Average Error: 15.4 → 0.0
Time: 9.9s
Precision: binary64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
↓
\[\begin{array}{l} t_0 := 0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\\ t_1 := 1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \mathbf{if}\;x \leq -0.027965099262194516:\\ \;\;\;\;\frac{t_0}{t_1}\\ \mathbf{elif}\;x \leq 0.028761853505194473:\\ \;\;\;\;\left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right) - \left(0.0859375 \cdot {x}^{4} + 0.056243896484375 \cdot {x}^{8}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(t_0\right)\right)}{t_1}\\ \end{array} \]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\begin{array}{l}
t_0 := 0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\\
t_1 := 1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\
\mathbf{if}\;x \leq -0.027965099262194516:\\
\;\;\;\;\frac{t_0}{t_1}\\

\mathbf{elif}\;x \leq 0.028761853505194473:\\
\;\;\;\;\left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right) - \left(0.0859375 \cdot {x}^{4} + 0.056243896484375 \cdot {x}^{8}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(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 (/ -0.5 (hypot 1.0 x))))
        (t_1 (+ 1.0 (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 x)))))))
   (if (<= x -0.027965099262194516)
     (/ t_0 t_1)
     (if (<= x 0.028761853505194473)
       (-
        (+ (* 0.0673828125 (pow x 6.0)) (* 0.125 (pow x 2.0)))
        (+ (* 0.0859375 (pow x 4.0)) (* 0.056243896484375 (pow x 8.0))))
       (/ (log1p (expm1 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 + (-0.5 / hypot(1.0, x));
	double t_1 = 1.0 + sqrt(0.5 + (0.5 / hypot(1.0, x)));
	double tmp;
	if (x <= -0.027965099262194516) {
		tmp = t_0 / t_1;
	} else if (x <= 0.028761853505194473) {
		tmp = ((0.0673828125 * pow(x, 6.0)) + (0.125 * pow(x, 2.0))) - ((0.0859375 * pow(x, 4.0)) + (0.056243896484375 * pow(x, 8.0)));
	} else {
		tmp = log1p(expm1(t_0)) / t_1;
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x < -0.027965099262194516

    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 sub-neg_binary640.1

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

    6. Simplified0.1

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

    if -0.027965099262194516 < x < 0.028761853505194473

    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

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

    if 0.028761853505194473 < 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. Simplified0.0

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

    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 simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.027965099262194516:\\ \;\;\;\;\frac{0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \mathbf{elif}\;x \leq 0.028761853505194473:\\ \;\;\;\;\left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right) - \left(0.0859375 \cdot {x}^{4} + 0.056243896484375 \cdot {x}^{8}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))