Toniolo and Linder, Equation (7)

Average Error: 42.5 → 9.3

Time: 42.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -3.9393699962868773 \cdot 10^{+127}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{1}{\sqrt{2}} \cdot \frac{t}{x \cdot x} - (t \cdot \left(\sqrt{2}\right) + \left(\frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x} + \frac{t}{x \cdot x}\right)\right))_*}\\ \mathbf{elif}\;t \le -8.498363187067144 \cdot 10^{-184}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot t\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{(\left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) \cdot 2 + \left(\frac{4 \cdot \left(t \cdot t\right)}{x}\right))_*}}\\ \mathbf{elif}\;t \le -4.539569288390012 \cdot 10^{-296}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{1}{\sqrt{2}} \cdot \frac{t}{x \cdot x} - (t \cdot \left(\sqrt{2}\right) + \left(\frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x} + \frac{t}{x \cdot x}\right)\right))_*}\\ \mathbf{elif}\;t \le 1.8148389345540428 \cdot 10^{-235}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{(\left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) \cdot 2 + \left(\frac{4 \cdot \left(t \cdot t\right)}{x}\right))_*}}\\ \mathbf{elif}\;t \le 1.835399898964734 \cdot 10^{-174}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{(\left(\frac{t}{\sqrt{2} \cdot x}\right) \cdot 2 + \left(\sqrt{2} \cdot t\right))_* + \frac{\frac{2}{x}}{x} \cdot \left(\frac{t}{\sqrt{2}} - \frac{\frac{t}{\sqrt{2}}}{2}\right)}\\ \mathbf{elif}\;t \le 2.732103184148166 \cdot 10^{+75}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot t\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{(\left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) \cdot 2 + \left(\frac{4 \cdot \left(t \cdot t\right)}{x}\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{(\left(\frac{t}{\sqrt{2} \cdot x}\right) \cdot 2 + \left(\sqrt{2} \cdot t\right))_* + \frac{\frac{2}{x}}{x} \cdot \left(\frac{t}{\sqrt{2}} - \frac{\frac{t}{\sqrt{2}}}{2}\right)}\\ \end{array}\]

C

double f(double x, double l, double t) {
        double r1582428 = 2.0;
        double r1582429 = sqrt(r1582428);
        double r1582430 = t;
        double r1582431 = r1582429 * r1582430;
        double r1582432 = x;
        double r1582433 = 1.0;
        double r1582434 = r1582432 + r1582433;
        double r1582435 = r1582432 - r1582433;
        double r1582436 = r1582434 / r1582435;
        double r1582437 = l;
        double r1582438 = r1582437 * r1582437;
        double r1582439 = r1582430 * r1582430;
        double r1582440 = r1582428 * r1582439;
        double r1582441 = r1582438 + r1582440;
        double r1582442 = r1582436 * r1582441;
        double r1582443 = r1582442 - r1582438;
        double r1582444 = sqrt(r1582443);
        double r1582445 = r1582431 / r1582444;
        return r1582445;
}

↓

double f(double x, double l, double t) {
        double r1582446 = t;
        double r1582447 = -3.9393699962868773e+127;
        bool r1582448 = r1582446 <= r1582447;
        double r1582449 = 2.0;
        double r1582450 = sqrt(r1582449);
        double r1582451 = r1582450 * r1582446;
        double r1582452 = 1.0;
        double r1582453 = r1582452 / r1582450;
        double r1582454 = x;
        double r1582455 = r1582454 * r1582454;
        double r1582456 = r1582446 / r1582455;
        double r1582457 = r1582453 * r1582456;
        double r1582458 = r1582449 / r1582450;
        double r1582459 = r1582446 / r1582454;
        double r1582460 = r1582459 + r1582456;
        double r1582461 = r1582458 * r1582460;
        double r1582462 = fma(r1582446, r1582450, r1582461);
        double r1582463 = r1582457 - r1582462;
        double r1582464 = r1582451 / r1582463;
        double r1582465 = -8.498363187067144e-184;
        bool r1582466 = r1582446 <= r1582465;
        double r1582467 = cbrt(r1582450);
        double r1582468 = r1582467 * r1582446;
        double r1582469 = r1582467 * r1582467;
        double r1582470 = r1582468 * r1582469;
        double r1582471 = l;
        double r1582472 = r1582471 / r1582454;
        double r1582473 = r1582446 * r1582446;
        double r1582474 = fma(r1582472, r1582471, r1582473);
        double r1582475 = 4.0;
        double r1582476 = r1582475 * r1582473;
        double r1582477 = r1582476 / r1582454;
        double r1582478 = fma(r1582474, r1582449, r1582477);
        double r1582479 = sqrt(r1582478);
        double r1582480 = r1582470 / r1582479;
        double r1582481 = -4.539569288390012e-296;
        bool r1582482 = r1582446 <= r1582481;
        double r1582483 = 1.8148389345540428e-235;
        bool r1582484 = r1582446 <= r1582483;
        double r1582485 = r1582451 / r1582479;
        double r1582486 = 1.835399898964734e-174;
        bool r1582487 = r1582446 <= r1582486;
        double r1582488 = r1582450 * r1582454;
        double r1582489 = r1582446 / r1582488;
        double r1582490 = fma(r1582489, r1582449, r1582451);
        double r1582491 = r1582449 / r1582454;
        double r1582492 = r1582491 / r1582454;
        double r1582493 = r1582446 / r1582450;
        double r1582494 = r1582493 / r1582449;
        double r1582495 = r1582493 - r1582494;
        double r1582496 = r1582492 * r1582495;
        double r1582497 = r1582490 + r1582496;
        double r1582498 = r1582451 / r1582497;
        double r1582499 = 2.732103184148166e+75;
        bool r1582500 = r1582446 <= r1582499;
        double r1582501 = r1582500 ? r1582480 : r1582498;
        double r1582502 = r1582487 ? r1582498 : r1582501;
        double r1582503 = r1582484 ? r1582485 : r1582502;
        double r1582504 = r1582482 ? r1582464 : r1582503;
        double r1582505 = r1582466 ? r1582480 : r1582504;
        double r1582506 = r1582448 ? r1582464 : r1582505;
        return r1582506;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

1. Split input into 4 regimes

2. if t < -3.9393699962868773e+127 or -8.498363187067144e-184 < t < -4.539569288390012e-296

   1. Initial program 57.0

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

   2. Taylor expanded around -inf 12.4

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

   3. Simplified 12.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\frac{1}{\sqrt{2}} \cdot \frac{t}{x \cdot x} - (t \cdot \left(\sqrt{2}\right) + \left(\frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x} + \frac{t}{x \cdot x}\right)\right))_*}}\]

   if -3.9393699962868773e+127 < t < -8.498363187067144e-184 or 1.835399898964734e-174 < t < 2.732103184148166e+75

   1. Initial program 28.1

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

   2. Taylor expanded around -inf 11.0

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

   3. Simplified 6.0

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

   5. Applied add-cube-cbrt 6.0

      \[\leadsto \frac{\color{blue}{\left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}\right)} \cdot t}{\sqrt{(\left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) \cdot 2 + \left(\frac{\left(t \cdot t\right) \cdot 4}{x}\right))_*}}\]

   6. Applied associate-*l* 5.9

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}}{\sqrt{(\left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) \cdot 2 + \left(\frac{\left(t \cdot t\right) \cdot 4}{x}\right))_*}}\]

   if -4.539569288390012e-296 < t < 1.8148389345540428e-235

   1. Initial program 61.7

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

   2. Taylor expanded around -inf 30.7

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

   3. Simplified 30.6

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

   5. Applied add-cube-cbrt 30.6

      \[\leadsto \frac{\color{blue}{\left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}\right)} \cdot t}{\sqrt{(\left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) \cdot 2 + \left(\frac{\left(t \cdot t\right) \cdot 4}{x}\right))_*}}\]

   6. Applied associate-*l* 30.6

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}}{\sqrt{(\left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) \cdot 2 + \left(\frac{\left(t \cdot t\right) \cdot 4}{x}\right))_*}}\]

   7. Taylor expanded around 0 30.6

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

   if 1.8148389345540428e-235 < t < 1.835399898964734e-174 or 2.732103184148166e+75 < t

   1. Initial program 49.0

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

   2. Taylor expanded around inf 7.8

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

   3. Simplified 7.8

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

4. Final simplification 9.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.9393699962868773 \cdot 10^{+127}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{1}{\sqrt{2}} \cdot \frac{t}{x \cdot x} - (t \cdot \left(\sqrt{2}\right) + \left(\frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x} + \frac{t}{x \cdot x}\right)\right))_*}\\ \mathbf{elif}\;t \le -8.498363187067144 \cdot 10^{-184}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot t\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{(\left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) \cdot 2 + \left(\frac{4 \cdot \left(t \cdot t\right)}{x}\right))_*}}\\ \mathbf{elif}\;t \le -4.539569288390012 \cdot 10^{-296}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{1}{\sqrt{2}} \cdot \frac{t}{x \cdot x} - (t \cdot \left(\sqrt{2}\right) + \left(\frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x} + \frac{t}{x \cdot x}\right)\right))_*}\\ \mathbf{elif}\;t \le 1.8148389345540428 \cdot 10^{-235}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{(\left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) \cdot 2 + \left(\frac{4 \cdot \left(t \cdot t\right)}{x}\right))_*}}\\ \mathbf{elif}\;t \le 1.835399898964734 \cdot 10^{-174}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{(\left(\frac{t}{\sqrt{2} \cdot x}\right) \cdot 2 + \left(\sqrt{2} \cdot t\right))_* + \frac{\frac{2}{x}}{x} \cdot \left(\frac{t}{\sqrt{2}} - \frac{\frac{t}{\sqrt{2}}}{2}\right)}\\ \mathbf{elif}\;t \le 2.732103184148166 \cdot 10^{+75}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot t\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{(\left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) \cdot 2 + \left(\frac{4 \cdot \left(t \cdot t\right)}{x}\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{(\left(\frac{t}{\sqrt{2} \cdot x}\right) \cdot 2 + \left(\sqrt{2} \cdot t\right))_* + \frac{\frac{2}{x}}{x} \cdot \left(\frac{t}{\sqrt{2}} - \frac{\frac{t}{\sqrt{2}}}{2}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 +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)))))