3frac (problem 3.3.3)

Average Error: 10.0 → 0.2

Time: 21.5s

Precision: 64

Math

\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -107.120037393270436609782336745411157608:\\ \;\;\;\;\left(\frac{\sqrt[3]{\sqrt{2}}}{\frac{x}{\sqrt{\sqrt{2}}}} \cdot \frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{\frac{x \cdot x}{\sqrt{\sqrt{2}}}} + \frac{2}{{x}^{7}}\right) + \frac{2}{{x}^{5}}\\ \mathbf{elif}\;x \le 139.9956438464524808296118862926959991455:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{3}}\right)\\ \end{array}\]

C

double f(double x) {
        double r146010 = 1.0;
        double r146011 = x;
        double r146012 = r146011 + r146010;
        double r146013 = r146010 / r146012;
        double r146014 = 2.0;
        double r146015 = r146014 / r146011;
        double r146016 = r146013 - r146015;
        double r146017 = r146011 - r146010;
        double r146018 = r146010 / r146017;
        double r146019 = r146016 + r146018;
        return r146019;
}

↓

double f(double x) {
        double r146020 = x;
        double r146021 = -107.12003739327044;
        bool r146022 = r146020 <= r146021;
        double r146023 = 2.0;
        double r146024 = sqrt(r146023);
        double r146025 = cbrt(r146024);
        double r146026 = sqrt(r146024);
        double r146027 = r146020 / r146026;
        double r146028 = r146025 / r146027;
        double r146029 = r146025 * r146025;
        double r146030 = r146020 * r146020;
        double r146031 = r146030 / r146026;
        double r146032 = r146029 / r146031;
        double r146033 = r146028 * r146032;
        double r146034 = 7.0;
        double r146035 = pow(r146020, r146034);
        double r146036 = r146023 / r146035;
        double r146037 = r146033 + r146036;
        double r146038 = 5.0;
        double r146039 = pow(r146020, r146038);
        double r146040 = r146023 / r146039;
        double r146041 = r146037 + r146040;
        double r146042 = 139.99564384645248;
        bool r146043 = r146020 <= r146042;
        double r146044 = 1.0;
        double r146045 = r146044 + r146020;
        double r146046 = r146044 / r146045;
        double r146047 = r146023 / r146020;
        double r146048 = r146046 - r146047;
        double r146049 = r146020 - r146044;
        double r146050 = r146044 / r146049;
        double r146051 = r146048 + r146050;
        double r146052 = 3.0;
        double r146053 = pow(r146020, r146052);
        double r146054 = r146023 / r146053;
        double r146055 = r146036 + r146054;
        double r146056 = r146040 + r146055;
        double r146057 = r146043 ? r146051 : r146056;
        double r146058 = r146022 ? r146041 : r146057;
        return r146058;
}

Error

Bits error versus x

Target

Original	10.0
Target	0.3
Herbie	0.2

\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

1. Split input into 3 regimes

2. if x < -107.12003739327044

   1. Initial program 19.6

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]

   2. Taylor expanded around inf 0.5

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{7}} + \left(2 \cdot \frac{1}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{5}}\right)}\]

   3. Simplified 0.5

      \[\leadsto \color{blue}{\left(\frac{2}{{x}^{3}} + \frac{2}{{x}^{7}}\right) + \frac{2}{{x}^{5}}}\]
   4. Using strategy rm

   5. Applied add-sqr-sqrt 0.8

      \[\leadsto \left(\frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{{x}^{3}} + \frac{2}{{x}^{7}}\right) + \frac{2}{{x}^{5}}\]

   6. Applied associate-/l* 0.7

      \[\leadsto \left(\color{blue}{\frac{\sqrt{2}}{\frac{{x}^{3}}{\sqrt{2}}}} + \frac{2}{{x}^{7}}\right) + \frac{2}{{x}^{5}}\]
   7. Using strategy rm

   8. Applied add-sqr-sqrt 0.7

      \[\leadsto \left(\frac{\sqrt{2}}{\frac{{x}^{3}}{\sqrt{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}}} + \frac{2}{{x}^{7}}\right) + \frac{2}{{x}^{5}}\]

   9. Applied sqrt-prod 0.5

      \[\leadsto \left(\frac{\sqrt{2}}{\frac{{x}^{3}}{\color{blue}{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}}}} + \frac{2}{{x}^{7}}\right) + \frac{2}{{x}^{5}}\]

   10. Applied add-cube-cbrt 1.2

       \[\leadsto \left(\frac{\sqrt{2}}{\frac{{\color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)}}^{3}}{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}}} + \frac{2}{{x}^{7}}\right) + \frac{2}{{x}^{5}}\]

   11. Applied unpow-prod-down 1.2

       \[\leadsto \left(\frac{\sqrt{2}}{\frac{\color{blue}{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{3} \cdot {\left(\sqrt[3]{x}\right)}^{3}}}{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}}} + \frac{2}{{x}^{7}}\right) + \frac{2}{{x}^{5}}\]

   12. Applied times-frac 1.1

       \[\leadsto \left(\frac{\sqrt{2}}{\color{blue}{\frac{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{3}}{\sqrt{\sqrt{2}}} \cdot \frac{{\left(\sqrt[3]{x}\right)}^{3}}{\sqrt{\sqrt{2}}}}} + \frac{2}{{x}^{7}}\right) + \frac{2}{{x}^{5}}\]

   13. Applied add-cube-cbrt 1.1

       \[\leadsto \left(\frac{\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}}}{\frac{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{3}}{\sqrt{\sqrt{2}}} \cdot \frac{{\left(\sqrt[3]{x}\right)}^{3}}{\sqrt{\sqrt{2}}}} + \frac{2}{{x}^{7}}\right) + \frac{2}{{x}^{5}}\]

   14. Applied times-frac 0.8

       \[\leadsto \left(\color{blue}{\frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{\frac{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{3}}{\sqrt{\sqrt{2}}}} \cdot \frac{\sqrt[3]{\sqrt{2}}}{\frac{{\left(\sqrt[3]{x}\right)}^{3}}{\sqrt{\sqrt{2}}}}} + \frac{2}{{x}^{7}}\right) + \frac{2}{{x}^{5}}\]

   15. Simplified 0.4

       \[\leadsto \left(\color{blue}{\frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{\frac{x \cdot x}{\sqrt{\sqrt{2}}}}} \cdot \frac{\sqrt[3]{\sqrt{2}}}{\frac{{\left(\sqrt[3]{x}\right)}^{3}}{\sqrt{\sqrt{2}}}} + \frac{2}{{x}^{7}}\right) + \frac{2}{{x}^{5}}\]

   16. Simplified 0.2

       \[\leadsto \left(\frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{\frac{x \cdot x}{\sqrt{\sqrt{2}}}} \cdot \color{blue}{\frac{\sqrt[3]{\sqrt{2}}}{\frac{x}{\sqrt{\sqrt{2}}}}} + \frac{2}{{x}^{7}}\right) + \frac{2}{{x}^{5}}\]

   if -107.12003739327044 < x < 139.99564384645248

   1. Initial program 0.0

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]

   if 139.99564384645248 < x

   1. Initial program 20.1

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]

   2. Taylor expanded around inf 0.6

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{7}} + \left(2 \cdot \frac{1}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{5}}\right)}\]

   3. Simplified 0.6

      \[\leadsto \color{blue}{\left(\frac{2}{{x}^{3}} + \frac{2}{{x}^{7}}\right) + \frac{2}{{x}^{5}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -107.120037393270436609782336745411157608:\\ \;\;\;\;\left(\frac{\sqrt[3]{\sqrt{2}}}{\frac{x}{\sqrt{\sqrt{2}}}} \cdot \frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{\frac{x \cdot x}{\sqrt{\sqrt{2}}}} + \frac{2}{{x}^{7}}\right) + \frac{2}{{x}^{5}}\\ \mathbf{elif}\;x \le 139.9956438464524808296118862926959991455:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{3}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (x)
  :name "3frac (problem 3.3.3)"

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))