Average Error: 15.3 → 0.0
Time: 3.5s
Precision: binary64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
↓
\[\begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.002:\\ \;\;\;\;\mathsf{fma}\left(0.125, x \cdot x, \mathsf{fma}\left(0.0673828125, {x}^{6}, \mathsf{fma}\left({x}^{4}, -0.0859375, {x}^{8} \cdot -0.056243896484375\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]
(FPCore (x)
 :precision binary64
 (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))
↓
(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.002)
   (fma
    0.125
    (* x x)
    (fma
     0.0673828125
     (pow x 6.0)
     (fma (pow x 4.0) -0.0859375 (* (pow x 8.0) -0.056243896484375))))
   (/
    (+ 0.5 (/ -0.5 (hypot 1.0 x)))
    (+ 1.0 (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 x))))))))
double code(double x) {
	return 1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x)))));
}
↓
double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.002) {
		tmp = fma(0.125, (x * x), fma(0.0673828125, pow(x, 6.0), fma(pow(x, 4.0), -0.0859375, (pow(x, 8.0) * -0.056243896484375))));
	} else {
		tmp = (0.5 + (-0.5 / hypot(1.0, x))) / (1.0 + sqrt((0.5 + (0.5 / hypot(1.0, x)))));
	}
	return tmp;
}

function code(x)
	return Float64(1.0 - sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / hypot(1.0, x))))))
end
↓
function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.002)
		tmp = fma(0.125, Float64(x * x), fma(0.0673828125, (x ^ 6.0), fma((x ^ 4.0), -0.0859375, Float64((x ^ 8.0) * -0.056243896484375))));
	else
		tmp = Float64(Float64(0.5 + Float64(-0.5 / hypot(1.0, x))) / Float64(1.0 + sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, x))))));
	end
	return tmp
end

code[x_] := N[(1.0 - N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]
↓
code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.002], N[(0.125 * N[(x * x), $MachinePrecision] + N[(0.0673828125 * N[Power[x, 6.0], $MachinePrecision] + N[(N[Power[x, 4.0], $MachinePrecision] * -0.0859375 + N[(N[Power[x, 8.0], $MachinePrecision] * -0.056243896484375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(-0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.002:\\
\;\;\;\;\mathsf{fma}\left(0.125, x \cdot x, \mathsf{fma}\left(0.0673828125, {x}^{6}, \mathsf{fma}\left({x}^{4}, -0.0859375, {x}^{8} \cdot -0.056243896484375\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\


\end{array}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if (hypot.f64 1 x) < 1.002

    1. Initial program 29.8

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]

    2. Simplified29.8

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

    3. Taylor expanded in x around 0 0.0

      \[\leadsto \color{blue}{\left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right) - \left(0.0859375 \cdot {x}^{4} + 0.056243896484375 \cdot {x}^{8}\right)} \]

    4. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, x \cdot x, \mathsf{fma}\left(0.0673828125, {x}^{6}, \mathsf{fma}\left({x}^{4}, -0.0859375, {x}^{8} \cdot -0.056243896484375\right)\right)\right)} \]

    if 1.002 < (hypot.f64 1 x)

    1. Initial program 1.0

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]

    2. Simplified1.0

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

    3. Applied egg-rr0.0

      \[\leadsto \color{blue}{\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.002:\\ \;\;\;\;\mathsf{fma}\left(0.125, x \cdot x, \mathsf{fma}\left(0.0673828125, {x}^{6}, \mathsf{fma}\left({x}^{4}, -0.0859375, {x}^{8} \cdot -0.056243896484375\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))