Average Error: 0.1 → 0.1
Time: 3.1s
Precision: binary64
\[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right), {\left({\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)\right)}^{-0.5}\right)}^{2}, x \cdot -0.70711\right) \]
(FPCore (x)
 :precision binary64
 (*
  0.70711
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x)))
↓
(FPCore (x)
 :precision binary64
 (fma
  (fma x 0.1913510371 1.6316775383)
  (pow (pow (fma x (fma x 0.04481 0.99229) 1.0) -0.5) 2.0)
  (* x -0.70711)))
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 fma(fma(x, 0.1913510371, 1.6316775383), pow(pow(fma(x, fma(x, 0.04481, 0.99229), 1.0), -0.5), 2.0), (x * -0.70711));
}

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 fma(fma(x, 0.1913510371, 1.6316775383), ((fma(x, fma(x, 0.04481, 0.99229), 1.0) ^ -0.5) ^ 2.0), Float64(x * -0.70711))
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[(N[(x * 0.1913510371 + 1.6316775383), $MachinePrecision] * N[Power[N[Power[N[(x * N[(x * 0.04481 + 0.99229), $MachinePrecision] + 1.0), $MachinePrecision], -0.5], $MachinePrecision], 2.0], $MachinePrecision] + N[(x * -0.70711), $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)
↓
\mathsf{fma}\left(\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right), {\left({\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)\right)}^{-0.5}\right)}^{2}, x \cdot -0.70711\right)

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-rr1.4

    \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)}\right)}^{3}} \]

  4. Applied egg-rr0.1

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

  5. Applied egg-rr0.1

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right), \color{blue}{{\left({\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)\right)}^{-0.5}\right)}^{2}}, x \cdot -0.70711\right) \]

  6. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right), {\left({\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)\right)}^{-0.5}\right)}^{2}, x \cdot -0.70711\right) \]

Reproduce

herbie shell --seed 2022210 
(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)))