Toniolo and Linder, Equation (7)

Average Error: 42.1 → 9.2

Time: 29.8s

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 -6.215042401537606 \cdot 10^{+109}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\left(\frac{\frac{2}{x}}{x}\right), \left(\frac{\frac{t}{\sqrt{2}}}{2}\right), \left(-\mathsf{fma}\left(\left(\frac{2}{x}\right), \left(\frac{t}{\sqrt{2}}\right), \left(\sqrt{2} \cdot t\right)\right)\right)\right)}\\ \mathbf{elif}\;t \le -2.0085308378865194 \cdot 10^{-143}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\left(t \cdot t\right), 2, \left(\mathsf{fma}\left(\left(\frac{t \cdot t}{x}\right), 4, \left(\frac{\ell \cdot \sqrt{2}}{x} \cdot \left(\ell \cdot \sqrt{2}\right)\right)\right)\right)\right)}}\\ \mathbf{elif}\;t \le -4.903859847993851 \cdot 10^{-271}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{2 \cdot t}{\left(2 \cdot \sqrt{2}\right) \cdot \left(x \cdot x\right)} - \mathsf{fma}\left(\left(\frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right), 2, \left(\mathsf{fma}\left(t, \left(\sqrt{2}\right), \left(\frac{t}{\sqrt{2}} \cdot \frac{2}{x}\right)\right)\right)\right)}\\ \mathbf{elif}\;t \le 1.2558204415951766 \cdot 10^{-249}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\left(t \cdot t\right), 2, \left(\mathsf{fma}\left(\left(\frac{t \cdot t}{x}\right), 4, \left(\frac{\ell \cdot \sqrt{2}}{x} \cdot \left(\ell \cdot \sqrt{2}\right)\right)\right)\right)\right)}}\\ \mathbf{elif}\;t \le 5.4024136247167366 \cdot 10^{-161}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\left(\frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right), 2, \left(\mathsf{fma}\left(t, \left(\sqrt{2}\right), \left(\frac{t}{\sqrt{2}} \cdot \frac{2}{x}\right)\right)\right)\right) - \frac{2 \cdot t}{\left(2 \cdot \sqrt{2}\right) \cdot \left(x \cdot x\right)}}\\ \mathbf{elif}\;t \le 4.930352686166445 \cdot 10^{+153}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\left(t \cdot t\right), 2, \left(\mathsf{fma}\left(\left(\frac{t \cdot t}{x}\right), 4, \left(\frac{\ell \cdot \sqrt{2}}{x} \cdot \left(\ell \cdot \sqrt{2}\right)\right)\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\left(\frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right), 2, \left(\mathsf{fma}\left(t, \left(\sqrt{2}\right), \left(\frac{t}{\sqrt{2}} \cdot \frac{2}{x}\right)\right)\right)\right) - \frac{2 \cdot t}{\left(2 \cdot \sqrt{2}\right) \cdot \left(x \cdot x\right)}}\\ \end{array}\]

C

double f(double x, double l, double t) {
        double r819223 = 2.0;
        double r819224 = sqrt(r819223);
        double r819225 = t;
        double r819226 = r819224 * r819225;
        double r819227 = x;
        double r819228 = 1.0;
        double r819229 = r819227 + r819228;
        double r819230 = r819227 - r819228;
        double r819231 = r819229 / r819230;
        double r819232 = l;
        double r819233 = r819232 * r819232;
        double r819234 = r819225 * r819225;
        double r819235 = r819223 * r819234;
        double r819236 = r819233 + r819235;
        double r819237 = r819231 * r819236;
        double r819238 = r819237 - r819233;
        double r819239 = sqrt(r819238);
        double r819240 = r819226 / r819239;
        return r819240;
}

↓

double f(double x, double l, double t) {
        double r819241 = t;
        double r819242 = -6.215042401537606e+109;
        bool r819243 = r819241 <= r819242;
        double r819244 = 2.0;
        double r819245 = sqrt(r819244);
        double r819246 = r819245 * r819241;
        double r819247 = x;
        double r819248 = r819244 / r819247;
        double r819249 = r819248 / r819247;
        double r819250 = r819241 / r819245;
        double r819251 = r819250 / r819244;
        double r819252 = fma(r819248, r819250, r819246);
        double r819253 = -r819252;
        double r819254 = fma(r819249, r819251, r819253);
        double r819255 = r819246 / r819254;
        double r819256 = -2.0085308378865194e-143;
        bool r819257 = r819241 <= r819256;
        double r819258 = r819241 * r819241;
        double r819259 = r819258 / r819247;
        double r819260 = 4.0;
        double r819261 = l;
        double r819262 = r819261 * r819245;
        double r819263 = r819262 / r819247;
        double r819264 = r819263 * r819262;
        double r819265 = fma(r819259, r819260, r819264);
        double r819266 = fma(r819258, r819244, r819265);
        double r819267 = sqrt(r819266);
        double r819268 = r819246 / r819267;
        double r819269 = -4.903859847993851e-271;
        bool r819270 = r819241 <= r819269;
        double r819271 = r819244 * r819241;
        double r819272 = r819244 * r819245;
        double r819273 = r819247 * r819247;
        double r819274 = r819272 * r819273;
        double r819275 = r819271 / r819274;
        double r819276 = r819250 / r819273;
        double r819277 = r819250 * r819248;
        double r819278 = fma(r819241, r819245, r819277);
        double r819279 = fma(r819276, r819244, r819278);
        double r819280 = r819275 - r819279;
        double r819281 = r819246 / r819280;
        double r819282 = 1.2558204415951766e-249;
        bool r819283 = r819241 <= r819282;
        double r819284 = 5.4024136247167366e-161;
        bool r819285 = r819241 <= r819284;
        double r819286 = r819279 - r819275;
        double r819287 = r819246 / r819286;
        double r819288 = 4.930352686166445e+153;
        bool r819289 = r819241 <= r819288;
        double r819290 = r819289 ? r819268 : r819287;
        double r819291 = r819285 ? r819287 : r819290;
        double r819292 = r819283 ? r819268 : r819291;
        double r819293 = r819270 ? r819281 : r819292;
        double r819294 = r819257 ? r819268 : r819293;
        double r819295 = r819243 ? r819255 : r819294;
        return r819295;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

1. Split input into 4 regimes

2. if t < -6.215042401537606e+109

   1. Initial program 52.6

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

   2. Taylor expanded around -inf 52.6

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

   3. Simplified 52.6

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

   5. Applied div-inv 52.6

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

   6. Applied times-frac 51.1

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

   7. Simplified 51.1

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

   8. Taylor expanded around -inf 3.1

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

   9. Simplified 3.1

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

   if -6.215042401537606e+109 < t < -2.0085308378865194e-143 or -4.903859847993851e-271 < t < 1.2558204415951766e-249 or 5.4024136247167366e-161 < t < 4.930352686166445e+153

   1. Initial program 27.2

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

   2. Taylor expanded around -inf 12.3

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

   3. Simplified 12.3

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

   5. Applied add-sqr-sqrt 12.3

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

   6. Applied *-un-lft-identity 12.3

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

   7. Applied times-frac 12.3

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

   8. Applied times-frac 7.3

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

   9. Simplified 7.3

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

   10. Simplified 7.3

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

   if -2.0085308378865194e-143 < t < -4.903859847993851e-271

   1. Initial program 58.8

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

   2. Taylor expanded around -inf 32.7

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

   3. Simplified 32.7

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

   if 1.2558204415951766e-249 < t < 5.4024136247167366e-161 or 4.930352686166445e+153 < t

   1. Initial program 61.8

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

   2. Taylor expanded around inf 10.3

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

   3. Simplified 10.3

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

4. Final simplification 9.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -6.215042401537606 \cdot 10^{+109}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\left(\frac{\frac{2}{x}}{x}\right), \left(\frac{\frac{t}{\sqrt{2}}}{2}\right), \left(-\mathsf{fma}\left(\left(\frac{2}{x}\right), \left(\frac{t}{\sqrt{2}}\right), \left(\sqrt{2} \cdot t\right)\right)\right)\right)}\\ \mathbf{elif}\;t \le -2.0085308378865194 \cdot 10^{-143}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\left(t \cdot t\right), 2, \left(\mathsf{fma}\left(\left(\frac{t \cdot t}{x}\right), 4, \left(\frac{\ell \cdot \sqrt{2}}{x} \cdot \left(\ell \cdot \sqrt{2}\right)\right)\right)\right)\right)}}\\ \mathbf{elif}\;t \le -4.903859847993851 \cdot 10^{-271}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{2 \cdot t}{\left(2 \cdot \sqrt{2}\right) \cdot \left(x \cdot x\right)} - \mathsf{fma}\left(\left(\frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right), 2, \left(\mathsf{fma}\left(t, \left(\sqrt{2}\right), \left(\frac{t}{\sqrt{2}} \cdot \frac{2}{x}\right)\right)\right)\right)}\\ \mathbf{elif}\;t \le 1.2558204415951766 \cdot 10^{-249}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\left(t \cdot t\right), 2, \left(\mathsf{fma}\left(\left(\frac{t \cdot t}{x}\right), 4, \left(\frac{\ell \cdot \sqrt{2}}{x} \cdot \left(\ell \cdot \sqrt{2}\right)\right)\right)\right)\right)}}\\ \mathbf{elif}\;t \le 5.4024136247167366 \cdot 10^{-161}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\left(\frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right), 2, \left(\mathsf{fma}\left(t, \left(\sqrt{2}\right), \left(\frac{t}{\sqrt{2}} \cdot \frac{2}{x}\right)\right)\right)\right) - \frac{2 \cdot t}{\left(2 \cdot \sqrt{2}\right) \cdot \left(x \cdot x\right)}}\\ \mathbf{elif}\;t \le 4.930352686166445 \cdot 10^{+153}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\left(t \cdot t\right), 2, \left(\mathsf{fma}\left(\left(\frac{t \cdot t}{x}\right), 4, \left(\frac{\ell \cdot \sqrt{2}}{x} \cdot \left(\ell \cdot \sqrt{2}\right)\right)\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\left(\frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right), 2, \left(\mathsf{fma}\left(t, \left(\sqrt{2}\right), \left(\frac{t}{\sqrt{2}} \cdot \frac{2}{x}\right)\right)\right)\right) - \frac{2 \cdot t}{\left(2 \cdot \sqrt{2}\right) \cdot \left(x \cdot x\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 +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)))))