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

Percentage Accurate: 99.9% → 99.9%

Time: 8.9s

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: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, -0.70711, \frac{x \cdot 0.1913510371 + 1.6316775383}{x \cdot 0.99229 + \left(0.04481 \cdot {x}^{2} + 1\right)}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  x
  -0.70711
  (/
   (+ (* x 0.1913510371) 1.6316775383)
   (+ (* x 0.99229) (+ (* 0.04481 (pow x 2.0)) 1.0)))))

C

double code(double x) {
	return fma(x, -0.70711, (((x * 0.1913510371) + 1.6316775383) / ((x * 0.99229) + ((0.04481 * pow(x, 2.0)) + 1.0))));
}

Julia

function code(x)
	return fma(x, -0.70711, Float64(Float64(Float64(x * 0.1913510371) + 1.6316775383) / Float64(Float64(x * 0.99229) + Float64(Float64(0.04481 * (x ^ 2.0)) + 1.0))))
end

Wolfram

code[x_] := N[(x * -0.70711 + N[(N[(N[(x * 0.1913510371), $MachinePrecision] + 1.6316775383), $MachinePrecision] / N[(N[(x * 0.99229), $MachinePrecision] + N[(N[(0.04481 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

   1. fma-udef 99.9%

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

   2. fma-udef 99.9%

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

   3. +-commutative 99.9%

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

   4. distribute-lft-in 99.9%

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

   5. associate-+l+ 99.9%

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

   6. *-commutative 99.9%

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

   7. *-commutative 99.9%

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

   8. associate-*l* 99.9%

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

   9. pow2 99.9%

      \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{x \cdot 0.99229 + \left(0.04481 \cdot \color{blue}{{x}^{2}} + 1\right)}\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\color{blue}{x \cdot 0.99229 + \left(0.04481 \cdot {x}^{2} + 1\right)}}\right) \]
7. Step-by-step derivation

   1. fma-udef 99.9%

      \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\color{blue}{x \cdot 0.1913510371 + 1.6316775383}}{x \cdot 0.99229 + \left(0.04481 \cdot {x}^{2} + 1\right)}\right) \]

8. Applied egg-rr 99.9%

   \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\color{blue}{x \cdot 0.1913510371 + 1.6316775383}}{x \cdot 0.99229 + \left(0.04481 \cdot {x}^{2} + 1\right)}\right) \]

9. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{x \cdot 0.1913510371 + 1.6316775383}{x \cdot 0.99229 + \left(0.04481 \cdot {x}^{2} + 1\right)}\right) \]
10. Add Preprocessing

Alternative 2: 99.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. expm1-log1p-u 76.1%

      \[\leadsto 0.70711 \cdot \left(\frac{2.30753 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot 0.27061\right)\right)}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

   2. expm1-udef 76.1%

      \[\leadsto 0.70711 \cdot \left(\frac{2.30753 + \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot 0.27061\right)} - 1\right)}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

4. Applied egg-rr 76.1%

   \[\leadsto 0.70711 \cdot \left(\frac{2.30753 + \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot 0.27061\right)} - 1\right)}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

5. Taylor expanded in x around 0 99.9%

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

   1. *-commutative 99.9%

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 3: 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 99.8%

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

3. Final simplification 99.8%

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

Alternative 4: 98.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0 98.3%

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

   1. *-commutative 98.3%

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

5. Simplified 98.3%

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

6. Final simplification 98.3%

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

Alternative 5: 98.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;1.6316775383\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 1.15))) (* x -0.70711) 1.6316775383))

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 1.15)) {
		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 <= (-1.05d0)) .or. (.not. (x <= 1.15d0))) 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 <= -1.05) || !(x <= 1.15)) {
		tmp = x * -0.70711;
	} else {
		tmp = 1.6316775383;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.05) or not (x <= 1.15):
		tmp = x * -0.70711
	else:
		tmp = 1.6316775383
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[Or[LessEqual[x, -1.05], N[Not[LessEqual[x, 1.15]], $MachinePrecision]], N[(x * -0.70711), $MachinePrecision], 1.6316775383]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.1499999999999999 < x

   1. Initial program 99.8%

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

   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. Add Preprocessing
   5. Step-by-step derivation

      1. fma-udef 99.8%

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

      2. fma-udef 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-lft-in 99.8%

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

      5. associate-+l+ 99.8%

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

      6. *-commutative 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*l* 99.8%

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

      9. pow2 99.8%

         \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{x \cdot 0.99229 + \left(0.04481 \cdot \color{blue}{{x}^{2}} + 1\right)}\right) \]

   6. Applied egg-rr 99.8%

      \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\color{blue}{x \cdot 0.99229 + \left(0.04481 \cdot {x}^{2} + 1\right)}}\right) \]

   7. Taylor expanded in x around inf 98.3%

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

      1. *-commutative 98.3%

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

   9. Simplified 98.3%

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

   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 97.5%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{2.30753} - x\right) \]
   4. Step-by-step derivation

      1. flip-- 97.5%

         \[\leadsto 0.70711 \cdot \color{blue}{\frac{2.30753 \cdot 2.30753 - x \cdot x}{2.30753 + x}} \]

      2. clear-num 97.5%

         \[\leadsto 0.70711 \cdot \color{blue}{\frac{1}{\frac{2.30753 + x}{2.30753 \cdot 2.30753 - x \cdot x}}} \]

      3. metadata-eval 96.0%

         \[\leadsto 0.70711 \cdot \frac{1}{\frac{2.30753 + x}{\color{blue}{5.3246947009} - x \cdot x}} \]

      4. unpow2 96.0%

         \[\leadsto 0.70711 \cdot \frac{1}{\frac{2.30753 + x}{5.3246947009 - \color{blue}{{x}^{2}}}} \]

   5. Applied egg-rr 96.0%

      \[\leadsto 0.70711 \cdot \color{blue}{\frac{1}{\frac{2.30753 + x}{5.3246947009 - {x}^{2}}}} \]

   6. Taylor expanded in x around 0 97.4%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;1.6316775383\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% 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. Add Preprocessing

3. Taylor expanded in x around 0 97.4%

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

4. Final simplification 97.4%

   \[\leadsto 0.70711 \cdot \left(2.30753 - x\right) \]
5. Add Preprocessing

Alternative 7: 50.4% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ 1.6316775383 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.6316775383)

C

double code(double x) {
	return 1.6316775383;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.6316775383d0
end function

Java

public static double code(double x) {
	return 1.6316775383;
}

Python

def code(x):
	return 1.6316775383

Julia

function code(x)
	return 1.6316775383
end

MATLAB

function tmp = code(x)
	tmp = 1.6316775383;
end

Wolfram

code[x_] := 1.6316775383

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0 97.4%

   \[\leadsto 0.70711 \cdot \left(\color{blue}{2.30753} - x\right) \]
4. Step-by-step derivation

   1. flip-- 77.4%

      \[\leadsto 0.70711 \cdot \color{blue}{\frac{2.30753 \cdot 2.30753 - x \cdot x}{2.30753 + x}} \]

   2. clear-num 77.3%

      \[\leadsto 0.70711 \cdot \color{blue}{\frac{1}{\frac{2.30753 + x}{2.30753 \cdot 2.30753 - x \cdot x}}} \]

   3. metadata-eval 76.6%

      \[\leadsto 0.70711 \cdot \frac{1}{\frac{2.30753 + x}{\color{blue}{5.3246947009} - x \cdot x}} \]

   4. unpow2 76.6%

      \[\leadsto 0.70711 \cdot \frac{1}{\frac{2.30753 + x}{5.3246947009 - \color{blue}{{x}^{2}}}} \]

5. Applied egg-rr 76.6%

   \[\leadsto 0.70711 \cdot \color{blue}{\frac{1}{\frac{2.30753 + x}{5.3246947009 - {x}^{2}}}} \]

6. Taylor expanded in x around 0 51.5%

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

7. Final simplification 51.5%

   \[\leadsto 1.6316775383 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024019 
(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)))