Kahan p9 Example

Average Error: 20.8 → 12.8

Time: 12.0s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -3.306610229065552697766015508018533353152 \cdot 10^{-6}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.040149902759510101461916731014611092466 \cdot 10^{-136}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{\frac{{x}^{4} - {y}^{4}}{\left(x + y\right) \cdot \left(x - y\right)}}\\ \mathbf{elif}\;y \le -4.002205660206990449557576569903637397741 \cdot 10^{-171}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le 2.245790742829554397735960283721886784496 \cdot 10^{-222}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 5.875513051791536100179706901856990854373 \cdot 10^{-203}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le 8.954732963342977511317338914679377322524 \cdot 10^{-75}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{\frac{{x}^{4} - {y}^{4}}{\left(x + y\right) \cdot \left(x - y\right)}}\\ \end{array}\]

C

double f(double x, double y) {
        double r81890 = x;
        double r81891 = y;
        double r81892 = r81890 - r81891;
        double r81893 = r81890 + r81891;
        double r81894 = r81892 * r81893;
        double r81895 = r81890 * r81890;
        double r81896 = r81891 * r81891;
        double r81897 = r81895 + r81896;
        double r81898 = r81894 / r81897;
        return r81898;
}

↓

double f(double x, double y) {
        double r81899 = y;
        double r81900 = -3.3066102290655527e-06;
        bool r81901 = r81899 <= r81900;
        double r81902 = -1.0;
        double r81903 = -1.0401499027595101e-136;
        bool r81904 = r81899 <= r81903;
        double r81905 = x;
        double r81906 = r81905 - r81899;
        double r81907 = r81905 + r81899;
        double r81908 = r81906 * r81907;
        double r81909 = 4.0;
        double r81910 = pow(r81905, r81909);
        double r81911 = pow(r81899, r81909);
        double r81912 = r81910 - r81911;
        double r81913 = r81907 * r81906;
        double r81914 = r81912 / r81913;
        double r81915 = r81908 / r81914;
        double r81916 = -4.0022056602069904e-171;
        bool r81917 = r81899 <= r81916;
        double r81918 = 2.2457907428295544e-222;
        bool r81919 = r81899 <= r81918;
        double r81920 = 1.0;
        double r81921 = 5.875513051791536e-203;
        bool r81922 = r81899 <= r81921;
        double r81923 = 8.954732963342978e-75;
        bool r81924 = r81899 <= r81923;
        double r81925 = r81924 ? r81920 : r81915;
        double r81926 = r81922 ? r81902 : r81925;
        double r81927 = r81919 ? r81920 : r81926;
        double r81928 = r81917 ? r81902 : r81927;
        double r81929 = r81904 ? r81915 : r81928;
        double r81930 = r81901 ? r81902 : r81929;
        return r81930;
}

Error

Bits error versus x

Bits error versus y

Target

Original	20.8
Target	0.0
Herbie	12.8

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

Derivation

1. Split input into 3 regimes

2. if y < -3.3066102290655527e-06 or -1.0401499027595101e-136 < y < -4.0022056602069904e-171 or 2.2457907428295544e-222 < y < 5.875513051791536e-203

   1. Initial program 30.0

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

   2. Taylor expanded around 0 4.6

      \[\leadsto \color{blue}{-1}\]

   if -3.3066102290655527e-06 < y < -1.0401499027595101e-136 or 8.954732963342978e-75 < 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 flip-+ 13.4

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

   4. Simplified 13.7

      \[\leadsto \frac{\left(x - y\right) \cdot \left(x + y\right)}{\frac{\color{blue}{{x}^{4} - {y}^{4}}}{x \cdot x - y \cdot y}}\]

   5. Simplified 13.8

      \[\leadsto \frac{\left(x - y\right) \cdot \left(x + y\right)}{\frac{{x}^{4} - {y}^{4}}{\color{blue}{\left(x + y\right) \cdot \left(x - y\right)}}}\]

   if -4.0022056602069904e-171 < y < 2.2457907428295544e-222 or 5.875513051791536e-203 < y < 8.954732963342978e-75

   1. Initial program 22.6

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

   2. Taylor expanded around inf 20.6

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

4. Final simplification 12.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3.306610229065552697766015508018533353152 \cdot 10^{-6}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.040149902759510101461916731014611092466 \cdot 10^{-136}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{\frac{{x}^{4} - {y}^{4}}{\left(x + y\right) \cdot \left(x - y\right)}}\\ \mathbf{elif}\;y \le -4.002205660206990449557576569903637397741 \cdot 10^{-171}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le 2.245790742829554397735960283721886784496 \cdot 10^{-222}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 5.875513051791536100179706901856990854373 \cdot 10^{-203}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le 8.954732963342977511317338914679377322524 \cdot 10^{-75}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{\frac{{x}^{4} - {y}^{4}}{\left(x + y\right) \cdot \left(x - y\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(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))))