Average Error: 15.2 → 0.1
Time: 5.6s
Precision: binary64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
↓
\[\begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ t_1 := 0.5 + t_0\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.000000161409677:\\ \;\;\;\;\mathsf{fma}\left(0.125, x \cdot x, 0.0673828125 \cdot {x}^{6}\right) - 0.0859375 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{1 - {t_1}^{1.5}}{\sqrt{t_1} + \left(t_0 + 1.5\right)}\right|\\ \end{array} \]
(FPCore (x)
 :precision binary64
 (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))) (t_1 (+ 0.5 t_0)))
   (if (<= (hypot 1.0 x) 1.000000161409677)
     (-
      (fma 0.125 (* x x) (* 0.0673828125 (pow x 6.0)))
      (* 0.0859375 (pow x 4.0)))
     (fabs (/ (- 1.0 (pow t_1 1.5)) (+ (sqrt t_1) (+ t_0 1.5)))))))
double code(double x) {
	return 1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x)))));
}
↓
double code(double x) {
	double t_0 = 0.5 / hypot(1.0, x);
	double t_1 = 0.5 + t_0;
	double tmp;
	if (hypot(1.0, x) <= 1.000000161409677) {
		tmp = fma(0.125, (x * x), (0.0673828125 * pow(x, 6.0))) - (0.0859375 * pow(x, 4.0));
	} else {
		tmp = fabs(((1.0 - pow(t_1, 1.5)) / (sqrt(t_1) + (t_0 + 1.5))));
	}
	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)
	t_0 = Float64(0.5 / hypot(1.0, x))
	t_1 = Float64(0.5 + t_0)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.000000161409677)
		tmp = Float64(fma(0.125, Float64(x * x), Float64(0.0673828125 * (x ^ 6.0))) - Float64(0.0859375 * (x ^ 4.0)));
	else
		tmp = abs(Float64(Float64(1.0 - (t_1 ^ 1.5)) / Float64(sqrt(t_1) + Float64(t_0 + 1.5))));
	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_] := Block[{t$95$0 = N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 + t$95$0), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.000000161409677], N[(N[(0.125 * N[(x * x), $MachinePrecision] + N[(0.0673828125 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[(1.0 - N[Power[t$95$1, 1.5], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$1], $MachinePrecision] + N[(t$95$0 + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\begin{array}{l}
t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\
t_1 := 0.5 + t_0\\
\mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.000000161409677:\\
\;\;\;\;\mathsf{fma}\left(0.125, x \cdot x, 0.0673828125 \cdot {x}^{6}\right) - 0.0859375 \cdot {x}^{4}\\

\mathbf{else}:\\
\;\;\;\;\left|\frac{1 - {t_1}^{1.5}}{\sqrt{t_1} + \left(t_0 + 1.5\right)}\right|\\


\end{array}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if (hypot.f64 1 x) < 1.00000016140967696

    1. Initial program 29.9

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

    2. Simplified29.9

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

    3. Taylor expanded in x around 0 0.0

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

    4. Simplified0.0

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

    if 1.00000016140967696 < (hypot.f64 1 x)

    1. Initial program 1.1

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

    2. Simplified1.1

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

    3. Applied egg-rr0.1

      \[\leadsto \color{blue}{\frac{1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5}}{1 + \left(\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}} \]

    4. Applied egg-rr0.1

      \[\leadsto \color{blue}{\left|\frac{1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)}\right|} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.000000161409677:\\ \;\;\;\;\mathsf{fma}\left(0.125, x \cdot x, 0.0673828125 \cdot {x}^{6}\right) - 0.0859375 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)}\right|\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))