Toniolo and Linder, Equation (7)

Average Error: 42.0 → 9.1

Time: 29.9s

Precision: 64

Math

\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.235121765864116 \cdot 10^{+102}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{t}{\sqrt{2}}, \frac{2}{\left(x \cdot x\right) \cdot 2}, \frac{-2}{\frac{x}{\frac{t}{\sqrt{2}}}}\right) - \sqrt{2} \cdot t}\\ \mathbf{elif}\;t \le -6.488460202761919 \cdot 10^{-152}:\\ \;\;\;\;\frac{\left(\sqrt{\sqrt{\sqrt{2}}} \cdot \left(t \cdot \sqrt{\sqrt{2}}\right)\right) \cdot \sqrt{\sqrt{\sqrt{2}}}}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(\frac{t \cdot t}{x}, 4, 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)\right)\right)}}\\ \mathbf{elif}\;t \le -1.8789547767602823 \cdot 10^{-244}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{t}{\sqrt{2}}, \frac{2}{\left(x \cdot x\right) \cdot 2}, \frac{-2}{\frac{x}{\frac{t}{\sqrt{2}}}}\right) - \sqrt{2} \cdot t}\\ \mathbf{elif}\;t \le 1.1951227663536998 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(\frac{t \cdot t}{x}, 4, 2 \cdot \frac{\frac{\ell \cdot \ell}{\sqrt{x}}}{\sqrt{x}}\right)\right)}}\\ \mathbf{elif}\;t \le 2.776222986329393 \cdot 10^{-161}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(-2, \frac{\frac{\frac{t}{\sqrt{2}}}{2}}{x \cdot x}, \mathsf{fma}\left(\frac{2}{x}, \frac{t}{\sqrt{2}}, \mathsf{fma}\left(t, \sqrt{2}, \frac{\frac{t}{\sqrt{2}}}{x} \cdot \frac{2}{x}\right)\right)\right)}\\ \mathbf{elif}\;t \le 1.4996133926227848 \cdot 10^{+124}:\\ \;\;\;\;\frac{\left(\sqrt{\sqrt{\sqrt{2}}} \cdot \left(t \cdot \sqrt{\sqrt{2}}\right)\right) \cdot \sqrt{\sqrt{\sqrt{2}}}}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(\frac{t \cdot t}{x}, 4, 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(-2, \frac{\frac{\frac{t}{\sqrt{2}}}{2}}{x \cdot x}, \mathsf{fma}\left(\frac{2}{x}, \frac{t}{\sqrt{2}}, \mathsf{fma}\left(t, \sqrt{2}, \frac{\frac{t}{\sqrt{2}}}{x} \cdot \frac{2}{x}\right)\right)\right)}\\ \end{array}\]

C

double f(double x, double l, double t) {
        double r1053267 = 2.0;
        double r1053268 = sqrt(r1053267);
        double r1053269 = t;
        double r1053270 = r1053268 * r1053269;
        double r1053271 = x;
        double r1053272 = 1.0;
        double r1053273 = r1053271 + r1053272;
        double r1053274 = r1053271 - r1053272;
        double r1053275 = r1053273 / r1053274;
        double r1053276 = l;
        double r1053277 = r1053276 * r1053276;
        double r1053278 = r1053269 * r1053269;
        double r1053279 = r1053267 * r1053278;
        double r1053280 = r1053277 + r1053279;
        double r1053281 = r1053275 * r1053280;
        double r1053282 = r1053281 - r1053277;
        double r1053283 = sqrt(r1053282);
        double r1053284 = r1053270 / r1053283;
        return r1053284;
}

↓

double f(double x, double l, double t) {
        double r1053285 = t;
        double r1053286 = -1.235121765864116e+102;
        bool r1053287 = r1053285 <= r1053286;
        double r1053288 = 2.0;
        double r1053289 = sqrt(r1053288);
        double r1053290 = r1053289 * r1053285;
        double r1053291 = r1053285 / r1053289;
        double r1053292 = x;
        double r1053293 = r1053292 * r1053292;
        double r1053294 = r1053293 * r1053288;
        double r1053295 = r1053288 / r1053294;
        double r1053296 = -2.0;
        double r1053297 = r1053292 / r1053291;
        double r1053298 = r1053296 / r1053297;
        double r1053299 = fma(r1053291, r1053295, r1053298);
        double r1053300 = r1053299 - r1053290;
        double r1053301 = r1053290 / r1053300;
        double r1053302 = -6.488460202761919e-152;
        bool r1053303 = r1053285 <= r1053302;
        double r1053304 = sqrt(r1053289);
        double r1053305 = sqrt(r1053304);
        double r1053306 = r1053285 * r1053304;
        double r1053307 = r1053305 * r1053306;
        double r1053308 = r1053307 * r1053305;
        double r1053309 = r1053285 * r1053285;
        double r1053310 = r1053309 / r1053292;
        double r1053311 = 4.0;
        double r1053312 = l;
        double r1053313 = r1053312 / r1053292;
        double r1053314 = r1053312 * r1053313;
        double r1053315 = r1053288 * r1053314;
        double r1053316 = fma(r1053310, r1053311, r1053315);
        double r1053317 = fma(r1053288, r1053309, r1053316);
        double r1053318 = sqrt(r1053317);
        double r1053319 = r1053308 / r1053318;
        double r1053320 = -1.8789547767602823e-244;
        bool r1053321 = r1053285 <= r1053320;
        double r1053322 = 1.1951227663536998e-222;
        bool r1053323 = r1053285 <= r1053322;
        double r1053324 = r1053312 * r1053312;
        double r1053325 = sqrt(r1053292);
        double r1053326 = r1053324 / r1053325;
        double r1053327 = r1053326 / r1053325;
        double r1053328 = r1053288 * r1053327;
        double r1053329 = fma(r1053310, r1053311, r1053328);
        double r1053330 = fma(r1053288, r1053309, r1053329);
        double r1053331 = sqrt(r1053330);
        double r1053332 = r1053290 / r1053331;
        double r1053333 = 2.776222986329393e-161;
        bool r1053334 = r1053285 <= r1053333;
        double r1053335 = r1053291 / r1053288;
        double r1053336 = r1053335 / r1053293;
        double r1053337 = r1053288 / r1053292;
        double r1053338 = r1053291 / r1053292;
        double r1053339 = r1053338 * r1053337;
        double r1053340 = fma(r1053285, r1053289, r1053339);
        double r1053341 = fma(r1053337, r1053291, r1053340);
        double r1053342 = fma(r1053296, r1053336, r1053341);
        double r1053343 = r1053290 / r1053342;
        double r1053344 = 1.4996133926227848e+124;
        bool r1053345 = r1053285 <= r1053344;
        double r1053346 = r1053345 ? r1053319 : r1053343;
        double r1053347 = r1053334 ? r1053343 : r1053346;
        double r1053348 = r1053323 ? r1053332 : r1053347;
        double r1053349 = r1053321 ? r1053301 : r1053348;
        double r1053350 = r1053303 ? r1053319 : r1053349;
        double r1053351 = r1053287 ? r1053301 : r1053350;
        return r1053351;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

1. Split input into 4 regimes

2. if t < -1.235121765864116e+102 or -6.488460202761919e-152 < t < -1.8789547767602823e-244

   1. Initial program 53.1

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

   2. Taylor expanded around inf 47.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \frac{{t}^{2}}{x}\right)}}}\]

   3. Simplified 47.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(\frac{t \cdot t}{x}, 4, \frac{\ell \cdot \ell}{x} \cdot 2\right)\right)}}}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity 47.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(\frac{t \cdot t}{x}, 4, \frac{\ell \cdot \ell}{\color{blue}{1 \cdot x}} \cdot 2\right)\right)}}\]

   6. Applied times-frac 45.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(\frac{t \cdot t}{x}, 4, \color{blue}{\left(\frac{\ell}{1} \cdot \frac{\ell}{x}\right)} \cdot 2\right)\right)}}\]

   7. Simplified 45.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(\frac{t \cdot t}{x}, 4, \left(\color{blue}{\ell} \cdot \frac{\ell}{x}\right) \cdot 2\right)\right)}}\]

   8. Taylor expanded around -inf 9.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \left(t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}}\]

   9. Simplified 9.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\mathsf{fma}\left(\frac{t}{\sqrt{2}}, \frac{2}{2 \cdot \left(x \cdot x\right)}, \frac{-2}{\frac{x}{\frac{t}{\sqrt{2}}}}\right) - \sqrt{2} \cdot t}}\]

   if -1.235121765864116e+102 < t < -6.488460202761919e-152 or 2.776222986329393e-161 < t < 1.4996133926227848e+124

   1. Initial program 24.7

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

   2. Taylor expanded around inf 9.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \frac{{t}^{2}}{x}\right)}}}\]

   3. Simplified 9.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(\frac{t \cdot t}{x}, 4, \frac{\ell \cdot \ell}{x} \cdot 2\right)\right)}}}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity 9.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(\frac{t \cdot t}{x}, 4, \frac{\ell \cdot \ell}{\color{blue}{1 \cdot x}} \cdot 2\right)\right)}}\]

   6. Applied times-frac 4.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(\frac{t \cdot t}{x}, 4, \color{blue}{\left(\frac{\ell}{1} \cdot \frac{\ell}{x}\right)} \cdot 2\right)\right)}}\]

   7. Simplified 4.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(\frac{t \cdot t}{x}, 4, \left(\color{blue}{\ell} \cdot \frac{\ell}{x}\right) \cdot 2\right)\right)}}\]
   8. Using strategy rm

   9. Applied add-sqr-sqrt 4.7

      \[\leadsto \frac{\sqrt{\color{blue}{\sqrt{2} \cdot \sqrt{2}}} \cdot t}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(\frac{t \cdot t}{x}, 4, \left(\ell \cdot \frac{\ell}{x}\right) \cdot 2\right)\right)}}\]

   10. Applied sqrt-prod 4.9

       \[\leadsto \frac{\color{blue}{\left(\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}\right)} \cdot t}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(\frac{t \cdot t}{x}, 4, \left(\ell \cdot \frac{\ell}{x}\right) \cdot 2\right)\right)}}\]

   11. Applied associate-*l* 4.8

       \[\leadsto \frac{\color{blue}{\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot t\right)}}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(\frac{t \cdot t}{x}, 4, \left(\ell \cdot \frac{\ell}{x}\right) \cdot 2\right)\right)}}\]
   12. Using strategy rm

   13. Applied add-sqr-sqrt 4.8

       \[\leadsto \frac{\sqrt{\sqrt{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}} \cdot \left(\sqrt{\sqrt{2}} \cdot t\right)}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(\frac{t \cdot t}{x}, 4, \left(\ell \cdot \frac{\ell}{x}\right) \cdot 2\right)\right)}}\]

   14. Applied sqrt-prod 4.8

       \[\leadsto \frac{\sqrt{\color{blue}{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}}} \cdot \left(\sqrt{\sqrt{2}} \cdot t\right)}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(\frac{t \cdot t}{x}, 4, \left(\ell \cdot \frac{\ell}{x}\right) \cdot 2\right)\right)}}\]

   15. Applied sqrt-prod 4.8

       \[\leadsto \frac{\color{blue}{\left(\sqrt{\sqrt{\sqrt{2}}} \cdot \sqrt{\sqrt{\sqrt{2}}}\right)} \cdot \left(\sqrt{\sqrt{2}} \cdot t\right)}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(\frac{t \cdot t}{x}, 4, \left(\ell \cdot \frac{\ell}{x}\right) \cdot 2\right)\right)}}\]

   16. Applied associate-*l* 4.7

       \[\leadsto \frac{\color{blue}{\sqrt{\sqrt{\sqrt{2}}} \cdot \left(\sqrt{\sqrt{\sqrt{2}}} \cdot \left(\sqrt{\sqrt{2}} \cdot t\right)\right)}}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(\frac{t \cdot t}{x}, 4, \left(\ell \cdot \frac{\ell}{x}\right) \cdot 2\right)\right)}}\]

   if -1.8789547767602823e-244 < t < 1.1951227663536998e-222

   1. Initial program 61.5

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

   2. Taylor expanded around inf 31.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \frac{{t}^{2}}{x}\right)}}}\]

   3. Simplified 31.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(\frac{t \cdot t}{x}, 4, \frac{\ell \cdot \ell}{x} \cdot 2\right)\right)}}}\]
   4. Using strategy rm

   5. Applied add-sqr-sqrt 31.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(\frac{t \cdot t}{x}, 4, \frac{\ell \cdot \ell}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \cdot 2\right)\right)}}\]

   6. Applied associate-/r* 31.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(\frac{t \cdot t}{x}, 4, \color{blue}{\frac{\frac{\ell \cdot \ell}{\sqrt{x}}}{\sqrt{x}}} \cdot 2\right)\right)}}\]

   if 1.1951227663536998e-222 < t < 2.776222986329393e-161 or 1.4996133926227848e+124 < t

   1. Initial program 55.6

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

   2. Taylor expanded around inf 7.9

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(2 \cdot \frac{t}{\sqrt{2} \cdot {x}^{2}} + \left(t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}}}\]

   3. Simplified 7.9

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\mathsf{fma}\left(-2, \frac{\frac{\frac{t}{\sqrt{2}}}{2}}{x \cdot x}, \mathsf{fma}\left(\frac{2}{x}, \frac{t}{\sqrt{2}}, \mathsf{fma}\left(t, \sqrt{2}, \frac{2}{x} \cdot \frac{\frac{t}{\sqrt{2}}}{x}\right)\right)\right)}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 9.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.235121765864116 \cdot 10^{+102}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{t}{\sqrt{2}}, \frac{2}{\left(x \cdot x\right) \cdot 2}, \frac{-2}{\frac{x}{\frac{t}{\sqrt{2}}}}\right) - \sqrt{2} \cdot t}\\ \mathbf{elif}\;t \le -6.488460202761919 \cdot 10^{-152}:\\ \;\;\;\;\frac{\left(\sqrt{\sqrt{\sqrt{2}}} \cdot \left(t \cdot \sqrt{\sqrt{2}}\right)\right) \cdot \sqrt{\sqrt{\sqrt{2}}}}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(\frac{t \cdot t}{x}, 4, 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)\right)\right)}}\\ \mathbf{elif}\;t \le -1.8789547767602823 \cdot 10^{-244}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{t}{\sqrt{2}}, \frac{2}{\left(x \cdot x\right) \cdot 2}, \frac{-2}{\frac{x}{\frac{t}{\sqrt{2}}}}\right) - \sqrt{2} \cdot t}\\ \mathbf{elif}\;t \le 1.1951227663536998 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(\frac{t \cdot t}{x}, 4, 2 \cdot \frac{\frac{\ell \cdot \ell}{\sqrt{x}}}{\sqrt{x}}\right)\right)}}\\ \mathbf{elif}\;t \le 2.776222986329393 \cdot 10^{-161}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(-2, \frac{\frac{\frac{t}{\sqrt{2}}}{2}}{x \cdot x}, \mathsf{fma}\left(\frac{2}{x}, \frac{t}{\sqrt{2}}, \mathsf{fma}\left(t, \sqrt{2}, \frac{\frac{t}{\sqrt{2}}}{x} \cdot \frac{2}{x}\right)\right)\right)}\\ \mathbf{elif}\;t \le 1.4996133926227848 \cdot 10^{+124}:\\ \;\;\;\;\frac{\left(\sqrt{\sqrt{\sqrt{2}}} \cdot \left(t \cdot \sqrt{\sqrt{2}}\right)\right) \cdot \sqrt{\sqrt{\sqrt{2}}}}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(\frac{t \cdot t}{x}, 4, 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(-2, \frac{\frac{\frac{t}{\sqrt{2}}}{2}}{x \cdot x}, \mathsf{fma}\left(\frac{2}{x}, \frac{t}{\sqrt{2}}, \mathsf{fma}\left(t, \sqrt{2}, \frac{\frac{t}{\sqrt{2}}}{x} \cdot \frac{2}{x}\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  (/ (* (sqrt 2) t) (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l)))))