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

Percentage Accurate: 99.9% → 99.9%

Time: 7.7s

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, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. div-inv 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

   4. +-commutative 99.8%

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

   5. +-commutative 99.8%

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

   6. fma-udef 99.8%

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

   7. fma-udef 99.8%

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

3. Applied egg-rr 99.8%

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

   1. fma-udef 99.8%

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

5. Applied egg-rr 99.8%

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

   1. fma-udef 99.8%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(0.70711 * N[(N[(N[(N[(x * 0.27061), $MachinePrecision] + 2.30753), $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%

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

Alternative 3: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \lor \neg \left(x \leq 3.5\right):\\ \;\;\;\;\left(\frac{4.2702753202410175}{x} - \frac{58.14938538768042}{x \cdot x}\right) + x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(\left(2.30753 + x \cdot -2.0191289437\right) - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -5.2) (not (<= x 3.5)))
   (+
    (- (/ 4.2702753202410175 x) (/ 58.14938538768042 (* x x)))
    (* x -0.70711))
   (* 0.70711 (- (+ 2.30753 (* x -2.0191289437)) x))))

C

double code(double x) {
	double tmp;
	if ((x <= -5.2) || !(x <= 3.5)) {
		tmp = ((4.2702753202410175 / x) - (58.14938538768042 / (x * x))) + (x * -0.70711);
	} else {
		tmp = 0.70711 * ((2.30753 + (x * -2.0191289437)) - x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-5.2d0)) .or. (.not. (x <= 3.5d0))) then
        tmp = ((4.2702753202410175d0 / x) - (58.14938538768042d0 / (x * x))) + (x * (-0.70711d0))
    else
        tmp = 0.70711d0 * ((2.30753d0 + (x * (-2.0191289437d0))) - x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -5.2) || !(x <= 3.5)) {
		tmp = ((4.2702753202410175 / x) - (58.14938538768042 / (x * x))) + (x * -0.70711);
	} else {
		tmp = 0.70711 * ((2.30753 + (x * -2.0191289437)) - x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -5.2) or not (x <= 3.5):
		tmp = ((4.2702753202410175 / x) - (58.14938538768042 / (x * x))) + (x * -0.70711)
	else:
		tmp = 0.70711 * ((2.30753 + (x * -2.0191289437)) - x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -5.2) || !(x <= 3.5))
		tmp = Float64(Float64(Float64(4.2702753202410175 / x) - Float64(58.14938538768042 / Float64(x * x))) + Float64(x * -0.70711));
	else
		tmp = Float64(0.70711 * Float64(Float64(2.30753 + Float64(x * -2.0191289437)) - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -5.2) || ~((x <= 3.5)))
		tmp = ((4.2702753202410175 / x) - (58.14938538768042 / (x * x))) + (x * -0.70711);
	else
		tmp = 0.70711 * ((2.30753 + (x * -2.0191289437)) - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -5.2], N[Not[LessEqual[x, 3.5]], $MachinePrecision]], N[(N[(N[(4.2702753202410175 / x), $MachinePrecision] - N[(58.14938538768042 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * -0.70711), $MachinePrecision]), $MachinePrecision], N[(0.70711 * N[(N[(2.30753 + N[(x * -2.0191289437), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.20000000000000018 or 3.5 < 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. Step-by-step derivation

      1. sub-neg 99.7%

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

      2. +-commutative 99.7%

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

      3. distribute-rgt-in 99.7%

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

      4. distribute-lft-neg-out 99.7%

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

      5. distribute-rgt-neg-in 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. fma-def 99.7%

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

      9. metadata-eval 99.7%

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

      10. associate-*l/ 99.7%

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

   3. Simplified 99.7%

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

      1. fma-udef 99.7%

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

      2. +-commutative 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x around inf 98.6%

      \[\leadsto \color{blue}{\left(4.2702753202410175 \cdot \frac{1}{x} - 58.14938538768042 \cdot \frac{1}{{x}^{2}}\right)} + x \cdot -0.70711 \]
   7. Step-by-step derivation

      1. associate-*r/ 98.6%

         \[\leadsto \left(\color{blue}{\frac{4.2702753202410175 \cdot 1}{x}} - 58.14938538768042 \cdot \frac{1}{{x}^{2}}\right) + x \cdot -0.70711 \]

      2. metadata-eval 98.6%

         \[\leadsto \left(\frac{\color{blue}{4.2702753202410175}}{x} - 58.14938538768042 \cdot \frac{1}{{x}^{2}}\right) + x \cdot -0.70711 \]

      3. associate-*r/ 98.6%

         \[\leadsto \left(\frac{4.2702753202410175}{x} - \color{blue}{\frac{58.14938538768042 \cdot 1}{{x}^{2}}}\right) + x \cdot -0.70711 \]

      4. metadata-eval 98.6%

         \[\leadsto \left(\frac{4.2702753202410175}{x} - \frac{\color{blue}{58.14938538768042}}{{x}^{2}}\right) + x \cdot -0.70711 \]

      5. unpow2 98.6%

         \[\leadsto \left(\frac{4.2702753202410175}{x} - \frac{58.14938538768042}{\color{blue}{x \cdot x}}\right) + x \cdot -0.70711 \]

   8. Simplified 98.6%

      \[\leadsto \color{blue}{\left(\frac{4.2702753202410175}{x} - \frac{58.14938538768042}{x \cdot x}\right)} + x \cdot -0.70711 \]

   if -5.20000000000000018 < x < 3.5

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.5%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\left(2.30753 + -2.0191289437 \cdot x\right)} - x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \lor \neg \left(x \leq 3.5\right):\\ \;\;\;\;\left(\frac{4.2702753202410175}{x} - \frac{58.14938538768042}{x \cdot x}\right) + x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(\left(2.30753 + x \cdot -2.0191289437\right) - x\right)\\ \end{array} \]

Alternative 4: 99.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \lor \neg \left(x \leq 0.75\right):\\ \;\;\;\;\frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(\left(2.30753 + x \cdot -2.0191289437\right) - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.1) (not (<= x 0.75)))
   (+ (/ 4.2702753202410175 x) (* x -0.70711))
   (* 0.70711 (- (+ 2.30753 (* x -2.0191289437)) x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.1) || !(x <= 0.75)) {
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	} else {
		tmp = 0.70711 * ((2.30753 + (x * -2.0191289437)) - x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.1d0)) .or. (.not. (x <= 0.75d0))) then
        tmp = (4.2702753202410175d0 / x) + (x * (-0.70711d0))
    else
        tmp = 0.70711d0 * ((2.30753d0 + (x * (-2.0191289437d0))) - x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.1) || !(x <= 0.75)) {
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	} else {
		tmp = 0.70711 * ((2.30753 + (x * -2.0191289437)) - x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.1) or not (x <= 0.75):
		tmp = (4.2702753202410175 / x) + (x * -0.70711)
	else:
		tmp = 0.70711 * ((2.30753 + (x * -2.0191289437)) - x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.1) || !(x <= 0.75))
		tmp = Float64(Float64(4.2702753202410175 / x) + Float64(x * -0.70711));
	else
		tmp = Float64(0.70711 * Float64(Float64(2.30753 + Float64(x * -2.0191289437)) - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.1) || ~((x <= 0.75)))
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	else
		tmp = 0.70711 * ((2.30753 + (x * -2.0191289437)) - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.1], N[Not[LessEqual[x, 0.75]], $MachinePrecision]], N[(N[(4.2702753202410175 / x), $MachinePrecision] + N[(x * -0.70711), $MachinePrecision]), $MachinePrecision], N[(0.70711 * N[(N[(2.30753 + N[(x * -2.0191289437), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1000000000000001 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 98.3%

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

      1. associate-*r/ 98.3%

         \[\leadsto \color{blue}{\frac{4.2702753202410175 \cdot 1}{x}} + -0.70711 \cdot x \]

      2. metadata-eval 98.3%

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

      3. *-commutative 98.3%

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

   4. Simplified 98.3%

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

   if -1.1000000000000001 < x < 0.75

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.5%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\left(2.30753 + -2.0191289437 \cdot x\right)} - x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \lor \neg \left(x \leq 0.75\right):\\ \;\;\;\;\frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(\left(2.30753 + x \cdot -2.0191289437\right) - x\right)\\ \end{array} \]

Alternative 5: 99.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \lor \neg \left(x \leq 0.75\right):\\ \;\;\;\;\frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2.134856267379707 + 1.6316775383\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.1) (not (<= x 0.75)))
   (+ (/ 4.2702753202410175 x) (* x -0.70711))
   (+ (* x -2.134856267379707) 1.6316775383)))

C

double code(double x) {
	double tmp;
	if ((x <= -1.1) || !(x <= 0.75)) {
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	} else {
		tmp = (x * -2.134856267379707) + 1.6316775383;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.1d0)) .or. (.not. (x <= 0.75d0))) then
        tmp = (4.2702753202410175d0 / x) + (x * (-0.70711d0))
    else
        tmp = (x * (-2.134856267379707d0)) + 1.6316775383d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.1) || !(x <= 0.75)) {
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	} else {
		tmp = (x * -2.134856267379707) + 1.6316775383;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.1) or not (x <= 0.75):
		tmp = (4.2702753202410175 / x) + (x * -0.70711)
	else:
		tmp = (x * -2.134856267379707) + 1.6316775383
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.1) || !(x <= 0.75))
		tmp = Float64(Float64(4.2702753202410175 / x) + Float64(x * -0.70711));
	else
		tmp = Float64(Float64(x * -2.134856267379707) + 1.6316775383);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.1) || ~((x <= 0.75)))
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	else
		tmp = (x * -2.134856267379707) + 1.6316775383;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.1], N[Not[LessEqual[x, 0.75]], $MachinePrecision]], N[(N[(4.2702753202410175 / x), $MachinePrecision] + N[(x * -0.70711), $MachinePrecision]), $MachinePrecision], N[(N[(x * -2.134856267379707), $MachinePrecision] + 1.6316775383), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1000000000000001 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 98.3%

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

      1. associate-*r/ 98.3%

         \[\leadsto \color{blue}{\frac{4.2702753202410175 \cdot 1}{x}} + -0.70711 \cdot x \]

      2. metadata-eval 98.3%

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

      3. *-commutative 98.3%

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

   4. Simplified 98.3%

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

   if -1.1000000000000001 < x < 0.75

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.5%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \lor \neg \left(x \leq 0.75\right):\\ \;\;\;\;\frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2.134856267379707 + 1.6316775383\\ \end{array} \]

Alternative 6: 98.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;x \cdot -2.134856267379707 + 1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.1)
   (* x -0.70711)
   (if (<= x 1.15) (+ (* x -2.134856267379707) 1.6316775383) (* x -0.70711))))

C

double code(double x) {
	double tmp;
	if (x <= -1.1) {
		tmp = x * -0.70711;
	} else if (x <= 1.15) {
		tmp = (x * -2.134856267379707) + 1.6316775383;
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.1d0)) then
        tmp = x * (-0.70711d0)
    else if (x <= 1.15d0) then
        tmp = (x * (-2.134856267379707d0)) + 1.6316775383d0
    else
        tmp = x * (-0.70711d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.1) {
		tmp = x * -0.70711;
	} else if (x <= 1.15) {
		tmp = (x * -2.134856267379707) + 1.6316775383;
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.1:
		tmp = x * -0.70711
	elif x <= 1.15:
		tmp = (x * -2.134856267379707) + 1.6316775383
	else:
		tmp = x * -0.70711
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.1)
		tmp = Float64(x * -0.70711);
	elseif (x <= 1.15)
		tmp = Float64(Float64(x * -2.134856267379707) + 1.6316775383);
	else
		tmp = Float64(x * -0.70711);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.1)
		tmp = x * -0.70711;
	elseif (x <= 1.15)
		tmp = (x * -2.134856267379707) + 1.6316775383;
	else
		tmp = x * -0.70711;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.1], N[(x * -0.70711), $MachinePrecision], If[LessEqual[x, 1.15], N[(N[(x * -2.134856267379707), $MachinePrecision] + 1.6316775383), $MachinePrecision], N[(x * -0.70711), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1000000000000001 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 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 -1.1000000000000001 < x < 1.1499999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.5%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;x \cdot -2.134856267379707 + 1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \]

Alternative 7: 98.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.4) (* x -0.70711) (if (<= x 1.2) 1.6316775383 (* x -0.70711))))

C

double code(double x) {
	double tmp;
	if (x <= -3.4) {
		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 <= (-3.4d0)) 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 <= -3.4) {
		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 <= -3.4:
		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 <= -3.4)
		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 <= -3.4)
		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, -3.4], 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 < -3.39999999999999991 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 -3.39999999999999991 < x < 1.19999999999999996

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 97.7%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \]

Alternative 8: 50.5% 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 46.9%

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

3. Final simplification 46.9%

   \[\leadsto 1.6316775383 \]

Reproduce

herbie shell --seed 2023171 
(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)))