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

Percentage Accurate: 99.8% → 99.9%

Time: 8.3s

Alternatives: 7

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. *-commutative 99.8%

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

   5. distribute-lft-neg-out 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-def 99.8%

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

   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. +-commutative 99.8%

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

   11. distribute-lft-in 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%

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

   13. associate-*r* 99.8%

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

   14. *-commutative 99.8%

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

   15. fma-def 99.8%

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

   16. metadata-eval 99.8%

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

   17. metadata-eval 99.8%

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

   18. +-commutative 99.8%

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

   19. fma-def 99.8%

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

   20. +-commutative 99.8%

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

   21. fma-def 99.8%

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

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: 98.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.06:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 2.8:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.06)
   (* x -0.70711)
   (if (<= x 2.8)
     (+ 1.6316775383 (* x -2.134856267379707))
     (/ 1.0 (/ 1.0 (* 0.70711 (- (/ 6.039053782637804 x) x)))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.06) {
		tmp = x * -0.70711;
	} else if (x <= 2.8) {
		tmp = 1.6316775383 + (x * -2.134856267379707);
	} else {
		tmp = 1.0 / (1.0 / (0.70711 * ((6.039053782637804 / x) - x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.06d0)) then
        tmp = x * (-0.70711d0)
    else if (x <= 2.8d0) then
        tmp = 1.6316775383d0 + (x * (-2.134856267379707d0))
    else
        tmp = 1.0d0 / (1.0d0 / (0.70711d0 * ((6.039053782637804d0 / x) - x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.06) {
		tmp = x * -0.70711;
	} else if (x <= 2.8) {
		tmp = 1.6316775383 + (x * -2.134856267379707);
	} else {
		tmp = 1.0 / (1.0 / (0.70711 * ((6.039053782637804 / x) - x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.06:
		tmp = x * -0.70711
	elif x <= 2.8:
		tmp = 1.6316775383 + (x * -2.134856267379707)
	else:
		tmp = 1.0 / (1.0 / (0.70711 * ((6.039053782637804 / x) - x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.06)
		tmp = Float64(x * -0.70711);
	elseif (x <= 2.8)
		tmp = Float64(1.6316775383 + Float64(x * -2.134856267379707));
	else
		tmp = Float64(1.0 / Float64(1.0 / Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.06)
		tmp = x * -0.70711;
	elseif (x <= 2.8)
		tmp = 1.6316775383 + (x * -2.134856267379707);
	else
		tmp = 1.0 / (1.0 / (0.70711 * ((6.039053782637804 / x) - x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.06], N[(x * -0.70711), $MachinePrecision], If[LessEqual[x, 2.8], N[(1.6316775383 + N[(x * -2.134856267379707), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.0600000000000001

   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 99.7%

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

      1. *-commutative 99.7%

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

   4. Simplified 99.7%

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

   if -1.0600000000000001 < x < 2.7999999999999998

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 98.7%

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

      1. *-commutative 98.7%

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

   4. Simplified 98.7%

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

   if 2.7999999999999998 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 98.5%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\frac{6.039053782637804}{x}} - x\right) \]
   3. Step-by-step derivation

      1. flip-- 47.7%

         \[\leadsto 0.70711 \cdot \color{blue}{\frac{\frac{6.039053782637804}{x} \cdot \frac{6.039053782637804}{x} - x \cdot x}{\frac{6.039053782637804}{x} + x}} \]

      2. div-inv 47.6%

         \[\leadsto 0.70711 \cdot \color{blue}{\left(\left(\frac{6.039053782637804}{x} \cdot \frac{6.039053782637804}{x} - x \cdot x\right) \cdot \frac{1}{\frac{6.039053782637804}{x} + x}\right)} \]

      3. frac-times 47.6%

         \[\leadsto 0.70711 \cdot \left(\left(\color{blue}{\frac{6.039053782637804 \cdot 6.039053782637804}{x \cdot x}} - x \cdot x\right) \cdot \frac{1}{\frac{6.039053782637804}{x} + x}\right) \]

      4. metadata-eval 47.6%

         \[\leadsto 0.70711 \cdot \left(\left(\frac{\color{blue}{36.47017058959197}}{x \cdot x} - x \cdot x\right) \cdot \frac{1}{\frac{6.039053782637804}{x} + x}\right) \]

      5. pow2 47.6%

         \[\leadsto 0.70711 \cdot \left(\left(\frac{36.47017058959197}{\color{blue}{{x}^{2}}} - x \cdot x\right) \cdot \frac{1}{\frac{6.039053782637804}{x} + x}\right) \]

      6. pow2 47.6%

         \[\leadsto 0.70711 \cdot \left(\left(\frac{36.47017058959197}{{x}^{2}} - \color{blue}{{x}^{2}}\right) \cdot \frac{1}{\frac{6.039053782637804}{x} + x}\right) \]

   4. Applied egg-rr 47.6%

      \[\leadsto 0.70711 \cdot \color{blue}{\left(\left(\frac{36.47017058959197}{{x}^{2}} - {x}^{2}\right) \cdot \frac{1}{\frac{6.039053782637804}{x} + x}\right)} \]
   5. Step-by-step derivation

      1. metadata-eval 47.6%

         \[\leadsto 0.70711 \cdot \left(\left(\frac{\color{blue}{6.039053782637804 \cdot 6.039053782637804}}{{x}^{2}} - {x}^{2}\right) \cdot \frac{1}{\frac{6.039053782637804}{x} + x}\right) \]

      2. unpow2 47.6%

         \[\leadsto 0.70711 \cdot \left(\left(\frac{6.039053782637804 \cdot 6.039053782637804}{\color{blue}{x \cdot x}} - {x}^{2}\right) \cdot \frac{1}{\frac{6.039053782637804}{x} + x}\right) \]

      3. frac-times 47.6%

         \[\leadsto 0.70711 \cdot \left(\left(\color{blue}{\frac{6.039053782637804}{x} \cdot \frac{6.039053782637804}{x}} - {x}^{2}\right) \cdot \frac{1}{\frac{6.039053782637804}{x} + x}\right) \]

   6. Applied egg-rr 47.6%

      \[\leadsto 0.70711 \cdot \left(\left(\color{blue}{\frac{6.039053782637804}{x} \cdot \frac{6.039053782637804}{x}} - {x}^{2}\right) \cdot \frac{1}{\frac{6.039053782637804}{x} + x}\right) \]
   7. Step-by-step derivation

      1. associate-*r* 47.7%

         \[\leadsto \color{blue}{\left(0.70711 \cdot \left(\frac{6.039053782637804}{x} \cdot \frac{6.039053782637804}{x} - {x}^{2}\right)\right) \cdot \frac{1}{\frac{6.039053782637804}{x} + x}} \]

      2. frac-times 47.7%

         \[\leadsto \left(0.70711 \cdot \left(\color{blue}{\frac{6.039053782637804 \cdot 6.039053782637804}{x \cdot x}} - {x}^{2}\right)\right) \cdot \frac{1}{\frac{6.039053782637804}{x} + x} \]

      3. metadata-eval 47.7%

         \[\leadsto \left(0.70711 \cdot \left(\frac{\color{blue}{36.47017058959197}}{x \cdot x} - {x}^{2}\right)\right) \cdot \frac{1}{\frac{6.039053782637804}{x} + x} \]

      4. unpow2 47.7%

         \[\leadsto \left(0.70711 \cdot \left(\frac{36.47017058959197}{\color{blue}{{x}^{2}}} - {x}^{2}\right)\right) \cdot \frac{1}{\frac{6.039053782637804}{x} + x} \]

      5. *-commutative 47.7%

         \[\leadsto \color{blue}{\left(\left(\frac{36.47017058959197}{{x}^{2}} - {x}^{2}\right) \cdot 0.70711\right)} \cdot \frac{1}{\frac{6.039053782637804}{x} + x} \]

      6. div-inv 47.6%

         \[\leadsto \color{blue}{\frac{\left(\frac{36.47017058959197}{{x}^{2}} - {x}^{2}\right) \cdot 0.70711}{\frac{6.039053782637804}{x} + x}} \]

      7. clear-num 47.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{6.039053782637804}{x} + x}{\left(\frac{36.47017058959197}{{x}^{2}} - {x}^{2}\right) \cdot 0.70711}}} \]

      8. clear-num 47.7%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\left(\frac{36.47017058959197}{{x}^{2}} - {x}^{2}\right) \cdot 0.70711}{\frac{6.039053782637804}{x} + x}}}} \]

      9. div-inv 47.7%

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\left(\left(\frac{36.47017058959197}{{x}^{2}} - {x}^{2}\right) \cdot 0.70711\right) \cdot \frac{1}{\frac{6.039053782637804}{x} + x}}}} \]

      10. *-commutative 47.7%

          \[\leadsto \frac{1}{\frac{1}{\color{blue}{\left(0.70711 \cdot \left(\frac{36.47017058959197}{{x}^{2}} - {x}^{2}\right)\right)} \cdot \frac{1}{\frac{6.039053782637804}{x} + x}}} \]

      11. metadata-eval 47.7%

          \[\leadsto \frac{1}{\frac{1}{\left(0.70711 \cdot \left(\frac{\color{blue}{6.039053782637804 \cdot 6.039053782637804}}{{x}^{2}} - {x}^{2}\right)\right) \cdot \frac{1}{\frac{6.039053782637804}{x} + x}}} \]

      12. unpow2 47.7%

          \[\leadsto \frac{1}{\frac{1}{\left(0.70711 \cdot \left(\frac{6.039053782637804 \cdot 6.039053782637804}{\color{blue}{x \cdot x}} - {x}^{2}\right)\right) \cdot \frac{1}{\frac{6.039053782637804}{x} + x}}} \]

      13. frac-times 47.7%

          \[\leadsto \frac{1}{\frac{1}{\left(0.70711 \cdot \left(\color{blue}{\frac{6.039053782637804}{x} \cdot \frac{6.039053782637804}{x}} - {x}^{2}\right)\right) \cdot \frac{1}{\frac{6.039053782637804}{x} + x}}} \]

   8. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.06:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 2.8:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)}}\\ \end{array} \]

Alternative 4: 98.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.06:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 2.8:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.06)
   (* x -0.70711)
   (if (<= x 2.8)
     (+ 1.6316775383 (* x -2.134856267379707))
     (* 0.70711 (- (/ 6.039053782637804 x) x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.06) {
		tmp = x * -0.70711;
	} else if (x <= 2.8) {
		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 <= (-1.06d0)) then
        tmp = x * (-0.70711d0)
    else if (x <= 2.8d0) 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 <= -1.06) {
		tmp = x * -0.70711;
	} else if (x <= 2.8) {
		tmp = 1.6316775383 + (x * -2.134856267379707);
	} else {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.06:
		tmp = x * -0.70711
	elif x <= 2.8:
		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 <= -1.06)
		tmp = Float64(x * -0.70711);
	elseif (x <= 2.8)
		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 <= -1.06)
		tmp = x * -0.70711;
	elseif (x <= 2.8)
		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, -1.06], N[(x * -0.70711), $MachinePrecision], If[LessEqual[x, 2.8], 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 < -1.0600000000000001

   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 99.7%

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

      1. *-commutative 99.7%

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

   4. Simplified 99.7%

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

   if -1.0600000000000001 < x < 2.7999999999999998

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 98.7%

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

      1. *-commutative 98.7%

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

   4. Simplified 98.7%

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

   if 2.7999999999999998 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 98.5%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.06:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 2.8:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \end{array} \]

Alternative 5: 98.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.06 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.06) (not (<= x 1.15)))
   (* x -0.70711)
   (+ 1.6316775383 (* x -2.134856267379707))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.06) || !(x <= 1.15)) {
		tmp = x * -0.70711;
	} 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 <= (-1.06d0)) .or. (.not. (x <= 1.15d0))) then
        tmp = x * (-0.70711d0)
    else
        tmp = 1.6316775383d0 + (x * (-2.134856267379707d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.06) || !(x <= 1.15)) {
		tmp = x * -0.70711;
	} else {
		tmp = 1.6316775383 + (x * -2.134856267379707);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.06) or not (x <= 1.15):
		tmp = x * -0.70711
	else:
		tmp = 1.6316775383 + (x * -2.134856267379707)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.06) || !(x <= 1.15))
		tmp = Float64(x * -0.70711);
	else
		tmp = Float64(1.6316775383 + Float64(x * -2.134856267379707));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.06) || ~((x <= 1.15)))
		tmp = x * -0.70711;
	else
		tmp = 1.6316775383 + (x * -2.134856267379707);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.06], N[Not[LessEqual[x, 1.15]], $MachinePrecision]], N[(x * -0.70711), $MachinePrecision], N[(1.6316775383 + N[(x * -2.134856267379707), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.0600000000000001 or 1.1499999999999999 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 99.2%

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

      1. *-commutative 99.2%

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

   4. Simplified 99.2%

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

   if -1.0600000000000001 < 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. Taylor expanded in x around 0 98.7%

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

      1. *-commutative 98.7%

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

   4. Simplified 98.7%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.06 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \end{array} \]

Alternative 6: 98.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.06 \lor \neg \left(x \leq 1.2\right):\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;1.6316775383\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.06) (not (<= x 1.2))) (* x -0.70711) 1.6316775383))

C

double code(double x) {
	double tmp;
	if ((x <= -1.06) || !(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 <= (-1.06d0)) .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 <= -1.06) || !(x <= 1.2)) {
		tmp = x * -0.70711;
	} else {
		tmp = 1.6316775383;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.06) or not (x <= 1.2):
		tmp = x * -0.70711
	else:
		tmp = 1.6316775383
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.06) || !(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 <= -1.06) || ~((x <= 1.2)))
		tmp = x * -0.70711;
	else
		tmp = 1.6316775383;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.06], N[Not[LessEqual[x, 1.2]], $MachinePrecision]], N[(x * -0.70711), $MachinePrecision], 1.6316775383]

Derivation

1. Split input into 2 regimes

2. if x < -1.0600000000000001 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 99.2%

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

      1. *-commutative 99.2%

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

   4. Simplified 99.2%

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

   if -1.0600000000000001 < 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 96.8%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.06 \lor \neg \left(x \leq 1.2\right):\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;1.6316775383\\ \end{array} \]

Alternative 7: 50.3% 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 50.5%

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

3. Final simplification 50.5%

   \[\leadsto 1.6316775383 \]

Reproduce

herbie shell --seed 2023322 
(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)))