?

Average Error: 0.1 → 0.1
Time: 1.7min
Precision: binary64
Cost: 1472

?

\[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
↓
\[0.70711 \cdot \left(\frac{2.30753}{\left(1 + \left(x \cdot 0.04481\right) \cdot x\right) + 0.99229 \cdot x} \cdot \left(\left(8.527142382025794 + x\right) \cdot 0.11727258150489918\right) - x\right) \]
(FPCore (x)
 :precision binary64
 (*
  0.70711
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x)))
↓
(FPCore (x)
 :precision binary64
 (*
  0.70711
  (-
   (*
    (/ 2.30753 (+ (+ 1.0 (* (* x 0.04481) x)) (* 0.99229 x)))
    (* (+ 8.527142382025794 x) 0.11727258150489918))
   x)))
double code(double x) {
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
}
↓
double code(double x) {
	return 0.70711 * (((2.30753 / ((1.0 + ((x * 0.04481) * x)) + (0.99229 * x))) * ((8.527142382025794 + x) * 0.11727258150489918)) - x);
}

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
↓
real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.70711d0 * (((2.30753d0 / ((1.0d0 + ((x * 0.04481d0) * x)) + (0.99229d0 * x))) * ((8.527142382025794d0 + x) * 0.11727258150489918d0)) - x)
end function

public static double code(double x) {
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
}
↓
public static double code(double x) {
	return 0.70711 * (((2.30753 / ((1.0 + ((x * 0.04481) * x)) + (0.99229 * x))) * ((8.527142382025794 + x) * 0.11727258150489918)) - x);
}

def code(x):
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x)
↓
def code(x):
	return 0.70711 * (((2.30753 / ((1.0 + ((x * 0.04481) * x)) + (0.99229 * x))) * ((8.527142382025794 + x) * 0.11727258150489918)) - x)

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
↓
function code(x)
	return Float64(0.70711 * Float64(Float64(Float64(2.30753 / Float64(Float64(1.0 + Float64(Float64(x * 0.04481) * x)) + Float64(0.99229 * x))) * Float64(Float64(8.527142382025794 + x) * 0.11727258150489918)) - x))
end

function tmp = code(x)
	tmp = 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
end
↓
function tmp = code(x)
	tmp = 0.70711 * (((2.30753 / ((1.0 + ((x * 0.04481) * x)) + (0.99229 * x))) * ((8.527142382025794 + x) * 0.11727258150489918)) - x);
end

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]
↓
code[x_] := N[(0.70711 * N[(N[(N[(2.30753 / N[(N[(1.0 + N[(N[(x * 0.04481), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + N[(0.99229 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(8.527142382025794 + x), $MachinePrecision] * 0.11727258150489918), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]

0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right)
↓
0.70711 \cdot \left(\frac{2.30753}{\left(1 + \left(x \cdot 0.04481\right) \cdot x\right) + 0.99229 \cdot x} \cdot \left(\left(8.527142382025794 + x\right) \cdot 0.11727258150489918\right) - x\right)

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 0.1

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

  2. Applied egg-rr0.1

    \[\leadsto 0.70711 \cdot \left(\color{blue}{2.30753 \cdot \frac{\frac{8.527142382025794 + x}{8.527142382025794}}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}} - x\right) \]

  3. Simplified0.1

    \[\leadsto 0.70711 \cdot \left(\color{blue}{\frac{2.30753}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)} \cdot \left(\left(8.527142382025794 + x\right) \cdot 0.11727258150489918\right)} - x\right) \]
    Proof

  4. Applied egg-rr0.1

    \[\leadsto 0.70711 \cdot \left(\frac{2.30753}{\color{blue}{\left(1 + \left(x \cdot 0.04481\right) \cdot x\right) + 0.99229 \cdot x}} \cdot \left(\left(8.527142382025794 + x\right) \cdot 0.11727258150489918\right) - x\right) \]

Alternatives

Alternative 1
Error0.1
Cost1344
\[0.70711 \cdot \left(\frac{\left(8.527142382025794 + x\right) \cdot 0.11727258150489918}{\frac{-1 - x \cdot \left(x \cdot 0.04481 + 0.99229\right)}{-2.30753}} - x\right) \]
Alternative 2
Error0.6
Cost1216
\[0.70711 \cdot \left(\frac{0.27061 \cdot \left(8.527142382025794 + x\right)}{1 + x \cdot \left(x \cdot 0.04481 + 0.99229\right)} - x\right) \]
Alternative 3
Error0.1
Cost1216
\[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
Alternative 4
Error1.1
Cost960
\[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + 0.99229 \cdot x} - x\right) \]
Alternative 5
Error1.2
Cost456
\[\begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.16:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \]
Alternative 6
Error1.6
Cost320
\[0.70711 \cdot \left(2.30753 - x\right) \]
Alternative 7
Error1.6
Cost320
\[\frac{x - 2.30753}{-1.4142071247754946} \]
Alternative 8
Error31.9
Cost64
\[1.6316775383 \]

Error

Reproduce?

herbie shell --seed 2023033 
(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)))