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

Percentage Accurate: 99.9% → 99.9%

Time: 11.8s

Alternatives: 13

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: 99.9% 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, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - x \cdot \left(-0.99229 + x \cdot -0.04481\right)\\ \left(x \cdot -0.70711 + \frac{1.6316775383}{t\_0}\right) + \frac{x \cdot 0.1913510371}{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- 1.0 (* x (+ -0.99229 (* x -0.04481))))))
   (+ (+ (* x -0.70711) (/ 1.6316775383 t_0)) (/ (* x 0.1913510371) t_0))))

C

double code(double x) {
	double t_0 = 1.0 - (x * (-0.99229 + (x * -0.04481)));
	return ((x * -0.70711) + (1.6316775383 / t_0)) + ((x * 0.1913510371) / t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = 1.0d0 - (x * ((-0.99229d0) + (x * (-0.04481d0))))
    code = ((x * (-0.70711d0)) + (1.6316775383d0 / t_0)) + ((x * 0.1913510371d0) / t_0)
end function

Java

public static double code(double x) {
	double t_0 = 1.0 - (x * (-0.99229 + (x * -0.04481)));
	return ((x * -0.70711) + (1.6316775383 / t_0)) + ((x * 0.1913510371) / t_0);
}

Python

def code(x):
	t_0 = 1.0 - (x * (-0.99229 + (x * -0.04481)))
	return ((x * -0.70711) + (1.6316775383 / t_0)) + ((x * 0.1913510371) / t_0)

Julia

function code(x)
	t_0 = Float64(1.0 - Float64(x * Float64(-0.99229 + Float64(x * -0.04481))))
	return Float64(Float64(Float64(x * -0.70711) + Float64(1.6316775383 / t_0)) + Float64(Float64(x * 0.1913510371) / t_0))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 - (x * (-0.99229 + (x * -0.04481)));
	tmp = ((x * -0.70711) + (1.6316775383 / t_0)) + ((x * 0.1913510371) / t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 - N[(x * N[(-0.99229 + N[(x * -0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(x * -0.70711), $MachinePrecision] + N[(1.6316775383 / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(x * 0.1913510371), $MachinePrecision] / t$95$0), $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. 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 99.9%

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

   1. clear-num N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \left(\frac{1}{\color{blue}{\frac{1 - x \cdot \left(\frac{-99229}{100000} + x \cdot \frac{-4481}{100000}\right)}{\frac{16316775383}{10000000000} + x \cdot \frac{1913510371}{10000000000}}}}\right)\right) \]

   2. associate-/r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \left(\frac{1}{1 - x \cdot \left(\frac{-99229}{100000} + x \cdot \frac{-4481}{100000}\right)} \cdot \color{blue}{\left(\frac{16316775383}{10000000000} + x \cdot \frac{1913510371}{10000000000}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{*.f64}\left(\left(\frac{1}{1 - x \cdot \left(\frac{-99229}{100000} + x \cdot \frac{-4481}{100000}\right)}\right), \color{blue}{\left(\frac{16316775383}{10000000000} + x \cdot \frac{1913510371}{10000000000}\right)}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(1 - x \cdot \left(\frac{-99229}{100000} + x \cdot \frac{-4481}{100000}\right)\right)\right), \left(\color{blue}{\frac{16316775383}{10000000000}} + x \cdot \frac{1913510371}{10000000000}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \left(x \cdot \left(\frac{-99229}{100000} + x \cdot \frac{-4481}{100000}\right)\right)\right)\right), \left(\frac{16316775383}{10000000000} + x \cdot \frac{1913510371}{10000000000}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-99229}{100000} + x \cdot \frac{-4481}{100000}\right)\right)\right)\right), \left(\frac{16316775383}{10000000000} + x \cdot \frac{1913510371}{10000000000}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-99229}{100000}, \left(x \cdot \frac{-4481}{100000}\right)\right)\right)\right)\right), \left(\frac{16316775383}{10000000000} + x \cdot \frac{1913510371}{10000000000}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-99229}{100000}, \mathsf{*.f64}\left(x, \frac{-4481}{100000}\right)\right)\right)\right)\right), \left(\frac{16316775383}{10000000000} + x \cdot \frac{1913510371}{10000000000}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-99229}{100000}, \mathsf{*.f64}\left(x, \frac{-4481}{100000}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \color{blue}{\left(x \cdot \frac{1913510371}{10000000000}\right)}\right)\right)\right) \]

   10. *-lowering-*.f64 99.9%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-99229}{100000}, \mathsf{*.f64}\left(x, \frac{-4481}{100000}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(x, \color{blue}{\frac{1913510371}{10000000000}}\right)\right)\right)\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto x \cdot -0.70711 + \color{blue}{\frac{1}{1 - x \cdot \left(-0.99229 + x \cdot -0.04481\right)} \cdot \left(1.6316775383 + x \cdot 0.1913510371\right)} \]
7. Step-by-step derivation

   1. distribute-lft-in N/A

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

   2. associate-+r+ N/A

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

   3. *-commutative N/A

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

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

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{-70711}{100000} + \frac{1}{1 - x \cdot \left(\frac{-99229}{100000} + x \cdot \frac{-4481}{100000}\right)} \cdot \frac{16316775383}{10000000000}\right), \color{blue}{\left(\left(x \cdot \frac{1913510371}{10000000000}\right) \cdot \frac{1}{1 - x \cdot \left(\frac{-99229}{100000} + x \cdot \frac{-4481}{100000}\right)}\right)}\right) \]

8. Applied egg-rr 99.9%

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

Alternative 2: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(x * -0.70711), $MachinePrecision] + N[(N[(1.0 / N[(1.0 - N[(x * N[(-0.99229 + N[(x * -0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.6316775383 + N[(x * 0.1913510371), $MachinePrecision]), $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. 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 99.9%

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

   1. clear-num N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \left(\frac{1}{\color{blue}{\frac{1 - x \cdot \left(\frac{-99229}{100000} + x \cdot \frac{-4481}{100000}\right)}{\frac{16316775383}{10000000000} + x \cdot \frac{1913510371}{10000000000}}}}\right)\right) \]

   2. associate-/r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \left(\frac{1}{1 - x \cdot \left(\frac{-99229}{100000} + x \cdot \frac{-4481}{100000}\right)} \cdot \color{blue}{\left(\frac{16316775383}{10000000000} + x \cdot \frac{1913510371}{10000000000}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{*.f64}\left(\left(\frac{1}{1 - x \cdot \left(\frac{-99229}{100000} + x \cdot \frac{-4481}{100000}\right)}\right), \color{blue}{\left(\frac{16316775383}{10000000000} + x \cdot \frac{1913510371}{10000000000}\right)}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(1 - x \cdot \left(\frac{-99229}{100000} + x \cdot \frac{-4481}{100000}\right)\right)\right), \left(\color{blue}{\frac{16316775383}{10000000000}} + x \cdot \frac{1913510371}{10000000000}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \left(x \cdot \left(\frac{-99229}{100000} + x \cdot \frac{-4481}{100000}\right)\right)\right)\right), \left(\frac{16316775383}{10000000000} + x \cdot \frac{1913510371}{10000000000}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-99229}{100000} + x \cdot \frac{-4481}{100000}\right)\right)\right)\right), \left(\frac{16316775383}{10000000000} + x \cdot \frac{1913510371}{10000000000}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-99229}{100000}, \left(x \cdot \frac{-4481}{100000}\right)\right)\right)\right)\right), \left(\frac{16316775383}{10000000000} + x \cdot \frac{1913510371}{10000000000}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-99229}{100000}, \mathsf{*.f64}\left(x, \frac{-4481}{100000}\right)\right)\right)\right)\right), \left(\frac{16316775383}{10000000000} + x \cdot \frac{1913510371}{10000000000}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-99229}{100000}, \mathsf{*.f64}\left(x, \frac{-4481}{100000}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \color{blue}{\left(x \cdot \frac{1913510371}{10000000000}\right)}\right)\right)\right) \]

   10. *-lowering-*.f64 99.9%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-99229}{100000}, \mathsf{*.f64}\left(x, \frac{-4481}{100000}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(x, \color{blue}{\frac{1913510371}{10000000000}}\right)\right)\right)\right) \]

6. Applied egg-rr 99.9%

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

Alternative 3: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5:\\ \;\;\;\;x \cdot -0.70711 + \frac{4.2702753202410175}{x}\\ \mathbf{elif}\;x \leq 1.16:\\ \;\;\;\;1.6316775383 + x \cdot \left(-2.134856267379707 + x \cdot 1.3436228731669864\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.5)
   (+ (* x -0.70711) (/ 4.2702753202410175 x))
   (if (<= x 1.16)
     (+ 1.6316775383 (* x (+ -2.134856267379707 (* x 1.3436228731669864))))
     (* x -0.70711))))

C

double code(double x) {
	double tmp;
	if (x <= -2.5) {
		tmp = (x * -0.70711) + (4.2702753202410175 / x);
	} else if (x <= 1.16) {
		tmp = 1.6316775383 + (x * (-2.134856267379707 + (x * 1.3436228731669864)));
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2.5d0)) then
        tmp = (x * (-0.70711d0)) + (4.2702753202410175d0 / x)
    else if (x <= 1.16d0) then
        tmp = 1.6316775383d0 + (x * ((-2.134856267379707d0) + (x * 1.3436228731669864d0)))
    else
        tmp = x * (-0.70711d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.5) {
		tmp = (x * -0.70711) + (4.2702753202410175 / x);
	} else if (x <= 1.16) {
		tmp = 1.6316775383 + (x * (-2.134856267379707 + (x * 1.3436228731669864)));
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.5:
		tmp = (x * -0.70711) + (4.2702753202410175 / x)
	elif x <= 1.16:
		tmp = 1.6316775383 + (x * (-2.134856267379707 + (x * 1.3436228731669864)))
	else:
		tmp = x * -0.70711
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.5)
		tmp = Float64(Float64(x * -0.70711) + Float64(4.2702753202410175 / x));
	elseif (x <= 1.16)
		tmp = Float64(1.6316775383 + Float64(x * Float64(-2.134856267379707 + Float64(x * 1.3436228731669864))));
	else
		tmp = Float64(x * -0.70711);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.5)
		tmp = (x * -0.70711) + (4.2702753202410175 / x);
	elseif (x <= 1.16)
		tmp = 1.6316775383 + (x * (-2.134856267379707 + (x * 1.3436228731669864)));
	else
		tmp = x * -0.70711;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.5], N[(N[(x * -0.70711), $MachinePrecision] + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.16], N[(1.6316775383 + N[(x * N[(-2.134856267379707 + N[(x * 1.3436228731669864), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * -0.70711), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.5

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. 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 99.9%

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

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 99.8%

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

   7. Simplified 99.8%

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

   if -2.5 < x < 1.15999999999999992

   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 98.0%

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

   5. Simplified 98.0%

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

   if 1.15999999999999992 < 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. *-commutative N/A

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

      2. *-lowering-*.f64 98.8%

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

   5. Simplified 98.8%

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

Alternative 4: 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]

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. Add Preprocessing

Alternative 5: 98.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.55:\\ \;\;\;\;x \cdot -0.70711 + \frac{4.2702753202410175}{x}\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;0.70711 \cdot \left(2.30753 + x \cdot -3.0191289437\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.55)
   (+ (* x -0.70711) (/ 4.2702753202410175 x))
   (if (<= x 1.15)
     (* 0.70711 (+ 2.30753 (* x -3.0191289437)))
     (* x -0.70711))))

C

double code(double x) {
	double tmp;
	if (x <= -2.55) {
		tmp = (x * -0.70711) + (4.2702753202410175 / x);
	} else if (x <= 1.15) {
		tmp = 0.70711 * (2.30753 + (x * -3.0191289437));
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2.55d0)) then
        tmp = (x * (-0.70711d0)) + (4.2702753202410175d0 / x)
    else if (x <= 1.15d0) then
        tmp = 0.70711d0 * (2.30753d0 + (x * (-3.0191289437d0)))
    else
        tmp = x * (-0.70711d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.55) {
		tmp = (x * -0.70711) + (4.2702753202410175 / x);
	} else if (x <= 1.15) {
		tmp = 0.70711 * (2.30753 + (x * -3.0191289437));
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.55:
		tmp = (x * -0.70711) + (4.2702753202410175 / x)
	elif x <= 1.15:
		tmp = 0.70711 * (2.30753 + (x * -3.0191289437))
	else:
		tmp = x * -0.70711
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.55)
		tmp = Float64(Float64(x * -0.70711) + Float64(4.2702753202410175 / x));
	elseif (x <= 1.15)
		tmp = Float64(0.70711 * Float64(2.30753 + Float64(x * -3.0191289437)));
	else
		tmp = Float64(x * -0.70711);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.55)
		tmp = (x * -0.70711) + (4.2702753202410175 / x);
	elseif (x <= 1.15)
		tmp = 0.70711 * (2.30753 + (x * -3.0191289437));
	else
		tmp = x * -0.70711;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.55], N[(N[(x * -0.70711), $MachinePrecision] + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.15], N[(0.70711 * N[(2.30753 + N[(x * -3.0191289437), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * -0.70711), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.5499999999999998

   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. 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 99.9%

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

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 99.8%

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

   7. Simplified 99.8%

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

   if -2.5499999999999998 < 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

      \[\leadsto \mathsf{*.f64}\left(\frac{70711}{100000}, \color{blue}{\left(\frac{230753}{100000} + \frac{-30191289437}{10000000000} \cdot x\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{70711}{100000}, \mathsf{+.f64}\left(\frac{230753}{100000}, \color{blue}{\left(\frac{-30191289437}{10000000000} \cdot x\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{70711}{100000}, \mathsf{+.f64}\left(\frac{230753}{100000}, \left(x \cdot \color{blue}{\frac{-30191289437}{10000000000}}\right)\right)\right) \]

      3. *-lowering-*.f64 97.3%

         \[\leadsto \mathsf{*.f64}\left(\frac{70711}{100000}, \mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \color{blue}{\frac{-30191289437}{10000000000}}\right)\right)\right) \]

   5. Simplified 97.3%

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

   if 1.1499999999999999 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 98.8%

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

   5. Simplified 98.8%

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

Alternative 6: 98.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;0.70711 \cdot \left(2.30753 + x \cdot -3.0191289437\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (* x -0.70711)
   (if (<= x 1.15)
     (* 0.70711 (+ 2.30753 (* x -3.0191289437)))
     (* x -0.70711))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = x * -0.70711;
	} else if (x <= 1.15) {
		tmp = 0.70711 * (2.30753 + (x * -3.0191289437));
	} 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 = 0.70711d0 * (2.30753d0 + (x * (-3.0191289437d0)))
    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 = 0.70711 * (2.30753 + (x * -3.0191289437));
	} 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 = 0.70711 * (2.30753 + (x * -3.0191289437))
	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(0.70711 * Float64(2.30753 + Float64(x * -3.0191289437)));
	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 = 0.70711 * (2.30753 + (x * -3.0191289437));
	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[(0.70711 * N[(2.30753 + N[(x * -3.0191289437), $MachinePrecision]), $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 99.9%

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

   3. Taylor expanded in x around 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 99.0%

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

   5. Simplified 99.0%

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

   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

      \[\leadsto \mathsf{*.f64}\left(\frac{70711}{100000}, \color{blue}{\left(\frac{230753}{100000} + \frac{-30191289437}{10000000000} \cdot x\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{70711}{100000}, \mathsf{+.f64}\left(\frac{230753}{100000}, \color{blue}{\left(\frac{-30191289437}{10000000000} \cdot x\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{70711}{100000}, \mathsf{+.f64}\left(\frac{230753}{100000}, \left(x \cdot \color{blue}{\frac{-30191289437}{10000000000}}\right)\right)\right) \]

      3. *-lowering-*.f64 97.3%

         \[\leadsto \mathsf{*.f64}\left(\frac{70711}{100000}, \mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \color{blue}{\frac{-30191289437}{10000000000}}\right)\right)\right) \]

   5. Simplified 97.3%

      \[\leadsto 0.70711 \cdot \color{blue}{\left(2.30753 + x \cdot -3.0191289437\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 98.6% 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 99.9%

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

   3. Taylor expanded in x around 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 99.0%

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

   5. Simplified 99.0%

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

   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 97.2%

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

   5. Simplified 97.2%

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

4. Final simplification 98.2%

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

Alternative 8: 98.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 98.2%

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

5. Simplified 98.2%

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

Alternative 9: 98.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.16:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05) (* x -0.70711) (if (<= x 1.16) 1.6316775383 (* x -0.70711))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = x * -0.70711;
	} else if (x <= 1.16) {
		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 <= (-1.05d0)) then
        tmp = x * (-0.70711d0)
    else if (x <= 1.16d0) 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 <= -1.05) {
		tmp = x * -0.70711;
	} else if (x <= 1.16) {
		tmp = 1.6316775383;
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = x * -0.70711
	elif x <= 1.16:
		tmp = 1.6316775383
	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 <= 1.16)
		tmp = 1.6316775383;
	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 <= 1.16)
		tmp = 1.6316775383;
	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, 1.16], 1.6316775383, N[(x * -0.70711), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.15999999999999992 < x

   1. Initial program 99.9%

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

   3. 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 99.0%

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

   5. Simplified 99.0%

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

   if -1.05000000000000004 < x < 1.15999999999999992

   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 95.4%

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

Alternative 10: 98.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ x \cdot -0.70711 + \frac{1}{0.6128661923249078 + x \cdot 0.5362685934910628} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (* x -0.70711) (/ 1.0 (+ 0.6128661923249078 (* x 0.5362685934910628)))))

C

double code(double x) {
	return (x * -0.70711) + (1.0 / (0.6128661923249078 + (x * 0.5362685934910628)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * (-0.70711d0)) + (1.0d0 / (0.6128661923249078d0 + (x * 0.5362685934910628d0)))
end function

Java

public static double code(double x) {
	return (x * -0.70711) + (1.0 / (0.6128661923249078 + (x * 0.5362685934910628)));
}

Python

def code(x):
	return (x * -0.70711) + (1.0 / (0.6128661923249078 + (x * 0.5362685934910628)))

Julia

function code(x)
	return Float64(Float64(x * -0.70711) + Float64(1.0 / Float64(0.6128661923249078 + Float64(x * 0.5362685934910628))))
end

MATLAB

function tmp = code(x)
	tmp = (x * -0.70711) + (1.0 / (0.6128661923249078 + (x * 0.5362685934910628)));
end

Wolfram

code[x_] := N[(N[(x * -0.70711), $MachinePrecision] + N[(1.0 / N[(0.6128661923249078 + N[(x * 0.5362685934910628), $MachinePrecision]), $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. 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 99.9%

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

   1. clear-num N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \left(\frac{1}{\color{blue}{\frac{1 - x \cdot \left(\frac{-99229}{100000} + x \cdot \frac{-4481}{100000}\right)}{\frac{16316775383}{10000000000} + x \cdot \frac{1913510371}{10000000000}}}}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 - x \cdot \left(\frac{-99229}{100000} + x \cdot \frac{-4481}{100000}\right)}{\frac{16316775383}{10000000000} + x \cdot \frac{1913510371}{10000000000}}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(1 - x \cdot \left(\frac{-99229}{100000} + x \cdot \frac{-4481}{100000}\right)\right), \color{blue}{\left(\frac{16316775383}{10000000000} + x \cdot \frac{1913510371}{10000000000}\right)}\right)\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(x \cdot \left(\frac{-99229}{100000} + x \cdot \frac{-4481}{100000}\right)\right)\right), \left(\color{blue}{\frac{16316775383}{10000000000}} + x \cdot \frac{1913510371}{10000000000}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-99229}{100000} + x \cdot \frac{-4481}{100000}\right)\right)\right), \left(\frac{16316775383}{10000000000} + x \cdot \frac{1913510371}{10000000000}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-99229}{100000}, \left(x \cdot \frac{-4481}{100000}\right)\right)\right)\right), \left(\frac{16316775383}{10000000000} + x \cdot \frac{1913510371}{10000000000}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-99229}{100000}, \mathsf{*.f64}\left(x, \frac{-4481}{100000}\right)\right)\right)\right), \left(\frac{16316775383}{10000000000} + x \cdot \frac{1913510371}{10000000000}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-99229}{100000}, \mathsf{*.f64}\left(x, \frac{-4481}{100000}\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \color{blue}{\left(x \cdot \frac{1913510371}{10000000000}\right)}\right)\right)\right)\right) \]

   9. *-lowering-*.f64 99.9%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-99229}{100000}, \mathsf{*.f64}\left(x, \frac{-4481}{100000}\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{16316775383}{10000000000}, \mathsf{*.f64}\left(x, \color{blue}{\frac{1913510371}{10000000000}}\right)\right)\right)\right)\right) \]

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{10000000000}{16316775383} + \frac{2019128943700000}{3765144869953399} \cdot x\right)}\right)\right) \]
8. Step-by-step derivation

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{10000000000}{16316775383}, \color{blue}{\left(\frac{2019128943700000}{3765144869953399} \cdot x\right)}\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{10000000000}{16316775383}, \left(x \cdot \color{blue}{\frac{2019128943700000}{3765144869953399}}\right)\right)\right)\right) \]

   3. *-lowering-*.f64 97.7%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{-70711}{100000}\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{10000000000}{16316775383}, \mathsf{*.f64}\left(x, \color{blue}{\frac{2019128943700000}{3765144869953399}}\right)\right)\right)\right) \]

9. Simplified 97.7%

   \[\leadsto x \cdot -0.70711 + \frac{1}{\color{blue}{0.6128661923249078 + x \cdot 0.5362685934910628}} \]
10. Add Preprocessing

Alternative 11: 98.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 98.2%

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

5. Simplified 98.2%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\frac{70711}{100000}, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\color{blue}{\frac{230753}{100000}}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \frac{99229}{100000}\right)\right)\right), x\right)\right) \]
7. Step-by-step derivation

8. Simplified 97.5%

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

Alternative 12: 50.0% 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 99.9%

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

3. Taylor expanded in x around 0

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

5. Simplified 47.6%

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

Alternative 13: 9.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ 0.1928378166664987 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.1928378166664987)

C

double code(double x) {
	return 0.1928378166664987;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.1928378166664987d0
end function

Java

public static double code(double x) {
	return 0.1928378166664987;
}

Python

def code(x):
	return 0.1928378166664987

Julia

function code(x)
	return 0.1928378166664987
end

MATLAB

function tmp = code(x)
	tmp = 0.1928378166664987;
end

Wolfram

code[x_] := 0.1928378166664987

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

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 98.2%

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

5. Simplified 98.2%

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

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x \cdot \left(\frac{1913510371}{9922900000} \cdot \frac{1}{x} - \frac{70711}{100000}\right)} \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto x \cdot \left(\frac{1913510371}{9922900000} \cdot \frac{1}{x} + \color{blue}{\left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right)}\right) \]

   2. metadata-eval N/A

      \[\leadsto x \cdot \left(\frac{1913510371}{9922900000} \cdot \frac{1}{x} + \frac{-70711}{100000}\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \left(\frac{1913510371}{9922900000} \cdot \frac{1}{x}\right) \cdot x + \color{blue}{\frac{-70711}{100000} \cdot x} \]

   4. associate-*l* N/A

      \[\leadsto \frac{1913510371}{9922900000} \cdot \left(\frac{1}{x} \cdot x\right) + \color{blue}{\frac{-70711}{100000}} \cdot x \]

   5. lft-mult-inverse N/A

      \[\leadsto \frac{1913510371}{9922900000} \cdot 1 + \frac{-70711}{100000} \cdot x \]

   6. metadata-eval N/A

      \[\leadsto \frac{1913510371}{9922900000} + \color{blue}{\frac{-70711}{100000}} \cdot x \]

   7. metadata-eval N/A

      \[\leadsto \frac{1913510371}{9922900000} + \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right) \cdot x \]

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

      \[\leadsto \frac{1913510371}{9922900000} + \left(\mathsf{neg}\left(\frac{70711}{100000} \cdot x\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \frac{1913510371}{9922900000} + \left(\mathsf{neg}\left(x \cdot \frac{70711}{100000}\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\frac{1913510371}{9922900000}, \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{70711}{100000}\right)\right)}\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\frac{1913510371}{9922900000}, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right)}\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\frac{1913510371}{9922900000}, \left(x \cdot \frac{-70711}{100000}\right)\right) \]

   13. *-lowering-*.f64 58.7%

       \[\leadsto \mathsf{+.f64}\left(\frac{1913510371}{9922900000}, \mathsf{*.f64}\left(x, \color{blue}{\frac{-70711}{100000}}\right)\right) \]

8. Simplified 58.7%

   \[\leadsto \color{blue}{0.1928378166664987 + x \cdot -0.70711} \]

9. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1913510371}{9922900000}} \]
10. Step-by-step derivation

11. Simplified 9.2%

    \[\leadsto \color{blue}{0.1928378166664987} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024158 
(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)))