Kahan p9 Example

Average Error: 20.0 → 4.8

Time: 2.2s

Precision: 64

\[0.0 \lt x \lt 1 \land y \lt 1\]

Math

\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]

↓

\[\begin{array}{l} \mathbf{if}\;y \le -1.14178066733525963 \cdot 10^{154}:\\ \;\;\;\;\sqrt[3]{-1}\\ \mathbf{elif}\;y \le -1.6515932751816961 \cdot 10^{-162}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\right)}^{3}}\\ \mathbf{elif}\;y \le 7.8035108112990794 \cdot 10^{-163}:\\ \;\;\;\;\sqrt[3]{1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\right)}^{3}}\\ \end{array}\]

C

double f(double x, double y) {
        double r74319 = x;
        double r74320 = y;
        double r74321 = r74319 - r74320;
        double r74322 = r74319 + r74320;
        double r74323 = r74321 * r74322;
        double r74324 = r74319 * r74319;
        double r74325 = r74320 * r74320;
        double r74326 = r74324 + r74325;
        double r74327 = r74323 / r74326;
        return r74327;
}

↓

double f(double x, double y) {
        double r74328 = y;
        double r74329 = -1.1417806673352596e+154;
        bool r74330 = r74328 <= r74329;
        double r74331 = -1.0;
        double r74332 = cbrt(r74331);
        double r74333 = -1.651593275181696e-162;
        bool r74334 = r74328 <= r74333;
        double r74335 = x;
        double r74336 = r74335 - r74328;
        double r74337 = r74335 + r74328;
        double r74338 = r74336 * r74337;
        double r74339 = r74335 * r74335;
        double r74340 = r74328 * r74328;
        double r74341 = r74339 + r74340;
        double r74342 = r74338 / r74341;
        double r74343 = 3.0;
        double r74344 = pow(r74342, r74343);
        double r74345 = cbrt(r74344);
        double r74346 = 7.80351081129908e-163;
        bool r74347 = r74328 <= r74346;
        double r74348 = 1.0;
        double r74349 = cbrt(r74348);
        double r74350 = r74347 ? r74349 : r74345;
        double r74351 = r74334 ? r74345 : r74350;
        double r74352 = r74330 ? r74332 : r74351;
        return r74352;
}

Error

Bits error versus x

Bits error versus y

Target

Original	20.0
Target	0.1
Herbie	4.8

\[\begin{array}{l} \mathbf{if}\;0.5 \lt \left|\frac{x}{y}\right| \lt 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if y < -1.1417806673352596e+154

   1. Initial program 64.0

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
   2. Using strategy rm

   3. Applied add-cbrt-cube 64.0

      \[\leadsto \frac{\left(x - y\right) \cdot \left(x + y\right)}{\color{blue}{\sqrt[3]{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}}}\]

   4. Applied add-cbrt-cube 64.0

      \[\leadsto \frac{\left(x - y\right) \cdot \color{blue}{\sqrt[3]{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)}}}{\sqrt[3]{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}}\]

   5. Applied add-cbrt-cube 64.0

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(x - y\right) \cdot \left(x - y\right)\right) \cdot \left(x - y\right)}} \cdot \sqrt[3]{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)}}{\sqrt[3]{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}}\]

   6. Applied cbrt-unprod 64.0

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\left(x - y\right) \cdot \left(x - y\right)\right) \cdot \left(x - y\right)\right) \cdot \left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)\right)}}}{\sqrt[3]{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}}\]

   7. Applied cbrt-undiv 64.0

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\left(\left(x - y\right) \cdot \left(x - y\right)\right) \cdot \left(x - y\right)\right) \cdot \left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)\right)}{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}}}\]

   8. Simplified 64.0

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\right)}^{3}}}\]

   9. Taylor expanded around 0 0

      \[\leadsto \sqrt[3]{{\color{blue}{-1}}^{3}}\]

   if -1.1417806673352596e+154 < y < -1.651593275181696e-162 or 7.80351081129908e-163 < y

   1. Initial program 0.0

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
   2. Using strategy rm

   3. Applied add-cbrt-cube 37.2

      \[\leadsto \frac{\left(x - y\right) \cdot \left(x + y\right)}{\color{blue}{\sqrt[3]{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}}}\]

   4. Applied add-cbrt-cube 37.5

      \[\leadsto \frac{\left(x - y\right) \cdot \color{blue}{\sqrt[3]{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)}}}{\sqrt[3]{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}}\]

   5. Applied add-cbrt-cube 37.7

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(x - y\right) \cdot \left(x - y\right)\right) \cdot \left(x - y\right)}} \cdot \sqrt[3]{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)}}{\sqrt[3]{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}}\]

   6. Applied cbrt-unprod 37.2

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\left(x - y\right) \cdot \left(x - y\right)\right) \cdot \left(x - y\right)\right) \cdot \left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)\right)}}}{\sqrt[3]{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}}\]

   7. Applied cbrt-undiv 37.1

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\left(\left(x - y\right) \cdot \left(x - y\right)\right) \cdot \left(x - y\right)\right) \cdot \left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)\right)}{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}}}\]

   8. Simplified 0.0

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\right)}^{3}}}\]

   if -1.651593275181696e-162 < y < 7.80351081129908e-163

   1. Initial program 29.4

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
   2. Using strategy rm

   3. Applied add-cbrt-cube 51.7

      \[\leadsto \frac{\left(x - y\right) \cdot \left(x + y\right)}{\color{blue}{\sqrt[3]{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}}}\]

   4. Applied add-cbrt-cube 51.7

      \[\leadsto \frac{\left(x - y\right) \cdot \color{blue}{\sqrt[3]{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)}}}{\sqrt[3]{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}}\]

   5. Applied add-cbrt-cube 51.8

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(x - y\right) \cdot \left(x - y\right)\right) \cdot \left(x - y\right)}} \cdot \sqrt[3]{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)}}{\sqrt[3]{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}}\]

   6. Applied cbrt-unprod 51.5

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\left(x - y\right) \cdot \left(x - y\right)\right) \cdot \left(x - y\right)\right) \cdot \left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)\right)}}}{\sqrt[3]{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}}\]

   7. Applied cbrt-undiv 51.5

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\left(\left(x - y\right) \cdot \left(x - y\right)\right) \cdot \left(x - y\right)\right) \cdot \left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)\right)}{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}}}\]

   8. Simplified 29.4

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\right)}^{3}}}\]

   9. Taylor expanded around inf 14.9

      \[\leadsto \sqrt[3]{{\color{blue}{1}}^{3}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 4.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.14178066733525963 \cdot 10^{154}:\\ \;\;\;\;\sqrt[3]{-1}\\ \mathbf{elif}\;y \le -1.6515932751816961 \cdot 10^{-162}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\right)}^{3}}\\ \mathbf{elif}\;y \le 7.8035108112990794 \cdot 10^{-163}:\\ \;\;\;\;\sqrt[3]{1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (< 0.0 x 1) (< y 1))

  :herbie-target
  (if (< 0.5 (fabs (/ x y)) 2) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1 (/ 2 (+ 1 (* (/ x y) (/ x y))))))

  (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))