Average Error: 15.3 → 0.1
Time: 6.7s
Precision: binary64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
↓
\[\begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ t_1 := 1 + \sqrt{0.5 + t_0}\\ \mathbf{if}\;x \leq -0.0025715001607349015:\\ \;\;\;\;\frac{0.5 - \log \left(e^{t_0}\right)}{t_1}\\ \mathbf{elif}\;x \leq 0.002254868162487943:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot 0.125, {x}^{4} \cdot -0.0859375\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(0.5 - t_0\right)\right)}{t_1}\\ \end{array} \]
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 := 1 + \sqrt{0.5 + t_0}\\
\mathbf{if}\;x \leq -0.0025715001607349015:\\
\;\;\;\;\frac{0.5 - \log \left(e^{t_0}\right)}{t_1}\\

\mathbf{elif}\;x \leq 0.002254868162487943:\\
\;\;\;\;\mathsf{fma}\left(x, x \cdot 0.125, {x}^{4} \cdot -0.0859375\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(0.5 - t_0\right)\right)}{t_1}\\


\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 (+ 1.0 (sqrt (+ 0.5 t_0)))))
   (if (<= x -0.0025715001607349015)
     (/ (- 0.5 (log (exp t_0))) t_1)
     (if (<= x 0.002254868162487943)
       (fma x (* x 0.125) (* (pow x 4.0) -0.0859375))
       (/ (log1p (expm1 (- 0.5 t_0))) t_1)))))
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 = 1.0 + sqrt((0.5 + t_0));
	double tmp;
	if (x <= -0.0025715001607349015) {
		tmp = (0.5 - log(exp(t_0))) / t_1;
	} else if (x <= 0.002254868162487943) {
		tmp = fma(x, (x * 0.125), (pow(x, 4.0) * -0.0859375));
	} else {
		tmp = log1p(expm1((0.5 - t_0))) / t_1;
	}
	return tmp;
}

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -0.00257150016073490152

    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 flip--_binary641.1

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

    4. Simplified0.1

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

    5. Applied add-log-exp_binary640.1

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

    if -0.00257150016073490152 < x < 0.00225486816248794296

    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. Applied flip--_binary6429.8

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

    4. Simplified29.8

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

    5. Taylor expanded in x around 0 0.1

      \[\leadsto \color{blue}{0.125 \cdot {x}^{2} - 0.0859375 \cdot {x}^{4}} \]

    6. Simplified0.0

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

    if 0.00225486816248794296 < 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 flip--_binary641.0

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

    4. Simplified0.1

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

    5. Applied log1p-expm1-u_binary640.1

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0025715001607349015:\\ \;\;\;\;\frac{0.5 - \log \left(e^{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \mathbf{elif}\;x \leq 0.002254868162487943:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot 0.125, {x}^{4} \cdot -0.0859375\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))