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

Percentage Accurate: 99.9% → 99.9%

Time: 10.0s

Alternatives: 8

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, 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 99.9%

   \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
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. associate-*l/ N/A

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

   12. /-lowering-/.f64 N/A

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

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. Add Preprocessing

Alternative 2: 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 3: 98.8% 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 99.8%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   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 97.8%

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

   5. Simplified 97.8%

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

   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 98.9%

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

   5. Simplified 98.9%

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

4. Final simplification 98.3%

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

Alternative 4: 98.5% 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 98.0%

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

5. Simplified 98.0%

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

Alternative 5: 98.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.5) (* x -0.70711) (if (<= x 1.2) 1.6316775383 (* x -0.70711))))

C

double code(double x) {
	double tmp;
	if (x <= -3.5) {
		tmp = x * -0.70711;
	} else if (x <= 1.2) {
		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 <= 1.2d0) 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 <= 1.2) {
		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 <= 1.2:
		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 <= 1.2)
		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 <= 1.2)
		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, 1.2], 1.6316775383, N[(x * -0.70711), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.5 or 1.19999999999999996 < 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 98.4%

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

   5. Simplified 98.4%

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

   if -3.5 < x < 1.19999999999999996

   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 97.7%

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

Alternative 6: 98.2% 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. associate-*l/ N/A

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

   12. /-lowering-/.f64 N/A

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

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

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

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 7: 97.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ x \cdot -0.70711 + 1.6316775383 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (* x -0.70711) 1.6316775383))

C

double code(double x) {
	return (x * -0.70711) + 1.6316775383;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * (-0.70711d0)) + 1.6316775383d0
end function

Java

public static double code(double x) {
	return (x * -0.70711) + 1.6316775383;
}

Python

def code(x):
	return (x * -0.70711) + 1.6316775383

Julia

function code(x)
	return Float64(Float64(x * -0.70711) + 1.6316775383)
end

MATLAB

function tmp = code(x)
	tmp = (x * -0.70711) + 1.6316775383;
end

Wolfram

code[x_] := N[(N[(x * -0.70711), $MachinePrecision] + 1.6316775383), $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. associate-*l/ N/A

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

   12. /-lowering-/.f64 N/A

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

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 0

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

7. Simplified 97.5%

   \[\leadsto x \cdot -0.70711 + \color{blue}{1.6316775383} \]
8. Add Preprocessing

Alternative 8: 50.8% 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 50.0%

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

Reproduce

herbie shell --seed 2024138 
(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)))