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

Percentage Accurate: 99.8% → 99.9%

Time: 10.3s

Alternatives: 10

Speedup: 1.0×

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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * (-0.70711d0)) + ((1.6316775383d0 + (x * 0.1913510371d0)) / (1.0d0 - (x * ((-0.99229d0) + (x * (-0.04481d0))))))
end function

Java

public static double code(double x) {
	return (x * -0.70711) + ((1.6316775383 + (x * 0.1913510371)) / (1.0 - (x * (-0.99229 + (x * -0.04481)))));
}

Python

def code(x):
	return (x * -0.70711) + ((1.6316775383 + (x * 0.1913510371)) / (1.0 - (x * (-0.99229 + (x * -0.04481)))))

Julia

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

MATLAB

function tmp = code(x)
	tmp = (x * -0.70711) + ((1.6316775383 + (x * 0.1913510371)) / (1.0 - (x * (-0.99229 + (x * -0.04481)))));
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{70711}{100000} \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}\right)}\right) \]

   6. neg-mul-1 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{70711}{100000} \cdot \left(-1 \cdot x\right)\right), \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} \cdot \frac{70711}{100000}\right)\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\frac{70711}{100000} \cdot -1\right) \cdot x\right), \left(\color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} \cdot \frac{70711}{100000}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(\frac{70711}{100000} \cdot -1\right)\right), \left(\color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} \cdot \frac{70711}{100000}\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{70711}{100000} \cdot -1\right)\right), \left(\color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} \cdot \frac{70711}{100000}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} \cdot \frac{70711}{100000}\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \left(\frac{70711}{100000} \cdot \color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}}\right)\right) \]

   12. associate-*r/ N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \left(\frac{\frac{70711}{100000} \cdot \left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}}\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\left(\frac{70711}{100000} \cdot \left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)\right), \color{blue}{\left(1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)}\right)\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{x \cdot -0.70711 + \frac{1.6316775383 + x \cdot 0.1913510371}{1 - x \cdot \left(-0.99229 + x \cdot -0.04481\right)}} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;x \leq 6.2:\\ \;\;\;\;1.6316775383 + x \cdot \left(-2.134856267379707 + x \cdot \left(1.3436228731669864 + x \cdot -1.2692862305735844\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711 + \frac{4.2702753202410175 + \frac{\frac{1192.3851440772235}{x} + -58.14938538768042}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (* 0.70711 (- (/ 6.039053782637804 x) x))
   (if (<= x 6.2)
     (+
      1.6316775383
      (*
       x
       (+
        -2.134856267379707
        (* x (+ 1.3436228731669864 (* x -1.2692862305735844))))))
     (+
      (* x -0.70711)
      (/
       (+
        4.2702753202410175
        (/ (+ (/ 1192.3851440772235 x) -58.14938538768042) x))
       x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else if (x <= 6.2) {
		tmp = 1.6316775383 + (x * (-2.134856267379707 + (x * (1.3436228731669864 + (x * -1.2692862305735844)))));
	} else {
		tmp = (x * -0.70711) + ((4.2702753202410175 + (((1192.3851440772235 / x) + -58.14938538768042) / x)) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.05d0)) then
        tmp = 0.70711d0 * ((6.039053782637804d0 / x) - x)
    else if (x <= 6.2d0) then
        tmp = 1.6316775383d0 + (x * ((-2.134856267379707d0) + (x * (1.3436228731669864d0 + (x * (-1.2692862305735844d0))))))
    else
        tmp = (x * (-0.70711d0)) + ((4.2702753202410175d0 + (((1192.3851440772235d0 / x) + (-58.14938538768042d0)) / x)) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else if (x <= 6.2) {
		tmp = 1.6316775383 + (x * (-2.134856267379707 + (x * (1.3436228731669864 + (x * -1.2692862305735844)))));
	} else {
		tmp = (x * -0.70711) + ((4.2702753202410175 + (((1192.3851440772235 / x) + -58.14938538768042) / x)) / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = 0.70711 * ((6.039053782637804 / x) - x)
	elif x <= 6.2:
		tmp = 1.6316775383 + (x * (-2.134856267379707 + (x * (1.3436228731669864 + (x * -1.2692862305735844)))))
	else:
		tmp = (x * -0.70711) + ((4.2702753202410175 + (((1192.3851440772235 / x) + -58.14938538768042) / x)) / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	elseif (x <= 6.2)
		tmp = Float64(1.6316775383 + Float64(x * Float64(-2.134856267379707 + Float64(x * Float64(1.3436228731669864 + Float64(x * -1.2692862305735844))))));
	else
		tmp = Float64(Float64(x * -0.70711) + Float64(Float64(4.2702753202410175 + Float64(Float64(Float64(1192.3851440772235 / x) + -58.14938538768042) / x)) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	elseif (x <= 6.2)
		tmp = 1.6316775383 + (x * (-2.134856267379707 + (x * (1.3436228731669864 + (x * -1.2692862305735844)))));
	else
		tmp = (x * -0.70711) + ((4.2702753202410175 + (((1192.3851440772235 / x) + -58.14938538768042) / x)) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.2], N[(1.6316775383 + N[(x * N[(-2.134856267379707 + N[(x * N[(1.3436228731669864 + N[(x * -1.2692862305735844), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * -0.70711), $MachinePrecision] + N[(N[(4.2702753202410175 + N[(N[(N[(1192.3851440772235 / x), $MachinePrecision] + -58.14938538768042), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004

   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

      \[\leadsto \mathsf{*.f64}\left(\frac{70711}{100000}, \mathsf{\_.f64}\left(\color{blue}{\left(\frac{\frac{27061}{4481}}{x}\right)}, x\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(\frac{70711}{100000}, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{27061}{4481}, x\right), x\right)\right) \]

   5. Simplified 99.8%

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

   if -1.05000000000000004 < x < 6.20000000000000018

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) + \frac{-2134856267379707}{1000000000000000}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(x, \left(\frac{-2134856267379707}{1000000000000000} + \color{blue}{x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-2134856267379707}{1000000000000000}, \color{blue}{\left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right)\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-2134856267379707}{1000000000000000}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right)}\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-2134856267379707}{1000000000000000}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{134362287316698645903}{100000000000000000000}, \color{blue}{\left(\frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right)}\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-2134856267379707}{1000000000000000}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{134362287316698645903}{100000000000000000000}, \left(x \cdot \color{blue}{\frac{-12692862305735843227608787}{10000000000000000000000000}}\right)\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 99.5%

          \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-2134856267379707}{1000000000000000}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{134362287316698645903}{100000000000000000000}, \mathsf{*.f64}\left(x, \color{blue}{\frac{-12692862305735843227608787}{10000000000000000000000000}}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 99.5%

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

   if 6.20000000000000018 < 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. Step-by-step derivation

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{70711}{100000} \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}\right)}\right) \]

      6. neg-mul-1 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{70711}{100000} \cdot \left(-1 \cdot x\right)\right), \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} \cdot \frac{70711}{100000}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\frac{70711}{100000} \cdot -1\right) \cdot x\right), \left(\color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} \cdot \frac{70711}{100000}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(\frac{70711}{100000} \cdot -1\right)\right), \left(\color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} \cdot \frac{70711}{100000}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{70711}{100000} \cdot -1\right)\right), \left(\color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} \cdot \frac{70711}{100000}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} \cdot \frac{70711}{100000}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \left(\frac{70711}{100000} \cdot \color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}}\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \left(\frac{\frac{70711}{100000} \cdot \left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\left(\frac{70711}{100000} \cdot \left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)\right), \color{blue}{\left(1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)}\right)\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x \cdot -0.70711 + \frac{1.6316775383 + x \cdot 0.1913510371}{1 - x \cdot \left(-0.99229 + x \cdot -0.04481\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \color{blue}{\left(\frac{\left(\frac{1913510371}{448100000} + \frac{\frac{335267464412236892}{281173802003125}}{{x}^{2}}\right) - \frac{3648757816023}{62748003125} \cdot \frac{1}{x}}{x}\right)}\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\left(\left(\frac{1913510371}{448100000} + \frac{\frac{335267464412236892}{281173802003125}}{{x}^{2}}\right) - \frac{3648757816023}{62748003125} \cdot \frac{1}{x}\right), \color{blue}{x}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\left(\frac{1913510371}{448100000} + \left(\frac{\frac{335267464412236892}{281173802003125}}{{x}^{2}} - \frac{3648757816023}{62748003125} \cdot \frac{1}{x}\right)\right), x\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\left(\frac{1913510371}{448100000} + \left(\frac{\frac{335267464412236892}{281173802003125}}{x \cdot x} - \frac{3648757816023}{62748003125} \cdot \frac{1}{x}\right)\right), x\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\left(\frac{1913510371}{448100000} + \left(\frac{\frac{\frac{335267464412236892}{281173802003125}}{x}}{x} - \frac{3648757816023}{62748003125} \cdot \frac{1}{x}\right)\right), x\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\left(\frac{1913510371}{448100000} + \left(\frac{\frac{\frac{335267464412236892}{281173802003125} \cdot 1}{x}}{x} - \frac{3648757816023}{62748003125} \cdot \frac{1}{x}\right)\right), x\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\left(\frac{1913510371}{448100000} + \left(\frac{\frac{335267464412236892}{281173802003125} \cdot \frac{1}{x}}{x} - \frac{3648757816023}{62748003125} \cdot \frac{1}{x}\right)\right), x\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\left(\frac{1913510371}{448100000} + \left(\frac{\frac{335267464412236892}{281173802003125} \cdot \frac{1}{x}}{x} - \frac{\frac{3648757816023}{62748003125} \cdot 1}{x}\right)\right), x\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\left(\frac{1913510371}{448100000} + \left(\frac{\frac{335267464412236892}{281173802003125} \cdot \frac{1}{x}}{x} - \frac{\frac{3648757816023}{62748003125}}{x}\right)\right), x\right)\right) \]

      9. div-sub N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\left(\frac{1913510371}{448100000} + \frac{\frac{335267464412236892}{281173802003125} \cdot \frac{1}{x} - \frac{3648757816023}{62748003125}}{x}\right), x\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1913510371}{448100000}, \left(\frac{\frac{335267464412236892}{281173802003125} \cdot \frac{1}{x} - \frac{3648757816023}{62748003125}}{x}\right)\right), x\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1913510371}{448100000}, \mathsf{/.f64}\left(\left(\frac{335267464412236892}{281173802003125} \cdot \frac{1}{x} - \frac{3648757816023}{62748003125}\right), x\right)\right), x\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1913510371}{448100000}, \mathsf{/.f64}\left(\left(\frac{335267464412236892}{281173802003125} \cdot \frac{1}{x} + \left(\mathsf{neg}\left(\frac{3648757816023}{62748003125}\right)\right)\right), x\right)\right), x\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1913510371}{448100000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{335267464412236892}{281173802003125} \cdot \frac{1}{x}\right), \left(\mathsf{neg}\left(\frac{3648757816023}{62748003125}\right)\right)\right), x\right)\right), x\right)\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1913510371}{448100000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{335267464412236892}{281173802003125} \cdot 1}{x}\right), \left(\mathsf{neg}\left(\frac{3648757816023}{62748003125}\right)\right)\right), x\right)\right), x\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1913510371}{448100000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{335267464412236892}{281173802003125}}{x}\right), \left(\mathsf{neg}\left(\frac{3648757816023}{62748003125}\right)\right)\right), x\right)\right), x\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1913510371}{448100000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{335267464412236892}{281173802003125}, x\right), \left(\mathsf{neg}\left(\frac{3648757816023}{62748003125}\right)\right)\right), x\right)\right), x\right)\right) \]

      17. metadata-eval 99.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1913510371}{448100000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{335267464412236892}{281173802003125}, x\right), \frac{-3648757816023}{62748003125}\right), x\right)\right), x\right)\right) \]

   7. Simplified 99.6%

      \[\leadsto x \cdot -0.70711 + \color{blue}{\frac{4.2702753202410175 + \frac{\frac{1192.3851440772235}{x} + -58.14938538768042}{x}}{x}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;x \leq 2.2:\\ \;\;\;\;1.6316775383 + x \cdot \left(-2.134856267379707 + x \cdot \left(1.3436228731669864 + x \cdot -1.2692862305735844\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711 + \frac{4.2702753202410175 + \frac{-58.14938538768042}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (* 0.70711 (- (/ 6.039053782637804 x) x))
   (if (<= x 2.2)
     (+
      1.6316775383
      (*
       x
       (+
        -2.134856267379707
        (* x (+ 1.3436228731669864 (* x -1.2692862305735844))))))
     (+
      (* x -0.70711)
      (/ (+ 4.2702753202410175 (/ -58.14938538768042 x)) x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else if (x <= 2.2) {
		tmp = 1.6316775383 + (x * (-2.134856267379707 + (x * (1.3436228731669864 + (x * -1.2692862305735844)))));
	} else {
		tmp = (x * -0.70711) + ((4.2702753202410175 + (-58.14938538768042 / x)) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.05d0)) then
        tmp = 0.70711d0 * ((6.039053782637804d0 / x) - x)
    else if (x <= 2.2d0) then
        tmp = 1.6316775383d0 + (x * ((-2.134856267379707d0) + (x * (1.3436228731669864d0 + (x * (-1.2692862305735844d0))))))
    else
        tmp = (x * (-0.70711d0)) + ((4.2702753202410175d0 + ((-58.14938538768042d0) / x)) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else if (x <= 2.2) {
		tmp = 1.6316775383 + (x * (-2.134856267379707 + (x * (1.3436228731669864 + (x * -1.2692862305735844)))));
	} else {
		tmp = (x * -0.70711) + ((4.2702753202410175 + (-58.14938538768042 / x)) / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = 0.70711 * ((6.039053782637804 / x) - x)
	elif x <= 2.2:
		tmp = 1.6316775383 + (x * (-2.134856267379707 + (x * (1.3436228731669864 + (x * -1.2692862305735844)))))
	else:
		tmp = (x * -0.70711) + ((4.2702753202410175 + (-58.14938538768042 / x)) / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	elseif (x <= 2.2)
		tmp = Float64(1.6316775383 + Float64(x * Float64(-2.134856267379707 + Float64(x * Float64(1.3436228731669864 + Float64(x * -1.2692862305735844))))));
	else
		tmp = Float64(Float64(x * -0.70711) + Float64(Float64(4.2702753202410175 + Float64(-58.14938538768042 / x)) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	elseif (x <= 2.2)
		tmp = 1.6316775383 + (x * (-2.134856267379707 + (x * (1.3436228731669864 + (x * -1.2692862305735844)))));
	else
		tmp = (x * -0.70711) + ((4.2702753202410175 + (-58.14938538768042 / x)) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.2], N[(1.6316775383 + N[(x * N[(-2.134856267379707 + N[(x * N[(1.3436228731669864 + N[(x * -1.2692862305735844), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * -0.70711), $MachinePrecision] + N[(N[(4.2702753202410175 + N[(-58.14938538768042 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004

   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

      \[\leadsto \mathsf{*.f64}\left(\frac{70711}{100000}, \mathsf{\_.f64}\left(\color{blue}{\left(\frac{\frac{27061}{4481}}{x}\right)}, x\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(\frac{70711}{100000}, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{27061}{4481}, x\right), x\right)\right) \]

   5. Simplified 99.8%

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

   if -1.05000000000000004 < x < 2.2000000000000002

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) + \frac{-2134856267379707}{1000000000000000}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(x, \left(\frac{-2134856267379707}{1000000000000000} + \color{blue}{x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-2134856267379707}{1000000000000000}, \color{blue}{\left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right)\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-2134856267379707}{1000000000000000}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right)}\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-2134856267379707}{1000000000000000}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{134362287316698645903}{100000000000000000000}, \color{blue}{\left(\frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right)}\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-2134856267379707}{1000000000000000}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{134362287316698645903}{100000000000000000000}, \left(x \cdot \color{blue}{\frac{-12692862305735843227608787}{10000000000000000000000000}}\right)\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 99.5%

          \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-2134856267379707}{1000000000000000}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{134362287316698645903}{100000000000000000000}, \mathsf{*.f64}\left(x, \color{blue}{\frac{-12692862305735843227608787}{10000000000000000000000000}}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 99.5%

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

   if 2.2000000000000002 < 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. Step-by-step derivation

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{70711}{100000} \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}\right)}\right) \]

      6. neg-mul-1 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{70711}{100000} \cdot \left(-1 \cdot x\right)\right), \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} \cdot \frac{70711}{100000}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\frac{70711}{100000} \cdot -1\right) \cdot x\right), \left(\color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} \cdot \frac{70711}{100000}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(\frac{70711}{100000} \cdot -1\right)\right), \left(\color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} \cdot \frac{70711}{100000}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{70711}{100000} \cdot -1\right)\right), \left(\color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} \cdot \frac{70711}{100000}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} \cdot \frac{70711}{100000}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \left(\frac{70711}{100000} \cdot \color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}}\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \left(\frac{\frac{70711}{100000} \cdot \left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\left(\frac{70711}{100000} \cdot \left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)\right), \color{blue}{\left(1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)}\right)\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x \cdot -0.70711 + \frac{1.6316775383 + x \cdot 0.1913510371}{1 - x \cdot \left(-0.99229 + x \cdot -0.04481\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \color{blue}{\left(\frac{\frac{1913510371}{448100000} - \frac{3648757816023}{62748003125} \cdot \frac{1}{x}}{x}\right)}\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\left(\frac{1913510371}{448100000} - \frac{3648757816023}{62748003125} \cdot \frac{1}{x}\right), \color{blue}{x}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\left(\frac{1913510371}{448100000} + \left(\mathsf{neg}\left(\frac{3648757816023}{62748003125} \cdot \frac{1}{x}\right)\right)\right), x\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1913510371}{448100000}, \left(\mathsf{neg}\left(\frac{3648757816023}{62748003125} \cdot \frac{1}{x}\right)\right)\right), x\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1913510371}{448100000}, \left(\mathsf{neg}\left(\frac{\frac{3648757816023}{62748003125} \cdot 1}{x}\right)\right)\right), x\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1913510371}{448100000}, \left(\mathsf{neg}\left(\frac{\frac{3648757816023}{62748003125}}{x}\right)\right)\right), x\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1913510371}{448100000}, \left(\frac{\mathsf{neg}\left(\frac{3648757816023}{62748003125}\right)}{x}\right)\right), x\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1913510371}{448100000}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{3648757816023}{62748003125}\right)\right), x\right)\right), x\right)\right) \]

      8. metadata-eval 99.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1913510371}{448100000}, \mathsf{/.f64}\left(\frac{-3648757816023}{62748003125}, x\right)\right), x\right)\right) \]

   7. Simplified 99.4%

      \[\leadsto x \cdot -0.70711 + \color{blue}{\frac{4.2702753202410175 + \frac{-58.14938538768042}{x}}{x}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;1.6316775383 + x \cdot \left(-2.134856267379707 + x \cdot 1.3436228731669864\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711 + \frac{4.2702753202410175 + \frac{-58.14938538768042}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (* 0.70711 (- (/ 6.039053782637804 x) x))
   (if (<= x 1.15)
     (+ 1.6316775383 (* x (+ -2.134856267379707 (* x 1.3436228731669864))))
     (+
      (* x -0.70711)
      (/ (+ 4.2702753202410175 (/ -58.14938538768042 x)) x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else if (x <= 1.15) {
		tmp = 1.6316775383 + (x * (-2.134856267379707 + (x * 1.3436228731669864)));
	} else {
		tmp = (x * -0.70711) + ((4.2702753202410175 + (-58.14938538768042 / x)) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.05d0)) then
        tmp = 0.70711d0 * ((6.039053782637804d0 / x) - x)
    else if (x <= 1.15d0) then
        tmp = 1.6316775383d0 + (x * ((-2.134856267379707d0) + (x * 1.3436228731669864d0)))
    else
        tmp = (x * (-0.70711d0)) + ((4.2702753202410175d0 + ((-58.14938538768042d0) / x)) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else if (x <= 1.15) {
		tmp = 1.6316775383 + (x * (-2.134856267379707 + (x * 1.3436228731669864)));
	} else {
		tmp = (x * -0.70711) + ((4.2702753202410175 + (-58.14938538768042 / x)) / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = 0.70711 * ((6.039053782637804 / x) - x)
	elif x <= 1.15:
		tmp = 1.6316775383 + (x * (-2.134856267379707 + (x * 1.3436228731669864)))
	else:
		tmp = (x * -0.70711) + ((4.2702753202410175 + (-58.14938538768042 / x)) / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	elseif (x <= 1.15)
		tmp = Float64(1.6316775383 + Float64(x * Float64(-2.134856267379707 + Float64(x * 1.3436228731669864))));
	else
		tmp = Float64(Float64(x * -0.70711) + Float64(Float64(4.2702753202410175 + Float64(-58.14938538768042 / x)) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	elseif (x <= 1.15)
		tmp = 1.6316775383 + (x * (-2.134856267379707 + (x * 1.3436228731669864)));
	else
		tmp = (x * -0.70711) + ((4.2702753202410175 + (-58.14938538768042 / x)) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.15], N[(1.6316775383 + N[(x * N[(-2.134856267379707 + N[(x * 1.3436228731669864), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * -0.70711), $MachinePrecision] + N[(N[(4.2702753202410175 + N[(-58.14938538768042 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004

   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

      \[\leadsto \mathsf{*.f64}\left(\frac{70711}{100000}, \mathsf{\_.f64}\left(\color{blue}{\left(\frac{\frac{27061}{4481}}{x}\right)}, x\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(\frac{70711}{100000}, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{27061}{4481}, x\right), x\right)\right) \]

   5. Simplified 99.8%

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

   if -1.05000000000000004 < x < 1.1499999999999999

   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

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \color{blue}{\left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(x, \left(\frac{134362287316698645903}{100000000000000000000} \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(x, \left(\frac{134362287316698645903}{100000000000000000000} \cdot x + \frac{-2134856267379707}{1000000000000000}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(x, \left(\frac{-2134856267379707}{1000000000000000} + \color{blue}{\frac{134362287316698645903}{100000000000000000000} \cdot x}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-2134856267379707}{1000000000000000}, \color{blue}{\left(\frac{134362287316698645903}{100000000000000000000} \cdot x\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-2134856267379707}{1000000000000000}, \left(x \cdot \color{blue}{\frac{134362287316698645903}{100000000000000000000}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 99.3%

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-2134856267379707}{1000000000000000}, \mathsf{*.f64}\left(x, \color{blue}{\frac{134362287316698645903}{100000000000000000000}}\right)\right)\right)\right) \]

   5. Simplified 99.3%

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

   if 1.1499999999999999 < 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. Step-by-step derivation

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{70711}{100000} \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}\right)}\right) \]

      6. neg-mul-1 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{70711}{100000} \cdot \left(-1 \cdot x\right)\right), \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} \cdot \frac{70711}{100000}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\frac{70711}{100000} \cdot -1\right) \cdot x\right), \left(\color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} \cdot \frac{70711}{100000}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(\frac{70711}{100000} \cdot -1\right)\right), \left(\color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} \cdot \frac{70711}{100000}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{70711}{100000} \cdot -1\right)\right), \left(\color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} \cdot \frac{70711}{100000}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} \cdot \frac{70711}{100000}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \left(\frac{70711}{100000} \cdot \color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}}\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \left(\frac{\frac{70711}{100000} \cdot \left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\left(\frac{70711}{100000} \cdot \left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right)\right), \color{blue}{\left(1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)}\right)\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x \cdot -0.70711 + \frac{1.6316775383 + x \cdot 0.1913510371}{1 - x \cdot \left(-0.99229 + x \cdot -0.04481\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \color{blue}{\left(\frac{\frac{1913510371}{448100000} - \frac{3648757816023}{62748003125} \cdot \frac{1}{x}}{x}\right)}\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\left(\frac{1913510371}{448100000} - \frac{3648757816023}{62748003125} \cdot \frac{1}{x}\right), \color{blue}{x}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\left(\frac{1913510371}{448100000} + \left(\mathsf{neg}\left(\frac{3648757816023}{62748003125} \cdot \frac{1}{x}\right)\right)\right), x\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1913510371}{448100000}, \left(\mathsf{neg}\left(\frac{3648757816023}{62748003125} \cdot \frac{1}{x}\right)\right)\right), x\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1913510371}{448100000}, \left(\mathsf{neg}\left(\frac{\frac{3648757816023}{62748003125} \cdot 1}{x}\right)\right)\right), x\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1913510371}{448100000}, \left(\mathsf{neg}\left(\frac{\frac{3648757816023}{62748003125}}{x}\right)\right)\right), x\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1913510371}{448100000}, \left(\frac{\mathsf{neg}\left(\frac{3648757816023}{62748003125}\right)}{x}\right)\right), x\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1913510371}{448100000}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{3648757816023}{62748003125}\right)\right), x\right)\right), x\right)\right) \]

      8. metadata-eval 99.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1913510371}{448100000}, \mathsf{/.f64}\left(\frac{-3648757816023}{62748003125}, x\right)\right), x\right)\right) \]

   7. Simplified 99.4%

      \[\leadsto x \cdot -0.70711 + \color{blue}{\frac{4.2702753202410175 + \frac{-58.14938538768042}{x}}{x}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.55:\\ \;\;\;\;1.6316775383 + x \cdot \left(-2.134856267379707 + x \cdot 1.3436228731669864\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 0.70711 (- (/ 6.039053782637804 x) x))))
   (if (<= x -1.05)
     t_0
     (if (<= x 1.55)
       (+ 1.6316775383 (* x (+ -2.134856267379707 (* x 1.3436228731669864))))
       t_0))))

C

double code(double x) {
	double t_0 = 0.70711 * ((6.039053782637804 / x) - x);
	double tmp;
	if (x <= -1.05) {
		tmp = t_0;
	} else if (x <= 1.55) {
		tmp = 1.6316775383 + (x * (-2.134856267379707 + (x * 1.3436228731669864)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.70711d0 * ((6.039053782637804d0 / x) - x)
    if (x <= (-1.05d0)) then
        tmp = t_0
    else if (x <= 1.55d0) then
        tmp = 1.6316775383d0 + (x * ((-2.134856267379707d0) + (x * 1.3436228731669864d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 0.70711 * ((6.039053782637804 / x) - x);
	double tmp;
	if (x <= -1.05) {
		tmp = t_0;
	} else if (x <= 1.55) {
		tmp = 1.6316775383 + (x * (-2.134856267379707 + (x * 1.3436228731669864)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.70711 * ((6.039053782637804 / x) - x)
	tmp = 0
	if x <= -1.05:
		tmp = t_0
	elif x <= 1.55:
		tmp = 1.6316775383 + (x * (-2.134856267379707 + (x * 1.3436228731669864)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x))
	tmp = 0.0
	if (x <= -1.05)
		tmp = t_0;
	elseif (x <= 1.55)
		tmp = Float64(1.6316775383 + Float64(x * Float64(-2.134856267379707 + Float64(x * 1.3436228731669864))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.70711 * ((6.039053782637804 / x) - x);
	tmp = 0.0;
	if (x <= -1.05)
		tmp = t_0;
	elseif (x <= 1.55)
		tmp = 1.6316775383 + (x * (-2.134856267379707 + (x * 1.3436228731669864)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.05], t$95$0, If[LessEqual[x, 1.55], N[(1.6316775383 + N[(x * N[(-2.134856267379707 + N[(x * 1.3436228731669864), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.55000000000000004 < 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

      \[\leadsto \mathsf{*.f64}\left(\frac{70711}{100000}, \mathsf{\_.f64}\left(\color{blue}{\left(\frac{\frac{27061}{4481}}{x}\right)}, x\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 99.5%

         \[\leadsto \mathsf{*.f64}\left(\frac{70711}{100000}, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{27061}{4481}, x\right), x\right)\right) \]

   5. Simplified 99.5%

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

   if -1.05000000000000004 < x < 1.55000000000000004

   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

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \color{blue}{\left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(x, \left(\frac{134362287316698645903}{100000000000000000000} \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(x, \left(\frac{134362287316698645903}{100000000000000000000} \cdot x + \frac{-2134856267379707}{1000000000000000}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(x, \left(\frac{-2134856267379707}{1000000000000000} + \color{blue}{\frac{134362287316698645903}{100000000000000000000} \cdot x}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-2134856267379707}{1000000000000000}, \color{blue}{\left(\frac{134362287316698645903}{100000000000000000000} \cdot x\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-2134856267379707}{1000000000000000}, \left(x \cdot \color{blue}{\frac{134362287316698645903}{100000000000000000000}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 99.3%

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-2134856267379707}{1000000000000000}, \mathsf{*.f64}\left(x, \color{blue}{\frac{134362287316698645903}{100000000000000000000}}\right)\right)\right)\right) \]

   5. Simplified 99.3%

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

Alternative 6: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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]

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. Add Preprocessing

Alternative 7: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.8:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 0.70711 (- (/ 6.039053782637804 x) x))))
   (if (<= x -1.05)
     t_0
     (if (<= x 2.8) (+ 1.6316775383 (* x -2.134856267379707)) t_0))))

C

double code(double x) {
	double t_0 = 0.70711 * ((6.039053782637804 / x) - x);
	double tmp;
	if (x <= -1.05) {
		tmp = t_0;
	} else if (x <= 2.8) {
		tmp = 1.6316775383 + (x * -2.134856267379707);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.70711d0 * ((6.039053782637804d0 / x) - x)
    if (x <= (-1.05d0)) then
        tmp = t_0
    else if (x <= 2.8d0) then
        tmp = 1.6316775383d0 + (x * (-2.134856267379707d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 0.70711 * ((6.039053782637804 / x) - x);
	double tmp;
	if (x <= -1.05) {
		tmp = t_0;
	} else if (x <= 2.8) {
		tmp = 1.6316775383 + (x * -2.134856267379707);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.70711 * ((6.039053782637804 / x) - x)
	tmp = 0
	if x <= -1.05:
		tmp = t_0
	elif x <= 2.8:
		tmp = 1.6316775383 + (x * -2.134856267379707)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x))
	tmp = 0.0
	if (x <= -1.05)
		tmp = t_0;
	elseif (x <= 2.8)
		tmp = Float64(1.6316775383 + Float64(x * -2.134856267379707));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.70711 * ((6.039053782637804 / x) - x);
	tmp = 0.0;
	if (x <= -1.05)
		tmp = t_0;
	elseif (x <= 2.8)
		tmp = 1.6316775383 + (x * -2.134856267379707);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.05], t$95$0, If[LessEqual[x, 2.8], N[(1.6316775383 + N[(x * -2.134856267379707), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 2.7999999999999998 < 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

      \[\leadsto \mathsf{*.f64}\left(\frac{70711}{100000}, \mathsf{\_.f64}\left(\color{blue}{\left(\frac{\frac{27061}{4481}}{x}\right)}, x\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 99.5%

         \[\leadsto \mathsf{*.f64}\left(\frac{70711}{100000}, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{27061}{4481}, x\right), x\right)\right) \]

   5. Simplified 99.5%

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

   if -1.05000000000000004 < x < 2.7999999999999998

   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

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \color{blue}{\left(\frac{-2134856267379707}{1000000000000000} \cdot x\right)}\right) \]

      2. *-lowering-*.f64 99.0%

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(\frac{-2134856267379707}{1000000000000000}, \color{blue}{x}\right)\right) \]

   5. Simplified 99.0%

      \[\leadsto \color{blue}{1.6316775383 + -2.134856267379707 \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;x \leq 2.8:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (* x -0.70711)
   (if (<= x 1.15) (+ 1.6316775383 (* x -2.134856267379707)) (* x -0.70711))))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.05d0)) then
        tmp = x * (-0.70711d0)
    else if (x <= 1.15d0) then
        tmp = 1.6316775383d0 + (x * (-2.134856267379707d0))
    else
        tmp = x * (-0.70711d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = x * -0.70711;
	} else if (x <= 1.15) {
		tmp = 1.6316775383 + (x * -2.134856267379707);
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = x * -0.70711
	elif x <= 1.15:
		tmp = 1.6316775383 + (x * -2.134856267379707)
	else:
		tmp = x * -0.70711
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(x * -0.70711);
	elseif (x <= 1.15)
		tmp = Float64(1.6316775383 + Float64(x * -2.134856267379707));
	else
		tmp = Float64(x * -0.70711);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = x * -0.70711;
	elseif (x <= 1.15)
		tmp = 1.6316775383 + (x * -2.134856267379707);
	else
		tmp = x * -0.70711;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(x * -0.70711), $MachinePrecision], If[LessEqual[x, 1.15], N[(1.6316775383 + N[(x * -2.134856267379707), $MachinePrecision]), $MachinePrecision], N[(x * -0.70711), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.1499999999999999 < 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. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\frac{-70711}{100000}} \]

      2. *-lowering-*.f64 99.2%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\frac{-70711}{100000}}\right) \]

   5. Simplified 99.2%

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

   if -1.05000000000000004 < x < 1.1499999999999999

   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

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \color{blue}{\left(\frac{-2134856267379707}{1000000000000000} \cdot x\right)}\right) \]

      2. *-lowering-*.f64 99.0%

         \[\leadsto \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(\frac{-2134856267379707}{1000000000000000}, \color{blue}{x}\right)\right) \]

   5. Simplified 99.0%

      \[\leadsto \color{blue}{1.6316775383 + -2.134856267379707 \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.5) (* x -0.70711) (if (<= x 1.15) 1.6316775383 (* x -0.70711))))

C

double code(double x) {
	double tmp;
	if (x <= -3.5) {
		tmp = x * -0.70711;
	} else if (x <= 1.15) {
		tmp = 1.6316775383;
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.5d0)) then
        tmp = x * (-0.70711d0)
    else if (x <= 1.15d0) then
        tmp = 1.6316775383d0
    else
        tmp = x * (-0.70711d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.5) {
		tmp = x * -0.70711;
	} else if (x <= 1.15) {
		tmp = 1.6316775383;
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.5:
		tmp = x * -0.70711
	elif x <= 1.15:
		tmp = 1.6316775383
	else:
		tmp = x * -0.70711
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.5)
		tmp = Float64(x * -0.70711);
	elseif (x <= 1.15)
		tmp = 1.6316775383;
	else
		tmp = Float64(x * -0.70711);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.5)
		tmp = x * -0.70711;
	elseif (x <= 1.15)
		tmp = 1.6316775383;
	else
		tmp = x * -0.70711;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.5], N[(x * -0.70711), $MachinePrecision], If[LessEqual[x, 1.15], 1.6316775383, N[(x * -0.70711), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.5 or 1.1499999999999999 < 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. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\frac{-70711}{100000}} \]

      2. *-lowering-*.f64 99.2%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\frac{-70711}{100000}}\right) \]

   5. Simplified 99.2%

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

   if -3.5 < x < 1.1499999999999999

   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. Simplified 98.0%

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

Alternative 10: 50.1% accurate, 19.0× 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. Simplified 51.6%

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

Reproduce

herbie shell --seed 2024161 
(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)))