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

Percentage Accurate: 99.8% → 99.9%

Time: 7.9s

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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. +-commutative 99.8%

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

   3. distribute-lft-in 99.8%

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

   4. neg-mul-1 99.8%

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

   5. associate-*r* 99.8%

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

   6. *-commutative 99.8%

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

   7. fma-def 99.8%

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

   8. metadata-eval 99.8%

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

   9. associate-*r/ 99.8%

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

   10. distribute-lft-in 99.8%

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

   11. +-commutative 99.8%

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

   12. *-commutative 99.8%

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

   13. associate-*l* 99.8%

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

   14. metadata-eval 99.8%

       \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{x \cdot \color{blue}{0.1913510371} + 0.70711 \cdot 2.30753}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right) \]

   15. metadata-eval 99.8%

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

   16. fma-def 99.8%

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

   17. metadata-eval 99.8%

       \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, \color{blue}{0.1913510371}, 0.70711 \cdot 2.30753\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right) \]

   18. metadata-eval 99.8%

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

   19. +-commutative 99.8%

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

   20. fma-def 99.8%

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

3. Simplified 99.8%

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

   1. fma-udef 99.8%

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

5. Applied egg-rr 99.8%

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

   1. fma-udef 99.8%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * ((x * 0.04481) + 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 * ((x * 0.04481d0) + 0.99229d0)))) - x)
end function

Java

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

Python

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

2. Final simplification 99.8%

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

Alternative 3: 99.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\frac{-4.2702753202410175}{-x} + x \cdot -0.70711\\ \mathbf{elif}\;x \leq 0.75:\\ \;\;\;\;0.70711 \cdot \left(\left(2.30753 + x \cdot -2.0191289437\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (+ (/ -4.2702753202410175 (- x)) (* x -0.70711))
   (if (<= x 0.75)
     (* 0.70711 (- (+ 2.30753 (* x -2.0191289437)) x))
     (* 0.70711 (- (/ 6.039053782637804 x) x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = (-4.2702753202410175 / -x) + (x * -0.70711);
	} else if (x <= 0.75) {
		tmp = 0.70711 * ((2.30753 + (x * -2.0191289437)) - x);
	} else {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.05d0)) then
        tmp = ((-4.2702753202410175d0) / -x) + (x * (-0.70711d0))
    else if (x <= 0.75d0) then
        tmp = 0.70711d0 * ((2.30753d0 + (x * (-2.0191289437d0))) - x)
    else
        tmp = 0.70711d0 * ((6.039053782637804d0 / x) - x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = (-4.2702753202410175 / -x) + (x * -0.70711);
	} else if (x <= 0.75) {
		tmp = 0.70711 * ((2.30753 + (x * -2.0191289437)) - x);
	} else {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = (-4.2702753202410175 / -x) + (x * -0.70711)
	elif x <= 0.75:
		tmp = 0.70711 * ((2.30753 + (x * -2.0191289437)) - x)
	else:
		tmp = 0.70711 * ((6.039053782637804 / x) - x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(Float64(-4.2702753202410175 / Float64(-x)) + Float64(x * -0.70711));
	elseif (x <= 0.75)
		tmp = Float64(0.70711 * Float64(Float64(2.30753 + Float64(x * -2.0191289437)) - x));
	else
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = (-4.2702753202410175 / -x) + (x * -0.70711);
	elseif (x <= 0.75)
		tmp = 0.70711 * ((2.30753 + (x * -2.0191289437)) - x);
	else
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(N[(-4.2702753202410175 / (-x)), $MachinePrecision] + N[(x * -0.70711), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.75], N[(0.70711 * N[(N[(2.30753 + N[(x * -2.0191289437), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 99.4%

      \[\leadsto \color{blue}{-0.70711 \cdot x + 4.2702753202410175 \cdot \frac{1}{x}} \]
   3. Step-by-step derivation

      1. un-div-inv 99.4%

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

      2. frac-2neg 99.4%

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

      3. metadata-eval 99.4%

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

   4. Applied egg-rr 99.4%

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

   if -1.05000000000000004 < x < 0.75

   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. Taylor expanded in x around 0 98.9%

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

   if 0.75 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 96.6%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\frac{6.039053782637804}{x}} - x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\frac{-4.2702753202410175}{-x} + x \cdot -0.70711\\ \mathbf{elif}\;x \leq 0.75:\\ \;\;\;\;0.70711 \cdot \left(\left(2.30753 + x \cdot -2.0191289437\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \end{array} \]

Alternative 4: 99.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 0.75\right):\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 0.75)))
   (* 0.70711 (- (/ 6.039053782637804 x) x))
   (+ 1.6316775383 (* x -2.134856267379707))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 0.75)) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else {
		tmp = 1.6316775383 + (x * -2.134856267379707);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.05d0)) .or. (.not. (x <= 0.75d0))) then
        tmp = 0.70711d0 * ((6.039053782637804d0 / x) - x)
    else
        tmp = 1.6316775383d0 + (x * (-2.134856267379707d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 0.75)) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else {
		tmp = 1.6316775383 + (x * -2.134856267379707);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.05) or not (x <= 0.75):
		tmp = 0.70711 * ((6.039053782637804 / x) - x)
	else:
		tmp = 1.6316775383 + (x * -2.134856267379707)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.05) || !(x <= 0.75))
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	else
		tmp = Float64(1.6316775383 + Float64(x * -2.134856267379707));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.05) || ~((x <= 0.75)))
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	else
		tmp = 1.6316775383 + (x * -2.134856267379707);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.05], N[Not[LessEqual[x, 0.75]], $MachinePrecision]], N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(1.6316775383 + N[(x * -2.134856267379707), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 0.75 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 98.2%

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

   if -1.05000000000000004 < x < 0.75

   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. Taylor expanded in x around 0 98.9%

      \[\leadsto \color{blue}{1.6316775383 + -2.134856267379707 \cdot x} \]
   3. Step-by-step derivation

      1. *-commutative 98.9%

         \[\leadsto 1.6316775383 + \color{blue}{x \cdot -2.134856267379707} \]

   4. Simplified 98.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 0.75\right):\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \end{array} \]

Alternative 5: 99.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\frac{-4.2702753202410175}{-x} + x \cdot -0.70711\\ \mathbf{elif}\;x \leq 0.75:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (+ (/ -4.2702753202410175 (- x)) (* x -0.70711))
   (if (<= x 0.75)
     (+ 1.6316775383 (* x -2.134856267379707))
     (* 0.70711 (- (/ 6.039053782637804 x) x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = (-4.2702753202410175 / -x) + (x * -0.70711);
	} else if (x <= 0.75) {
		tmp = 1.6316775383 + (x * -2.134856267379707);
	} else {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.05d0)) then
        tmp = ((-4.2702753202410175d0) / -x) + (x * (-0.70711d0))
    else if (x <= 0.75d0) then
        tmp = 1.6316775383d0 + (x * (-2.134856267379707d0))
    else
        tmp = 0.70711d0 * ((6.039053782637804d0 / x) - x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = (-4.2702753202410175 / -x) + (x * -0.70711);
	} else if (x <= 0.75) {
		tmp = 1.6316775383 + (x * -2.134856267379707);
	} else {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = (-4.2702753202410175 / -x) + (x * -0.70711)
	elif x <= 0.75:
		tmp = 1.6316775383 + (x * -2.134856267379707)
	else:
		tmp = 0.70711 * ((6.039053782637804 / x) - x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(Float64(-4.2702753202410175 / Float64(-x)) + Float64(x * -0.70711));
	elseif (x <= 0.75)
		tmp = Float64(1.6316775383 + Float64(x * -2.134856267379707));
	else
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = (-4.2702753202410175 / -x) + (x * -0.70711);
	elseif (x <= 0.75)
		tmp = 1.6316775383 + (x * -2.134856267379707);
	else
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(N[(-4.2702753202410175 / (-x)), $MachinePrecision] + N[(x * -0.70711), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.75], N[(1.6316775383 + N[(x * -2.134856267379707), $MachinePrecision]), $MachinePrecision], N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 99.4%

      \[\leadsto \color{blue}{-0.70711 \cdot x + 4.2702753202410175 \cdot \frac{1}{x}} \]
   3. Step-by-step derivation

      1. un-div-inv 99.4%

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

      2. frac-2neg 99.4%

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

      3. metadata-eval 99.4%

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

   4. Applied egg-rr 99.4%

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

   if -1.05000000000000004 < x < 0.75

   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. Taylor expanded in x around 0 98.9%

      \[\leadsto \color{blue}{1.6316775383 + -2.134856267379707 \cdot x} \]
   3. Step-by-step derivation

      1. *-commutative 98.9%

         \[\leadsto 1.6316775383 + \color{blue}{x \cdot -2.134856267379707} \]

   4. Simplified 98.9%

      \[\leadsto \color{blue}{1.6316775383 + x \cdot -2.134856267379707} \]

   if 0.75 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 96.6%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\frac{6.039053782637804}{x}} - x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\frac{-4.2702753202410175}{-x} + x \cdot -0.70711\\ \mathbf{elif}\;x \leq 0.75:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \end{array} \]

Alternative 6: 98.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \lor \neg \left(x \leq 1.2\right):\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;1.6316775383\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -3.5) (not (<= x 1.2))) (* x -0.70711) 1.6316775383))

C

double code(double x) {
	double tmp;
	if ((x <= -3.5) || !(x <= 1.2)) {
		tmp = x * -0.70711;
	} else {
		tmp = 1.6316775383;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-3.5d0)) .or. (.not. (x <= 1.2d0))) then
        tmp = x * (-0.70711d0)
    else
        tmp = 1.6316775383d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -3.5) || !(x <= 1.2)) {
		tmp = x * -0.70711;
	} else {
		tmp = 1.6316775383;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -3.5) or not (x <= 1.2):
		tmp = x * -0.70711
	else:
		tmp = 1.6316775383
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -3.5) || !(x <= 1.2))
		tmp = Float64(x * -0.70711);
	else
		tmp = 1.6316775383;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -3.5) || ~((x <= 1.2)))
		tmp = x * -0.70711;
	else
		tmp = 1.6316775383;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -3.5], N[Not[LessEqual[x, 1.2]], $MachinePrecision]], N[(x * -0.70711), $MachinePrecision], 1.6316775383]

Derivation

1. Split input into 2 regimes

2. if x < -3.5 or 1.19999999999999996 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 98.1%

      \[\leadsto \color{blue}{-0.70711 \cdot x} \]
   3. Step-by-step derivation

      1. *-commutative 98.1%

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

   4. Simplified 98.1%

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

   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. Taylor expanded in x around 0 96.9%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \lor \neg \left(x \leq 1.2\right):\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;1.6316775383\\ \end{array} \]

Alternative 7: 97.6% accurate, 3.8× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* 0.70711 (- 2.30753 x)))

C

double code(double x) {
	return 0.70711 * (2.30753 - x);
}

Fortran

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

Java

public static double code(double x) {
	return 0.70711 * (2.30753 - x);
}

Python

def code(x):
	return 0.70711 * (2.30753 - x)

Julia

function code(x)
	return Float64(0.70711 * Float64(2.30753 - x))
end

MATLAB

function tmp = code(x)
	tmp = 0.70711 * (2.30753 - x);
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0 96.5%

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

3. Final simplification 96.5%

   \[\leadsto 0.70711 \cdot \left(2.30753 - x\right) \]

Alternative 8: 51.0% 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 99.8%

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

2. Taylor expanded in x around 0 50.8%

   \[\leadsto \color{blue}{1.6316775383} \]

3. Final simplification 50.8%

   \[\leadsto 1.6316775383 \]

Reproduce

herbie shell --seed 2023336 
(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)))