2nthrt (problem 3.4.6)

Average Error: 29.2 → 22.1

Time: 17.5s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

• could not determine a ground truth for program body

  1. x = 1.818673981998486e+237
  2. n = 3.109809281420871e-25

Math

\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -9.277743339242776164382043874840818047014 \cdot 10^{-13}:\\ \;\;\;\;\left(2 \cdot \log \left({\left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^{\frac{1}{3}}\right) + \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 8.740688882085781214297279027767987281218 \cdot 10^{-15}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \log \left({\left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^{\frac{1}{3}}\right) + \mathsf{fma}\left(\frac{1}{3}, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}, -{x}^{\left(\frac{1}{n}\right)}\right)\\ \end{array}\]

C

double f(double x, double n) {
        double r59527 = x;
        double r59528 = 1.0;
        double r59529 = r59527 + r59528;
        double r59530 = n;
        double r59531 = r59528 / r59530;
        double r59532 = pow(r59529, r59531);
        double r59533 = pow(r59527, r59531);
        double r59534 = r59532 - r59533;
        return r59534;
}

↓

double f(double x, double n) {
        double r59535 = 1.0;
        double r59536 = n;
        double r59537 = r59535 / r59536;
        double r59538 = -9.277743339242776e-13;
        bool r59539 = r59537 <= r59538;
        double r59540 = 2.0;
        double r59541 = x;
        double r59542 = r59541 + r59535;
        double r59543 = pow(r59542, r59537);
        double r59544 = exp(r59543);
        double r59545 = 0.3333333333333333;
        double r59546 = pow(r59544, r59545);
        double r59547 = log(r59546);
        double r59548 = r59540 * r59547;
        double r59549 = cbrt(r59544);
        double r59550 = log(r59549);
        double r59551 = r59548 + r59550;
        double r59552 = pow(r59541, r59537);
        double r59553 = r59551 - r59552;
        double r59554 = 8.740688882085781e-15;
        bool r59555 = r59537 <= r59554;
        double r59556 = r59535 / r59541;
        double r59557 = 1.0;
        double r59558 = r59557 / r59536;
        double r59559 = log(r59541);
        double r59560 = -r59559;
        double r59561 = pow(r59536, r59540);
        double r59562 = r59560 / r59561;
        double r59563 = r59558 - r59562;
        double r59564 = r59556 * r59563;
        double r59565 = 0.5;
        double r59566 = pow(r59541, r59540);
        double r59567 = r59566 * r59536;
        double r59568 = r59565 / r59567;
        double r59569 = r59564 - r59568;
        double r59570 = -r59552;
        double r59571 = fma(r59545, r59543, r59570);
        double r59572 = r59548 + r59571;
        double r59573 = r59555 ? r59569 : r59572;
        double r59574 = r59539 ? r59553 : r59573;
        return r59574;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1.0 n) < -9.277743339242776e-13

   1. Initial program 1.3

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
   2. Using strategy rm

   3. Applied add-log-exp 1.3

      \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 1.4

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

   6. Applied log-prod 1.4

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

   7. Simplified 1.4

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

   9. Applied pow1/3 1.4

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

   if -9.277743339242776e-13 < (/ 1.0 n) < 8.740688882085781e-15

   1. Initial program 45.4

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]

   2. Taylor expanded around inf 33.2

      \[\leadsto \color{blue}{1 \cdot \frac{1}{x \cdot n} - \left(0.5 \cdot \frac{1}{{x}^{2} \cdot n} + 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)}\]

   3. Simplified 32.6

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}}\]

   if 8.740688882085781e-15 < (/ 1.0 n)

   1. Initial program 24.6

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
   2. Using strategy rm

   3. Applied add-log-exp 24.6

      \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 26.0

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

   6. Applied log-prod 26.0

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

   7. Simplified 26.0

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

   9. Applied pow1/3 25.3

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

   11. Applied associate--l+ 25.3

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

   12. Simplified 24.8

       \[\leadsto 2 \cdot \log \left({\left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^{\frac{1}{3}}\right) + \color{blue}{\mathsf{fma}\left(\frac{1}{3}, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}, -{x}^{\left(\frac{1}{n}\right)}\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 22.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -9.277743339242776164382043874840818047014 \cdot 10^{-13}:\\ \;\;\;\;\left(2 \cdot \log \left({\left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^{\frac{1}{3}}\right) + \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 8.740688882085781214297279027767987281218 \cdot 10^{-15}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \log \left({\left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^{\frac{1}{3}}\right) + \mathsf{fma}\left(\frac{1}{3}, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}, -{x}^{\left(\frac{1}{n}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))