sqrtexp (problem 3.4.4)

Average Error: 4.5 → 0.9

Time: 19.5s

Precision: 64

• could not determine a ground truth for program body

  1. x = 3.9540437202843566e+32

Math

\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -6.751810080739973559159999205335871800477 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\frac{\sqrt{{\left(e^{x}\right)}^{2}} - \sqrt{1}}{e^{x} - 1}} \cdot \sqrt{\sqrt{1} + \left|\sqrt[3]{{\left(e^{x}\right)}^{2}}\right| \cdot \sqrt{\sqrt[3]{{\left(e^{x}\right)}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} + \left(\frac{3}{16} \cdot \frac{x \cdot x}{\sqrt{2}} + \frac{\frac{1}{2}}{\frac{\sqrt{2}}{x}}\right)\\ \end{array}\]

C

double f(double x) {
        double r28428 = 2.0;
        double r28429 = x;
        double r28430 = r28428 * r28429;
        double r28431 = exp(r28430);
        double r28432 = 1.0;
        double r28433 = r28431 - r28432;
        double r28434 = exp(r28429);
        double r28435 = r28434 - r28432;
        double r28436 = r28433 / r28435;
        double r28437 = sqrt(r28436);
        return r28437;
}

↓

double f(double x) {
        double r28438 = x;
        double r28439 = -6.751810080739974e-16;
        bool r28440 = r28438 <= r28439;
        double r28441 = exp(r28438);
        double r28442 = 2.0;
        double r28443 = pow(r28441, r28442);
        double r28444 = sqrt(r28443);
        double r28445 = 1.0;
        double r28446 = sqrt(r28445);
        double r28447 = r28444 - r28446;
        double r28448 = r28441 - r28445;
        double r28449 = r28447 / r28448;
        double r28450 = sqrt(r28449);
        double r28451 = cbrt(r28443);
        double r28452 = fabs(r28451);
        double r28453 = sqrt(r28451);
        double r28454 = r28452 * r28453;
        double r28455 = r28446 + r28454;
        double r28456 = sqrt(r28455);
        double r28457 = r28450 * r28456;
        double r28458 = 2.0;
        double r28459 = sqrt(r28458);
        double r28460 = 0.1875;
        double r28461 = r28438 * r28438;
        double r28462 = r28461 / r28459;
        double r28463 = r28460 * r28462;
        double r28464 = 0.5;
        double r28465 = r28459 / r28438;
        double r28466 = r28464 / r28465;
        double r28467 = r28463 + r28466;
        double r28468 = r28459 + r28467;
        double r28469 = r28440 ? r28457 : r28468;
        return r28469;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -6.751810080739974e-16

   1. Initial program 0.8

      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]

   2. Simplified 0.5

      \[\leadsto \color{blue}{\sqrt{\frac{{\left(e^{x}\right)}^{2} - 1}{e^{x} - 1}}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 0.5

      \[\leadsto \sqrt{\frac{{\left(e^{x}\right)}^{2} - 1}{\color{blue}{1 \cdot \left(e^{x} - 1\right)}}}\]

   5. Applied add-sqr-sqrt 0.5

      \[\leadsto \sqrt{\frac{{\left(e^{x}\right)}^{2} - \color{blue}{\sqrt{1} \cdot \sqrt{1}}}{1 \cdot \left(e^{x} - 1\right)}}\]

   6. Applied add-sqr-sqrt 0.5

      \[\leadsto \sqrt{\frac{\color{blue}{\sqrt{{\left(e^{x}\right)}^{2}} \cdot \sqrt{{\left(e^{x}\right)}^{2}}} - \sqrt{1} \cdot \sqrt{1}}{1 \cdot \left(e^{x} - 1\right)}}\]

   7. Applied difference-of-squares 0.0

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

   8. Applied times-frac 0.0

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

   9. Applied sqrt-prod 0.0

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

   10. Simplified 0.0

       \[\leadsto \color{blue}{\sqrt{\sqrt{1} + \sqrt{{\left(e^{x}\right)}^{2}}}} \cdot \sqrt{\frac{\sqrt{{\left(e^{x}\right)}^{2}} - \sqrt{1}}{e^{x} - 1}}\]
   11. Using strategy rm

   12. Applied add-cube-cbrt 0.0

       \[\leadsto \sqrt{\sqrt{1} + \sqrt{\color{blue}{\left(\sqrt[3]{{\left(e^{x}\right)}^{2}} \cdot \sqrt[3]{{\left(e^{x}\right)}^{2}}\right) \cdot \sqrt[3]{{\left(e^{x}\right)}^{2}}}}} \cdot \sqrt{\frac{\sqrt{{\left(e^{x}\right)}^{2}} - \sqrt{1}}{e^{x} - 1}}\]

   13. Applied sqrt-prod 0.0

       \[\leadsto \sqrt{\sqrt{1} + \color{blue}{\sqrt{\sqrt[3]{{\left(e^{x}\right)}^{2}} \cdot \sqrt[3]{{\left(e^{x}\right)}^{2}}} \cdot \sqrt{\sqrt[3]{{\left(e^{x}\right)}^{2}}}}} \cdot \sqrt{\frac{\sqrt{{\left(e^{x}\right)}^{2}} - \sqrt{1}}{e^{x} - 1}}\]

   14. Simplified 0.0

       \[\leadsto \sqrt{\sqrt{1} + \color{blue}{\left|\sqrt[3]{{\left(e^{x}\right)}^{2}}\right|} \cdot \sqrt{\sqrt[3]{{\left(e^{x}\right)}^{2}}}} \cdot \sqrt{\frac{\sqrt{{\left(e^{x}\right)}^{2}} - \sqrt{1}}{e^{x} - 1}}\]

   if -6.751810080739974e-16 < x

   1. Initial program 37.3

      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]

   2. Simplified 33.3

      \[\leadsto \color{blue}{\sqrt{\frac{{\left(e^{x}\right)}^{2} - 1}{e^{x} - 1}}}\]

   3. Taylor expanded around 0 9.3

      \[\leadsto \color{blue}{\left(\sqrt{2} + \left(\frac{1}{4} \cdot \frac{{x}^{2}}{\sqrt{2}} + \frac{1}{2} \cdot \frac{x}{\sqrt{2}}\right)\right) - \frac{1}{8} \cdot \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}}\]

   4. Simplified 9.3

      \[\leadsto \color{blue}{\left(\frac{\frac{1}{2}}{\frac{\sqrt{2}}{x}} + \frac{x \cdot x}{\sqrt{2}} \cdot \left(\frac{1}{4} - \frac{1}{16}\right)\right) + \sqrt{2}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6.751810080739973559159999205335871800477 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\frac{\sqrt{{\left(e^{x}\right)}^{2}} - \sqrt{1}}{e^{x} - 1}} \cdot \sqrt{\sqrt{1} + \left|\sqrt[3]{{\left(e^{x}\right)}^{2}}\right| \cdot \sqrt{\sqrt[3]{{\left(e^{x}\right)}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} + \left(\frac{3}{16} \cdot \frac{x \cdot x}{\sqrt{2}} + \frac{\frac{1}{2}}{\frac{\sqrt{2}}{x}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))