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

Percentage Accurate: 99.9% → 99.9%

Time: 9.4s

Alternatives: 8

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 99.9%

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

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. rgt-mult-inverse N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

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

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

   8. *-commutative N/A

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

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

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

   10. associate-*r* N/A

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

   11. rgt-mult-inverse N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval 99.9

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

7. Applied rewrites 99.9%

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

Alternative 2: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\\ \mathbf{if}\;t\_0 \leq -20000 \lor \neg \left(t\_0 \leq 4\right):\\ \;\;\;\;\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175 - \frac{58.14938538768042}{x}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right), x, 1.6316775383\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (-
          (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481)))))
          x)))
   (if (or (<= t_0 -20000.0) (not (<= t_0 4.0)))
     (fma -0.70711 x (/ (- 4.2702753202410175 (/ 58.14938538768042 x)) x))
     (fma (fma 1.3436228731669864 x -2.134856267379707) x 1.6316775383))))

C

double code(double x) {
	double t_0 = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
	double tmp;
	if ((t_0 <= -20000.0) || !(t_0 <= 4.0)) {
		tmp = fma(-0.70711, x, ((4.2702753202410175 - (58.14938538768042 / x)) / x));
	} else {
		tmp = fma(fma(1.3436228731669864, x, -2.134856267379707), x, 1.6316775383);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
	tmp = 0.0
	if ((t_0 <= -20000.0) || !(t_0 <= 4.0))
		tmp = fma(-0.70711, x, Float64(Float64(4.2702753202410175 - Float64(58.14938538768042 / x)) / x));
	else
		tmp = fma(fma(1.3436228731669864, x, -2.134856267379707), x, 1.6316775383);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = 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]}, If[Or[LessEqual[t$95$0, -20000.0], N[Not[LessEqual[t$95$0, 4.0]], $MachinePrecision]], N[(-0.70711 * x + N[(N[(4.2702753202410175 - N[(58.14938538768042 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(1.3436228731669864 * x + -2.134856267379707), $MachinePrecision] * x + 1.6316775383), $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) < -2e4 or 4 < (-.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

   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 inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 99.5

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

   7. Applied rewrites 99.5%

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

   if -2e4 < (-.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) < 4

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{16316775383}{10000000000} + x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right)} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq -20000 \lor \neg \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq 4\right):\\ \;\;\;\;\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175 - \frac{58.14938538768042}{x}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right), x, 1.6316775383\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\\ \mathbf{if}\;t\_0 \leq -20000 \lor \neg \left(t\_0 \leq 4\right):\\ \;\;\;\;\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right), x, 1.6316775383\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (-
          (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481)))))
          x)))
   (if (or (<= t_0 -20000.0) (not (<= t_0 4.0)))
     (fma -0.70711 x (/ 4.2702753202410175 x))
     (fma (fma 1.3436228731669864 x -2.134856267379707) x 1.6316775383))))

C

double code(double x) {
	double t_0 = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
	double tmp;
	if ((t_0 <= -20000.0) || !(t_0 <= 4.0)) {
		tmp = fma(-0.70711, x, (4.2702753202410175 / x));
	} else {
		tmp = fma(fma(1.3436228731669864, x, -2.134856267379707), x, 1.6316775383);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
	tmp = 0.0
	if ((t_0 <= -20000.0) || !(t_0 <= 4.0))
		tmp = fma(-0.70711, x, Float64(4.2702753202410175 / x));
	else
		tmp = fma(fma(1.3436228731669864, x, -2.134856267379707), x, 1.6316775383);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = 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]}, If[Or[LessEqual[t$95$0, -20000.0], N[Not[LessEqual[t$95$0, 4.0]], $MachinePrecision]], N[(-0.70711 * x + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision], N[(N[(1.3436228731669864 * x + -2.134856267379707), $MachinePrecision] * x + 1.6316775383), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.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) < -2e4 or 4 < (-.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 99.3%

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

   if -2e4 < (-.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) < 4

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{16316775383}{10000000000} + x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right)} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq -20000 \lor \neg \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq 4\right):\\ \;\;\;\;\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right), x, 1.6316775383\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\\ \mathbf{if}\;t\_0 \leq -20000:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;t\_0 \leq 4:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (-
          (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481)))))
          x)))
   (if (<= t_0 -20000.0)
     (* 0.70711 (- (/ 6.039053782637804 x) x))
     (if (<= t_0 4.0)
       (fma (fma 1.3436228731669864 x -2.134856267379707) x 1.6316775383)
       (fma -0.70711 x (/ 4.2702753202410175 x))))))

C

double code(double x) {
	double t_0 = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
	double tmp;
	if (t_0 <= -20000.0) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else if (t_0 <= 4.0) {
		tmp = fma(fma(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(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
	tmp = 0.0
	if (t_0 <= -20000.0)
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	elseif (t_0 <= 4.0)
		tmp = fma(fma(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[(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]}, If[LessEqual[t$95$0, -20000.0], N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4.0], N[(N[(1.3436228731669864 * 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) < -2e4

   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 99.3

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

   5. Applied rewrites 99.3%

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

   if -2e4 < (-.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) < 4

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{16316775383}{10000000000} + x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right)} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   if 4 < (-.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 99.2%

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

Alternative 5: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\\ \mathbf{if}\;t\_0 \leq -20000 \lor \neg \left(t\_0 \leq 4\right):\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right), x, 1.6316775383\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (-
          (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481)))))
          x)))
   (if (or (<= t_0 -20000.0) (not (<= t_0 4.0)))
     (* -0.70711 x)
     (fma (fma 1.3436228731669864 x -2.134856267379707) x 1.6316775383))))

C

double code(double x) {
	double t_0 = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
	double tmp;
	if ((t_0 <= -20000.0) || !(t_0 <= 4.0)) {
		tmp = -0.70711 * x;
	} else {
		tmp = fma(fma(1.3436228731669864, x, -2.134856267379707), x, 1.6316775383);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
	tmp = 0.0
	if ((t_0 <= -20000.0) || !(t_0 <= 4.0))
		tmp = Float64(-0.70711 * x);
	else
		tmp = fma(fma(1.3436228731669864, x, -2.134856267379707), x, 1.6316775383);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = 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]}, If[Or[LessEqual[t$95$0, -20000.0], N[Not[LessEqual[t$95$0, 4.0]], $MachinePrecision]], N[(-0.70711 * x), $MachinePrecision], N[(N[(1.3436228731669864 * x + -2.134856267379707), $MachinePrecision] * x + 1.6316775383), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.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) < -2e4 or 4 < (-.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

      \[\leadsto \color{blue}{\frac{-70711}{100000} \cdot x} \]
   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 -2e4 < (-.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) < 4

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{16316775383}{10000000000} + x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right)} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq -20000 \lor \neg \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq 4\right):\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right), x, 1.6316775383\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\\ \mathbf{if}\;t\_0 \leq -20000 \lor \neg \left(t\_0 \leq 4\right):\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2.134856267379707, x, 1.6316775383\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (-
          (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481)))))
          x)))
   (if (or (<= t_0 -20000.0) (not (<= t_0 4.0)))
     (* -0.70711 x)
     (fma -2.134856267379707 x 1.6316775383))))

C

double code(double x) {
	double t_0 = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
	double tmp;
	if ((t_0 <= -20000.0) || !(t_0 <= 4.0)) {
		tmp = -0.70711 * x;
	} else {
		tmp = fma(-2.134856267379707, x, 1.6316775383);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
	tmp = 0.0
	if ((t_0 <= -20000.0) || !(t_0 <= 4.0))
		tmp = Float64(-0.70711 * x);
	else
		tmp = fma(-2.134856267379707, x, 1.6316775383);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = 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]}, If[Or[LessEqual[t$95$0, -20000.0], N[Not[LessEqual[t$95$0, 4.0]], $MachinePrecision]], N[(-0.70711 * x), $MachinePrecision], N[(-2.134856267379707 * x + 1.6316775383), $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) < -2e4 or 4 < (-.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

      \[\leadsto \color{blue}{\frac{-70711}{100000} \cdot x} \]
   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 -2e4 < (-.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) < 4

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq -20000 \lor \neg \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq 4\right):\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2.134856267379707, x, 1.6316775383\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\\ \mathbf{if}\;t\_0 \leq -20000 \lor \neg \left(t\_0 \leq 4\right):\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{else}:\\ \;\;\;\;1.6316775383\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (-
          (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481)))))
          x)))
   (if (or (<= t_0 -20000.0) (not (<= t_0 4.0))) (* -0.70711 x) 1.6316775383)))

C

double code(double x) {
	double t_0 = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
	double tmp;
	if ((t_0 <= -20000.0) || !(t_0 <= 4.0)) {
		tmp = -0.70711 * x;
	} else {
		tmp = 1.6316775383;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x
    if ((t_0 <= (-20000.0d0)) .or. (.not. (t_0 <= 4.0d0))) then
        tmp = (-0.70711d0) * x
    else
        tmp = 1.6316775383d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
	double tmp;
	if ((t_0 <= -20000.0) || !(t_0 <= 4.0)) {
		tmp = -0.70711 * x;
	} else {
		tmp = 1.6316775383;
	}
	return tmp;
}

Python

def code(x):
	t_0 = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x
	tmp = 0
	if (t_0 <= -20000.0) or not (t_0 <= 4.0):
		tmp = -0.70711 * x
	else:
		tmp = 1.6316775383
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
	tmp = 0.0
	if ((t_0 <= -20000.0) || !(t_0 <= 4.0))
		tmp = Float64(-0.70711 * x);
	else
		tmp = 1.6316775383;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
	tmp = 0.0;
	if ((t_0 <= -20000.0) || ~((t_0 <= 4.0)))
		tmp = -0.70711 * x;
	else
		tmp = 1.6316775383;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = 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]}, If[Or[LessEqual[t$95$0, -20000.0], N[Not[LessEqual[t$95$0, 4.0]], $MachinePrecision]], N[(-0.70711 * x), $MachinePrecision], 1.6316775383]]

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) < -2e4 or 4 < (-.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

      \[\leadsto \color{blue}{\frac{-70711}{100000} \cdot x} \]
   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 -2e4 < (-.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) < 4

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.2%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq -20000 \lor \neg \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq 4\right):\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{else}:\\ \;\;\;\;1.6316775383\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.1% 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 99.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 52.0%

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

Reproduce

herbie shell --seed 2024318 
(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)))