites 99.4%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq 0.5:\\ \;\;\;\;\left(\frac{6.039053782637804}{x} - x\right) \cdot 0.70711\\ \mathbf{elif}\;\frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq 5:\\ \;\;\;\;1.6316775383 + \mathsf{fma}\left(\mathsf{fma}\left(-1.2692862305735844, x, 1.3436228731669864\right), x, -2.134856267379707\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x\\ \mathbf{if}\;t\_0 \leq 0.5:\\ \;\;\;\;\left(\frac{6.039053782637804}{x} - x\right) \cdot 0.70711\\ \mathbf{elif}\;t\_0 \leq 5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.2692862305735844, x, 1.3436228731669864\right), x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (-
          (/ (+ (* 0.27061 x) 2.30753) (+ (* (+ (* 0.04481 x) 0.99229) x) 1.0))
          x)))
   (if (<= t_0 0.5)
     (* (- (/ 6.039053782637804 x) x) 0.70711)
     (if (<= t_0 5.0)
       (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 = (((0.27061 * x) + 2.30753) / ((((0.04481 * x) + 0.99229) * x) + 1.0)) - x;
	double tmp;
	if (t_0 <= 0.5) {
		tmp = ((6.039053782637804 / x) - x) * 0.70711;
	} else if (t_0 <= 5.0) {
		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(Float64(Float64(Float64(0.27061 * x) + 2.30753) / Float64(Float64(Float64(Float64(0.04481 * x) + 0.99229) * x) + 1.0)) - x)
	tmp = 0.0
	if (t_0 <= 0.5)
		tmp = Float64(Float64(Float64(6.039053782637804 / x) - x) * 0.70711);
	elseif (t_0 <= 5.0)
		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[(N[(N[(N[(0.27061 * x), $MachinePrecision] + 2.30753), $MachinePrecision] / N[(N[(N[(N[(0.04481 * x), $MachinePrecision] + 0.99229), $MachinePrecision] * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[t$95$0, 0.5], N[(N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision] * 0.70711), $MachinePrecision], If[LessEqual[t$95$0, 5.0], 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 (/.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) < 0.5

   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 97.7

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

   5. Applied rewrites 97.7%

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

   if 0.5 < (-.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) < 5

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if 5 < (-.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 99.8%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. remove-double-neg N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. neg-mul-1 N/A

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

      14. remove-double-neg N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

   5. Applied rewrites 99.4%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq 0.5:\\ \;\;\;\;\left(\frac{6.039053782637804}{x} - x\right) \cdot 0.70711\\ \mathbf{elif}\;\frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq 5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.2692862305735844, x, 1.3436228731669864\right), x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ t_1 := \frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x\\ \mathbf{if}\;t\_1 \leq 0.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.2692862305735844, x, 1.3436228731669864\right), x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma -0.70711 x (/ 4.2702753202410175 x)))
        (t_1
         (-
          (/ (+ (* 0.27061 x) 2.30753) (+ (* (+ (* 0.04481 x) 0.99229) x) 1.0))
          x)))
   (if (<= t_1 0.5)
     t_0
     (if (<= t_1 5.0)
       (fma
        (fma
         (fma -1.2692862305735844 x 1.3436228731669864)
         x
         -2.134856267379707)
        x
        1.6316775383)
       t_0))))

C

double code(double x) {
	double t_0 = fma(-0.70711, x, (4.2702753202410175 / x));
	double t_1 = (((0.27061 * x) + 2.30753) / ((((0.04481 * x) + 0.99229) * x) + 1.0)) - x;
	double tmp;
	if (t_1 <= 0.5) {
		tmp = t_0;
	} else if (t_1 <= 5.0) {
		tmp = fma(fma(fma(-1.2692862305735844, x, 1.3436228731669864), x, -2.134856267379707), x, 1.6316775383);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(-0.70711, x, Float64(4.2702753202410175 / x))
	t_1 = Float64(Float64(Float64(Float64(0.27061 * x) + 2.30753) / Float64(Float64(Float64(Float64(0.04481 * x) + 0.99229) * x) + 1.0)) - x)
	tmp = 0.0
	if (t_1 <= 0.5)
		tmp = t_0;
	elseif (t_1 <= 5.0)
		tmp = fma(fma(fma(-1.2692862305735844, x, 1.3436228731669864), x, -2.134856267379707), x, 1.6316775383);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.70711 * x + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(0.27061 * x), $MachinePrecision] + 2.30753), $MachinePrecision] / N[(N[(N[(N[(0.04481 * x), $MachinePrecision] + 0.99229), $MachinePrecision] * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[t$95$1, 0.5], t$95$0, If[LessEqual[t$95$1, 5.0], N[(N[(N[(-1.2692862305735844 * x + 1.3436228731669864), $MachinePrecision] * x + -2.134856267379707), $MachinePrecision] * x + 1.6316775383), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (-.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) < 0.5 or 5 < (-.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 99.7%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. remove-double-neg N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. neg-mul-1 N/A

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

      14. remove-double-neg N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

   5. Applied rewrites 98.7%

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

   if 0.5 < (-.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) < 5

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 97.7

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

   5. Applied rewrites 97.7%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ \mathbf{elif}\;\frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq 5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.2692862305735844, x, 1.3436228731669864\right), x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ t_1 := \frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x\\ \mathbf{if}\;t\_1 \leq -4000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.900161040244073, x, -3.0191289437\right), x, 2.30753\right) \cdot 0.70711\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma -0.70711 x (/ 4.2702753202410175 x)))
        (t_1
         (-
          (/ (+ (* 0.27061 x) 2.30753) (+ (* (+ (* 0.04481 x) 0.99229) x) 1.0))
          x)))
   (if (<= t_1 -4000.0)
     t_0
     (if (<= t_1 5.0)
       (* (fma (fma 1.900161040244073 x -3.0191289437) x 2.30753) 0.70711)
       t_0))))

C

double code(double x) {
	double t_0 = fma(-0.70711, x, (4.2702753202410175 / x));
	double t_1 = (((0.27061 * x) + 2.30753) / ((((0.04481 * x) + 0.99229) * x) + 1.0)) - x;
	double tmp;
	if (t_1 <= -4000.0) {
		tmp = t_0;
	} else if (t_1 <= 5.0) {
		tmp = fma(fma(1.900161040244073, x, -3.0191289437), x, 2.30753) * 0.70711;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(-0.70711, x, Float64(4.2702753202410175 / x))
	t_1 = Float64(Float64(Float64(Float64(0.27061 * x) + 2.30753) / Float64(Float64(Float64(Float64(0.04481 * x) + 0.99229) * x) + 1.0)) - x)
	tmp = 0.0
	if (t_1 <= -4000.0)
		tmp = t_0;
	elseif (t_1 <= 5.0)
		tmp = Float64(fma(fma(1.900161040244073, x, -3.0191289437), x, 2.30753) * 0.70711);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.70711 * x + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(0.27061 * x), $MachinePrecision] + 2.30753), $MachinePrecision] / N[(N[(N[(N[(0.04481 * x), $MachinePrecision] + 0.99229), $MachinePrecision] * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[t$95$1, -4000.0], t$95$0, If[LessEqual[t$95$1, 5.0], N[(N[(N[(1.900161040244073 * x + -3.0191289437), $MachinePrecision] * x + 2.30753), $MachinePrecision] * 0.70711), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (-.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) < -4e3 or 5 < (-.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 99.7%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. remove-double-neg N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. neg-mul-1 N/A

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

      14. remove-double-neg N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

   5. Applied rewrites 99.3%

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

   if -4e3 < (-.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) < 5

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{70711}{100000} \cdot \left(\color{blue}{\left(\frac{230753}{100000} + \frac{-20191289437}{10000000000} \cdot x\right)} - x\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\color{blue}{\left(\frac{-20191289437}{10000000000} \cdot x + \frac{230753}{100000}\right)} - x\right) \]

      2. lower-fma.f64 95.3

         \[\leadsto 0.70711 \cdot \left(\color{blue}{\mathsf{fma}\left(-2.0191289437, x, 2.30753\right)} - x\right) \]

   5. Applied rewrites 95.3%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\mathsf{fma}\left(-2.0191289437, x, 2.30753\right)} - x\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{230753}{100000} + x \cdot \left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{30191289437}{10000000000}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(x \cdot \left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{30191289437}{10000000000}\right) + \frac{230753}{100000}\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\color{blue}{\left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{30191289437}{10000000000}\right) \cdot x} + \frac{230753}{100000}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{30191289437}{10000000000}, x, \frac{230753}{100000}\right)} \]

      4. sub-neg N/A

         \[\leadsto \frac{70711}{100000} \cdot \mathsf{fma}\left(\color{blue}{\frac{1900161040244073}{1000000000000000} \cdot x + \left(\mathsf{neg}\left(\frac{30191289437}{10000000000}\right)\right)}, x, \frac{230753}{100000}\right) \]

      5. metadata-eval N/A

         \[\leadsto \frac{70711}{100000} \cdot \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x + \color{blue}{\frac{-30191289437}{10000000000}}, x, \frac{230753}{100000}\right) \]

      6. lower-fma.f64 96.5

         \[\leadsto 0.70711 \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(1.900161040244073, x, -3.0191289437\right)}, x, 2.30753\right) \]

   8. Applied rewrites 96.5%

      \[\leadsto 0.70711 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(1.900161040244073, x, -3.0191289437\right), x, 2.30753\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq -4000:\\ \;\;\;\;\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ \mathbf{elif}\;\frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq 5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.900161040244073, x, -3.0191289437\right), x, 2.30753\right) \cdot 0.70711\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ t_1 := \frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x\\ \mathbf{if}\;t\_1 \leq -4000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma -0.70711 x (/ 4.2702753202410175 x)))
        (t_1
         (-
          (/ (+ (* 0.27061 x) 2.30753) (+ (* (+ (* 0.04481 x) 0.99229) x) 1.0))
          x)))
   (if (<= t_1 -4000.0)
     t_0
     (if (<= t_1 5.0)
       (fma (fma 1.3436228731669864 x -2.134856267379707) x 1.6316775383)
       t_0))))

C

double code(double x) {
	double t_0 = fma(-0.70711, x, (4.2702753202410175 / x));
	double t_1 = (((0.27061 * x) + 2.30753) / ((((0.04481 * x) + 0.99229) * x) + 1.0)) - x;
	double tmp;
	if (t_1 <= -4000.0) {
		tmp = t_0;
	} else if (t_1 <= 5.0) {
		tmp = fma(fma(1.3436228731669864, x, -2.134856267379707), x, 1.6316775383);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(-0.70711, x, Float64(4.2702753202410175 / x))
	t_1 = Float64(Float64(Float64(Float64(0.27061 * x) + 2.30753) / Float64(Float64(Float64(Float64(0.04481 * x) + 0.99229) * x) + 1.0)) - x)
	tmp = 0.0
	if (t_1 <= -4000.0)
		tmp = t_0;
	elseif (t_1 <= 5.0)
		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[(-0.70711 * x + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(0.27061 * x), $MachinePrecision] + 2.30753), $MachinePrecision] / N[(N[(N[(N[(0.04481 * x), $MachinePrecision] + 0.99229), $MachinePrecision] * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[t$95$1, -4000.0], t$95$0, If[LessEqual[t$95$1, 5.0], N[(N[(1.3436228731669864 * x + -2.134856267379707), $MachinePrecision] * x + 1.6316775383), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (-.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) < -4e3 or 5 < (-.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 99.7%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. remove-double-neg N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. neg-mul-1 N/A

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

      14. remove-double-neg N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

   5. Applied rewrites 99.3%

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

   if -4e3 < (-.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) < 5

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 96.4

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

   5. Applied rewrites 96.4%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq -4000:\\ \;\;\;\;\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ \mathbf{elif}\;\frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq 5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x\\ \mathbf{if}\;t\_0 \leq -4000:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;t\_0 \leq 5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (-
          (/ (+ (* 0.27061 x) 2.30753) (+ (* (+ (* 0.04481 x) 0.99229) x) 1.0))
          x)))
   (if (<= t_0 -4000.0)
     (* x -0.70711)
     (if (<= t_0 5.0)
       (fma (fma 1.3436228731669864 x -2.134856267379707) x 1.6316775383)
       (* x -0.70711)))))

C

double code(double x) {
	double t_0 = (((0.27061 * x) + 2.30753) / ((((0.04481 * x) + 0.99229) * x) + 1.0)) - x;
	double tmp;
	if (t_0 <= -4000.0) {
		tmp = x * -0.70711;
	} else if (t_0 <= 5.0) {
		tmp = fma(fma(1.3436228731669864, x, -2.134856267379707), x, 1.6316775383);
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(Float64(Float64(0.27061 * x) + 2.30753) / Float64(Float64(Float64(Float64(0.04481 * x) + 0.99229) * x) + 1.0)) - x)
	tmp = 0.0
	if (t_0 <= -4000.0)
		tmp = Float64(x * -0.70711);
	elseif (t_0 <= 5.0)
		tmp = fma(fma(1.3436228731669864, x, -2.134856267379707), x, 1.6316775383);
	else
		tmp = Float64(x * -0.70711);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(N[(0.27061 * x), $MachinePrecision] + 2.30753), $MachinePrecision] / N[(N[(N[(N[(0.04481 * x), $MachinePrecision] + 0.99229), $MachinePrecision] * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[t$95$0, -4000.0], N[(x * -0.70711), $MachinePrecision], If[LessEqual[t$95$0, 5.0], N[(N[(1.3436228731669864 * x + -2.134856267379707), $MachinePrecision] * x + 1.6316775383), $MachinePrecision], N[(x * -0.70711), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.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) < -4e3 or 5 < (-.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 99.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if -4e3 < (-.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) < 5

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 96.4

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

   5. Applied rewrites 96.4%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq -4000:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;\frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq 5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x\\ \mathbf{if}\;t\_0 \leq -4000:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;t\_0 \leq 5:\\ \;\;\;\;\mathsf{fma}\left(-3.0191289437, x, 2.30753\right) \cdot 0.70711\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (-
          (/ (+ (* 0.27061 x) 2.30753) (+ (* (+ (* 0.04481 x) 0.99229) x) 1.0))
          x)))
   (if (<= t_0 -4000.0)
     (* x -0.70711)
     (if (<= t_0 5.0)
       (* (fma -3.0191289437 x 2.30753) 0.70711)
       (* x -0.70711)))))

C

double code(double x) {
	double t_0 = (((0.27061 * x) + 2.30753) / ((((0.04481 * x) + 0.99229) * x) + 1.0)) - x;
	double tmp;
	if (t_0 <= -4000.0) {
		tmp = x * -0.70711;
	} else if (t_0 <= 5.0) {
		tmp = fma(-3.0191289437, x, 2.30753) * 0.70711;
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(Float64(Float64(0.27061 * x) + 2.30753) / Float64(Float64(Float64(Float64(0.04481 * x) + 0.99229) * x) + 1.0)) - x)
	tmp = 0.0
	if (t_0 <= -4000.0)
		tmp = Float64(x * -0.70711);
	elseif (t_0 <= 5.0)
		tmp = Float64(fma(-3.0191289437, x, 2.30753) * 0.70711);
	else
		tmp = Float64(x * -0.70711);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(N[(0.27061 * x), $MachinePrecision] + 2.30753), $MachinePrecision] / N[(N[(N[(N[(0.04481 * x), $MachinePrecision] + 0.99229), $MachinePrecision] * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[t$95$0, -4000.0], N[(x * -0.70711), $MachinePrecision], If[LessEqual[t$95$0, 5.0], N[(N[(-3.0191289437 * x + 2.30753), $MachinePrecision] * 0.70711), $MachinePrecision], N[(x * -0.70711), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.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) < -4e3 or 5 < (-.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 99.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if -4e3 < (-.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) < 5

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{70711}{100000} \cdot \left(\color{blue}{\left(\frac{230753}{100000} + \frac{-20191289437}{10000000000} \cdot x\right)} - x\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\color{blue}{\left(\frac{-20191289437}{10000000000} \cdot x + \frac{230753}{100000}\right)} - x\right) \]

      2. lower-fma.f64 95.3

         \[\leadsto 0.70711 \cdot \left(\color{blue}{\mathsf{fma}\left(-2.0191289437, x, 2.30753\right)} - x\right) \]

   5. Applied rewrites 95.3%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\mathsf{fma}\left(-2.0191289437, x, 2.30753\right)} - x\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{230753}{100000} + \frac{-30191289437}{10000000000} \cdot x\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 95.4

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

   8. Applied rewrites 95.4%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq -4000:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;\frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq 5:\\ \;\;\;\;\mathsf{fma}\left(-3.0191289437, x, 2.30753\right) \cdot 0.70711\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x\\ \mathbf{if}\;t\_0 \leq -4000:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;t\_0 \leq 5:\\ \;\;\;\;\mathsf{fma}\left(-2.134856267379707, x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (-
          (/ (+ (* 0.27061 x) 2.30753) (+ (* (+ (* 0.04481 x) 0.99229) x) 1.0))
          x)))
   (if (<= t_0 -4000.0)
     (* x -0.70711)
     (if (<= t_0 5.0)
       (fma -2.134856267379707 x 1.6316775383)
       (* x -0.70711)))))

C

double code(double x) {
	double t_0 = (((0.27061 * x) + 2.30753) / ((((0.04481 * x) + 0.99229) * x) + 1.0)) - x;
	double tmp;
	if (t_0 <= -4000.0) {
		tmp = x * -0.70711;
	} else if (t_0 <= 5.0) {
		tmp = fma(-2.134856267379707, x, 1.6316775383);
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(Float64(Float64(0.27061 * x) + 2.30753) / Float64(Float64(Float64(Float64(0.04481 * x) + 0.99229) * x) + 1.0)) - x)
	tmp = 0.0
	if (t_0 <= -4000.0)
		tmp = Float64(x * -0.70711);
	elseif (t_0 <= 5.0)
		tmp = fma(-2.134856267379707, x, 1.6316775383);
	else
		tmp = Float64(x * -0.70711);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(N[(0.27061 * x), $MachinePrecision] + 2.30753), $MachinePrecision] / N[(N[(N[(N[(0.04481 * x), $MachinePrecision] + 0.99229), $MachinePrecision] * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[t$95$0, -4000.0], N[(x * -0.70711), $MachinePrecision], If[LessEqual[t$95$0, 5.0], N[(-2.134856267379707 * x + 1.6316775383), $MachinePrecision], N[(x * -0.70711), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.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) < -4e3 or 5 < (-.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 99.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if -4e3 < (-.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) < 5

   1. Initial program 99.8%

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

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

      2. lower-fma.f64 95.3

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

   5. Applied rewrites 95.3%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq -4000:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;\frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq 5:\\ \;\;\;\;\mathsf{fma}\left(-2.134856267379707, x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x\\ \mathbf{if}\;t\_0 \leq -4000:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;t\_0 \leq 5:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (-
          (/ (+ (* 0.27061 x) 2.30753) (+ (* (+ (* 0.04481 x) 0.99229) x) 1.0))
          x)))
   (if (<= t_0 -4000.0)
     (* x -0.70711)
     (if (<= t_0 5.0) 1.6316775383 (* x -0.70711)))))

C

double code(double x) {
	double t_0 = (((0.27061 * x) + 2.30753) / ((((0.04481 * x) + 0.99229) * x) + 1.0)) - x;
	double tmp;
	if (t_0 <= -4000.0) {
		tmp = x * -0.70711;
	} else if (t_0 <= 5.0) {
		tmp = 1.6316775383;
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((0.27061d0 * x) + 2.30753d0) / ((((0.04481d0 * x) + 0.99229d0) * x) + 1.0d0)) - x
    if (t_0 <= (-4000.0d0)) then
        tmp = x * (-0.70711d0)
    else if (t_0 <= 5.0d0) then
        tmp = 1.6316775383d0
    else
        tmp = x * (-0.70711d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (((0.27061 * x) + 2.30753) / ((((0.04481 * x) + 0.99229) * x) + 1.0)) - x;
	double tmp;
	if (t_0 <= -4000.0) {
		tmp = x * -0.70711;
	} else if (t_0 <= 5.0) {
		tmp = 1.6316775383;
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (((0.27061 * x) + 2.30753) / ((((0.04481 * x) + 0.99229) * x) + 1.0)) - x
	tmp = 0
	if t_0 <= -4000.0:
		tmp = x * -0.70711
	elif t_0 <= 5.0:
		tmp = 1.6316775383
	else:
		tmp = x * -0.70711
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(Float64(Float64(0.27061 * x) + 2.30753) / Float64(Float64(Float64(Float64(0.04481 * x) + 0.99229) * x) + 1.0)) - x)
	tmp = 0.0
	if (t_0 <= -4000.0)
		tmp = Float64(x * -0.70711);
	elseif (t_0 <= 5.0)
		tmp = 1.6316775383;
	else
		tmp = Float64(x * -0.70711);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (((0.27061 * x) + 2.30753) / ((((0.04481 * x) + 0.99229) * x) + 1.0)) - x;
	tmp = 0.0;
	if (t_0 <= -4000.0)
		tmp = x * -0.70711;
	elseif (t_0 <= 5.0)
		tmp = 1.6316775383;
	else
		tmp = x * -0.70711;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(N[(0.27061 * x), $MachinePrecision] + 2.30753), $MachinePrecision] / N[(N[(N[(N[(0.04481 * x), $MachinePrecision] + 0.99229), $MachinePrecision] * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[t$95$0, -4000.0], N[(x * -0.70711), $MachinePrecision], If[LessEqual[t$95$0, 5.0], 1.6316775383, N[(x * -0.70711), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.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) < -4e3 or 5 < (-.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 99.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if -4e3 < (-.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) < 5

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 93.0%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq -4000:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;\frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq 5:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

   \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{1 + \frac{99229}{100000} \cdot x}} - x\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

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

   2. metadata-eval N/A

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

   3. lft-mult-inverse N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f64 N/A

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

   6. associate-*l* N/A

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

   7. lft-mult-inverse N/A

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

   8. metadata-eval 97.3

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

5. Applied rewrites 97.3%

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

6. Final simplification 97.3%

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

Alternative 12: 98.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(-0.70711 * x + N[(N[(-0.1913510371 * x + -1.6316775383), $MachinePrecision] / 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. Add Preprocessing

4. Applied rewrites 99.8%

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

5. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

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

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

   3. *-commutative N/A

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

   4. metadata-eval N/A

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

   5. rgt-mult-inverse N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. unsub-neg N/A

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

   9. *-commutative N/A

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

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

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

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

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

   14. mul-1-neg N/A

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

   15. *-commutative N/A

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

   16. associate-*r* N/A

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

   17. mul-1-neg N/A

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

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

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

   19. rgt-mult-inverse N/A

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

   20. metadata-eval N/A

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

   21. metadata-eval 99.8

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, \frac{\mathsf{fma}\left(\frac{-1913510371}{10000000000}, x, \frac{-16316775383}{10000000000}\right)}{\mathsf{fma}\left(\color{blue}{\frac{-99229}{100000}}, x, -1\right)}\right) \]
9. Step-by-step derivation

10. Applied rewrites 97.3%

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

Alternative 13: 98.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(-0.70711 * x + 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. Add Preprocessing

4. Applied rewrites 99.8%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, \frac{\color{blue}{\frac{-16316775383}{10000000000}}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4481}{100000}, x, \frac{-99229}{100000}\right), x, -1\right)}\right) \]
6. Step-by-step derivation

7. Applied rewrites 96.7%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, \frac{\frac{-16316775383}{10000000000}}{\mathsf{fma}\left(\color{blue}{\frac{-99229}{100000}}, x, -1\right)}\right) \]
9. Step-by-step derivation

10. Applied rewrites 96.7%

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

Alternative 14: 49.5% 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

real(8) function code(x)
    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. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Applied rewrites 43.7%

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

Reproduce

herbie shell --seed 2024285 
(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)))