sqrtexp (problem 3.4.4)

Average Error: 4.5 → 1.0

Time: 27.8s

Precision: 64

• could not determine a ground truth for program body

  1. x = 7.843670218358717e+76

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -9.587317954112346857544454308807111121368 \cdot 10^{-14}:\\ \;\;\;\;\sqrt{\frac{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\left(e^{0.5 \cdot x} - \sqrt{\sqrt{1}}\right) \cdot \left(e^{0.5 \cdot x} + \sqrt{\sqrt{1}}\right)\right)}{\left(\sqrt{e^{x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{x}} - \sqrt{1}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{\frac{\sqrt{2}}{x}} \cdot \left(0.25 - \frac{0.125}{2}\right) + \frac{0.5}{\frac{\sqrt{2}}{x}}\right) + \sqrt{2}\\ \end{array}\]

C

double f(double x) {
        double r2375920 = 2.0;
        double r2375921 = x;
        double r2375922 = r2375920 * r2375921;
        double r2375923 = exp(r2375922);
        double r2375924 = 1.0;
        double r2375925 = r2375923 - r2375924;
        double r2375926 = exp(r2375921);
        double r2375927 = r2375926 - r2375924;
        double r2375928 = r2375925 / r2375927;
        double r2375929 = sqrt(r2375928);
        return r2375929;
}

↓

double f(double x) {
        double r2375930 = x;
        double r2375931 = -9.587317954112347e-14;
        bool r2375932 = r2375930 <= r2375931;
        double r2375933 = 2.0;
        double r2375934 = r2375933 * r2375930;
        double r2375935 = exp(r2375934);
        double r2375936 = sqrt(r2375935);
        double r2375937 = 1.0;
        double r2375938 = sqrt(r2375937);
        double r2375939 = r2375936 + r2375938;
        double r2375940 = 0.5;
        double r2375941 = r2375940 * r2375930;
        double r2375942 = exp(r2375941);
        double r2375943 = sqrt(r2375938);
        double r2375944 = r2375942 - r2375943;
        double r2375945 = r2375942 + r2375943;
        double r2375946 = r2375944 * r2375945;
        double r2375947 = r2375939 * r2375946;
        double r2375948 = exp(r2375930);
        double r2375949 = sqrt(r2375948);
        double r2375950 = r2375949 + r2375938;
        double r2375951 = r2375949 - r2375938;
        double r2375952 = r2375950 * r2375951;
        double r2375953 = r2375947 / r2375952;
        double r2375954 = sqrt(r2375953);
        double r2375955 = sqrt(r2375933);
        double r2375956 = r2375955 / r2375930;
        double r2375957 = r2375930 / r2375956;
        double r2375958 = 0.25;
        double r2375959 = 0.125;
        double r2375960 = r2375959 / r2375933;
        double r2375961 = r2375958 - r2375960;
        double r2375962 = r2375957 * r2375961;
        double r2375963 = r2375940 / r2375956;
        double r2375964 = r2375962 + r2375963;
        double r2375965 = r2375964 + r2375955;
        double r2375966 = r2375932 ? r2375954 : r2375965;
        return r2375966;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -9.587317954112347e-14

   1. Initial program 0.7

      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 0.7

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

   4. Applied add-sqr-sqrt 0.6

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

   5. Applied difference-of-squares 0.2

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

   6. Applied associate-/l* 0.2

      \[\leadsto \sqrt{\color{blue}{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{\frac{e^{x} - 1}{\sqrt{e^{2 \cdot x}} - \sqrt{1}}}}}\]
   7. Using strategy rm

   8. Applied add-sqr-sqrt 0.2

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

   9. Applied sqrt-prod 0.2

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

   10. Applied add-sqr-sqrt 0.2

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

   11. Applied sqrt-prod 0.5

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

   12. Applied difference-of-squares 0.7

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

   13. Applied add-sqr-sqrt 0.7

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

   14. Applied add-sqr-sqrt 0.6

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

   15. Applied difference-of-squares 0.1

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

   16. Applied times-frac 0.1

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

   17. Taylor expanded around -inf 0.2

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

   if -9.587317954112347e-14 < x

   1. Initial program 36.5

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

   2. Taylor expanded around 0 7.5

      \[\leadsto \color{blue}{\left(\sqrt{2} + \left(0.5 \cdot \frac{x}{\sqrt{2}} + 0.25 \cdot \frac{{x}^{2}}{\sqrt{2}}\right)\right) - 0.125 \cdot \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}}\]

   3. Simplified 7.5

      \[\leadsto \color{blue}{\sqrt{2} + \left(\frac{0.5}{\frac{\sqrt{2}}{x}} + \frac{x}{\frac{\sqrt{2}}{x}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9.587317954112346857544454308807111121368 \cdot 10^{-14}:\\ \;\;\;\;\sqrt{\frac{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\left(e^{0.5 \cdot x} - \sqrt{\sqrt{1}}\right) \cdot \left(e^{0.5 \cdot x} + \sqrt{\sqrt{1}}\right)\right)}{\left(\sqrt{e^{x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{x}} - \sqrt{1}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{\frac{\sqrt{2}}{x}} \cdot \left(0.25 - \frac{0.125}{2}\right) + \frac{0.5}{\frac{\sqrt{2}}{x}}\right) + \sqrt{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))