Toniolo and Linder, Equation (10+)

Average Error: 32.4 → 2.7

Time: 1.1m

Precision: 64

Math

\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

↓

\[\frac{2}{\frac{\left(\frac{\sin k \cdot t}{\ell} \cdot 2\right) \cdot \frac{\sin k \cdot t}{\ell}}{\cos k} \cdot t - \frac{{\left(\frac{1}{{-1}^{3}}\right)}^{1}}{\frac{\cos k}{\frac{k \cdot \sin k}{\ell}}} \cdot \left(\frac{k \cdot \sin k}{\ell} \cdot t\right)}\]

C

double f(double t, double l, double k) {
        double r3775250 = 2.0;
        double r3775251 = t;
        double r3775252 = 3.0;
        double r3775253 = pow(r3775251, r3775252);
        double r3775254 = l;
        double r3775255 = r3775254 * r3775254;
        double r3775256 = r3775253 / r3775255;
        double r3775257 = k;
        double r3775258 = sin(r3775257);
        double r3775259 = r3775256 * r3775258;
        double r3775260 = tan(r3775257);
        double r3775261 = r3775259 * r3775260;
        double r3775262 = 1.0;
        double r3775263 = r3775257 / r3775251;
        double r3775264 = pow(r3775263, r3775250);
        double r3775265 = r3775262 + r3775264;
        double r3775266 = r3775265 + r3775262;
        double r3775267 = r3775261 * r3775266;
        double r3775268 = r3775250 / r3775267;
        return r3775268;
}

↓

double f(double t, double l, double k) {
        double r3775269 = 2.0;
        double r3775270 = k;
        double r3775271 = sin(r3775270);
        double r3775272 = t;
        double r3775273 = r3775271 * r3775272;
        double r3775274 = l;
        double r3775275 = r3775273 / r3775274;
        double r3775276 = r3775275 * r3775269;
        double r3775277 = r3775276 * r3775275;
        double r3775278 = cos(r3775270);
        double r3775279 = r3775277 / r3775278;
        double r3775280 = r3775279 * r3775272;
        double r3775281 = 1.0;
        double r3775282 = -1.0;
        double r3775283 = 3.0;
        double r3775284 = pow(r3775282, r3775283);
        double r3775285 = r3775281 / r3775284;
        double r3775286 = 1.0;
        double r3775287 = pow(r3775285, r3775286);
        double r3775288 = r3775270 * r3775271;
        double r3775289 = r3775288 / r3775274;
        double r3775290 = r3775278 / r3775289;
        double r3775291 = r3775287 / r3775290;
        double r3775292 = r3775289 * r3775272;
        double r3775293 = r3775291 * r3775292;
        double r3775294 = r3775280 - r3775293;
        double r3775295 = r3775269 / r3775294;
        return r3775295;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Initial program 32.4

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

2. Taylor expanded around -inf 31.3

   \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{t}^{3} \cdot {\left(\sin k\right)}^{2}}{\cos k \cdot {\ell}^{2}} - {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{t \cdot \left({k}^{2} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}}}\]

3. Simplified 19.8

   \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot t}{\frac{\cos k}{t}} \cdot \left(\left(\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot 2\right) - \frac{t \cdot {\left(\frac{1}{{-1}^{3}}\right)}^{1}}{\frac{\cos k}{\frac{\sin k \cdot k}{\ell} \cdot \frac{\sin k \cdot k}{\ell}}}}}\]
4. Using strategy rm

5. Applied div-inv 19.8

   \[\leadsto \frac{2}{\frac{t \cdot t}{\color{blue}{\cos k \cdot \frac{1}{t}}} \cdot \left(\left(\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot 2\right) - \frac{t \cdot {\left(\frac{1}{{-1}^{3}}\right)}^{1}}{\frac{\cos k}{\frac{\sin k \cdot k}{\ell} \cdot \frac{\sin k \cdot k}{\ell}}}}\]

6. Applied times-frac 19.8

   \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\cos k} \cdot \frac{t}{\frac{1}{t}}\right)} \cdot \left(\left(\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot 2\right) - \frac{t \cdot {\left(\frac{1}{{-1}^{3}}\right)}^{1}}{\frac{\cos k}{\frac{\sin k \cdot k}{\ell} \cdot \frac{\sin k \cdot k}{\ell}}}}\]

7. Applied associate-*l* 17.1

   \[\leadsto \frac{2}{\color{blue}{\frac{t}{\cos k} \cdot \left(\frac{t}{\frac{1}{t}} \cdot \left(\left(\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot 2\right)\right)} - \frac{t \cdot {\left(\frac{1}{{-1}^{3}}\right)}^{1}}{\frac{\cos k}{\frac{\sin k \cdot k}{\ell} \cdot \frac{\sin k \cdot k}{\ell}}}}\]

8. Simplified 6.7

   \[\leadsto \frac{2}{\frac{t}{\cos k} \cdot \color{blue}{\left(\left(\left(t \cdot \frac{\sin k}{\ell}\right) \cdot \left(t \cdot \frac{\sin k}{\ell}\right)\right) \cdot 2\right)} - \frac{t \cdot {\left(\frac{1}{{-1}^{3}}\right)}^{1}}{\frac{\cos k}{\frac{\sin k \cdot k}{\ell} \cdot \frac{\sin k \cdot k}{\ell}}}}\]
9. Using strategy rm

10. Applied *-un-lft-identity 6.7

    \[\leadsto \frac{2}{\frac{t}{\cos k} \cdot \left(\left(\left(t \cdot \frac{\sin k}{\ell}\right) \cdot \left(t \cdot \frac{\sin k}{\ell}\right)\right) \cdot 2\right) - \frac{t \cdot {\left(\frac{1}{{-1}^{3}}\right)}^{1}}{\frac{\color{blue}{1 \cdot \cos k}}{\frac{\sin k \cdot k}{\ell} \cdot \frac{\sin k \cdot k}{\ell}}}}\]

11. Applied times-frac 6.6

    \[\leadsto \frac{2}{\frac{t}{\cos k} \cdot \left(\left(\left(t \cdot \frac{\sin k}{\ell}\right) \cdot \left(t \cdot \frac{\sin k}{\ell}\right)\right) \cdot 2\right) - \frac{t \cdot {\left(\frac{1}{{-1}^{3}}\right)}^{1}}{\color{blue}{\frac{1}{\frac{\sin k \cdot k}{\ell}} \cdot \frac{\cos k}{\frac{\sin k \cdot k}{\ell}}}}}\]

12. Applied times-frac 3.7

    \[\leadsto \frac{2}{\frac{t}{\cos k} \cdot \left(\left(\left(t \cdot \frac{\sin k}{\ell}\right) \cdot \left(t \cdot \frac{\sin k}{\ell}\right)\right) \cdot 2\right) - \color{blue}{\frac{t}{\frac{1}{\frac{\sin k \cdot k}{\ell}}} \cdot \frac{{\left(\frac{1}{{-1}^{3}}\right)}^{1}}{\frac{\cos k}{\frac{\sin k \cdot k}{\ell}}}}}\]

13. Simplified 3.7

    \[\leadsto \frac{2}{\frac{t}{\cos k} \cdot \left(\left(\left(t \cdot \frac{\sin k}{\ell}\right) \cdot \left(t \cdot \frac{\sin k}{\ell}\right)\right) \cdot 2\right) - \color{blue}{\left(t \cdot \frac{\sin k \cdot k}{\ell}\right)} \cdot \frac{{\left(\frac{1}{{-1}^{3}}\right)}^{1}}{\frac{\cos k}{\frac{\sin k \cdot k}{\ell}}}}\]
14. Using strategy rm

15. Applied div-inv 3.7

    \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \frac{1}{\cos k}\right)} \cdot \left(\left(\left(t \cdot \frac{\sin k}{\ell}\right) \cdot \left(t \cdot \frac{\sin k}{\ell}\right)\right) \cdot 2\right) - \left(t \cdot \frac{\sin k \cdot k}{\ell}\right) \cdot \frac{{\left(\frac{1}{{-1}^{3}}\right)}^{1}}{\frac{\cos k}{\frac{\sin k \cdot k}{\ell}}}}\]

16. Applied associate-*l* 3.7

    \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{1}{\cos k} \cdot \left(\left(\left(t \cdot \frac{\sin k}{\ell}\right) \cdot \left(t \cdot \frac{\sin k}{\ell}\right)\right) \cdot 2\right)\right)} - \left(t \cdot \frac{\sin k \cdot k}{\ell}\right) \cdot \frac{{\left(\frac{1}{{-1}^{3}}\right)}^{1}}{\frac{\cos k}{\frac{\sin k \cdot k}{\ell}}}}\]

17. Simplified 2.7

    \[\leadsto \frac{2}{t \cdot \color{blue}{\frac{\frac{\sin k \cdot t}{\ell} \cdot \left(2 \cdot \frac{\sin k \cdot t}{\ell}\right)}{\cos k}} - \left(t \cdot \frac{\sin k \cdot k}{\ell}\right) \cdot \frac{{\left(\frac{1}{{-1}^{3}}\right)}^{1}}{\frac{\cos k}{\frac{\sin k \cdot k}{\ell}}}}\]

18. Final simplification 2.7

    \[\leadsto \frac{2}{\frac{\left(\frac{\sin k \cdot t}{\ell} \cdot 2\right) \cdot \frac{\sin k \cdot t}{\ell}}{\cos k} \cdot t - \frac{{\left(\frac{1}{{-1}^{3}}\right)}^{1}}{\frac{\cos k}{\frac{k \cdot \sin k}{\ell}}} \cdot \left(\frac{k \cdot \sin k}{\ell} \cdot t\right)}\]

Reproduce

herbie shell --seed 2019200 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))