Average Error: 0.1 → 0.1
Time: 3.9s
Precision: binary64
\[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
↓
\[\begin{array}{l} t_0 := \log \left(\sqrt{e^{\frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}}}\right)\\ \mathsf{fma}\left(x, -0.70711, t_0 + t_0\right) \end{array} \]
(FPCore (x)
 :precision binary64
 (*
  0.70711
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x)))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0
         (log
          (sqrt
           (exp
            (/
             (fma x 0.1913510371 1.6316775383)
             (fma x (fma x 0.04481 0.99229) 1.0)))))))
   (fma x -0.70711 (+ t_0 t_0))))
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) {
	double t_0 = log(sqrt(exp((fma(x, 0.1913510371, 1.6316775383) / fma(x, fma(x, 0.04481, 0.99229), 1.0)))));
	return fma(x, -0.70711, (t_0 + t_0));
}

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)
	t_0 = log(sqrt(exp(Float64(fma(x, 0.1913510371, 1.6316775383) / fma(x, fma(x, 0.04481, 0.99229), 1.0)))))
	return fma(x, -0.70711, Float64(t_0 + t_0))
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_] := Block[{t$95$0 = N[Log[N[Sqrt[N[Exp[N[(N[(x * 0.1913510371 + 1.6316775383), $MachinePrecision] / N[(x * N[(x * 0.04481 + 0.99229), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, N[(x * -0.70711 + N[(t$95$0 + t$95$0), $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)
↓
\begin{array}{l}
t_0 := \log \left(\sqrt{e^{\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)\\
\mathsf{fma}\left(x, -0.70711, t_0 + t_0\right)
\end{array}

Error

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. Simplified0.1

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

  3. Applied egg-rr0.1

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2022182 
(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)))