Average Error: 0.1 → 0.1
Time: 2.3s
Precision: binary64
\[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
↓
\[0.70711 \cdot \left(\mathsf{fma}\left(x, 0.27061, 2.30753\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)} - x\right) \]
(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
  (-
   (* (fma x 0.27061 2.30753) (/ 1.0 (fma x (fma x 0.04481 0.99229) 1.0)))
   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 * ((fma(x, 0.27061, 2.30753) * (1.0 / fma(x, fma(x, 0.04481, 0.99229), 1.0))) - 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(fma(x, 0.27061, 2.30753) * Float64(1.0 / fma(x, fma(x, 0.04481, 0.99229), 1.0))) - 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[(x * 0.27061 + 2.30753), $MachinePrecision] * N[(1.0 / N[(x * N[(x * 0.04481 + 0.99229), $MachinePrecision] + 1.0), $MachinePrecision]), $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(\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)

Error

Bits error versus x

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 div-inv_binary640.1

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

  3. Simplified0.1

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

  4. Applied *-un-lft-identity_binary640.1

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

  5. Applied *-un-lft-identity_binary640.1

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

  6. Applied times-frac_binary640.1

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

  7. Applied associate-*r*_binary640.1

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

  8. Simplified0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2022129 
(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)))