Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Average Error: 5.8 → 0.1

Time: 7.7s

Precision: 64

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

Math

\[x \cdot \left(1 + y \cdot y\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;y \le -6.68515282831365304 \cdot 10^{28}:\\ \;\;\;\;\sqrt{\left(\frac{0.125}{{y}^{3}} - y\right) - \frac{0.5}{y}} \cdot \left(\sqrt{\mathsf{hypot}\left(y, \sqrt{1}\right)} \cdot \left(\mathsf{hypot}\left(y, \sqrt{1}\right) \cdot x\right)\right)\\ \mathbf{elif}\;y \le 4.4508025748694133 \cdot 10^{153}:\\ \;\;\;\;\mathsf{fma}\left(y, y, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y + \left(\frac{0.5}{y} - \frac{0.125}{{y}^{3}}\right)} \cdot \left(\sqrt{\mathsf{hypot}\left(y, \sqrt{1}\right)} \cdot \left(\mathsf{hypot}\left(y, \sqrt{1}\right) \cdot x\right)\right)\\ \end{array}\]

C

double f(double x, double y) {
        double r491081 = x;
        double r491082 = 1.0;
        double r491083 = y;
        double r491084 = r491083 * r491083;
        double r491085 = r491082 + r491084;
        double r491086 = r491081 * r491085;
        return r491086;
}

↓

double f(double x, double y) {
        double r491087 = y;
        double r491088 = -6.685152828313653e+28;
        bool r491089 = r491087 <= r491088;
        double r491090 = 0.125;
        double r491091 = 3.0;
        double r491092 = pow(r491087, r491091);
        double r491093 = r491090 / r491092;
        double r491094 = r491093 - r491087;
        double r491095 = 0.5;
        double r491096 = r491095 / r491087;
        double r491097 = r491094 - r491096;
        double r491098 = sqrt(r491097);
        double r491099 = 1.0;
        double r491100 = sqrt(r491099);
        double r491101 = hypot(r491087, r491100);
        double r491102 = sqrt(r491101);
        double r491103 = x;
        double r491104 = r491101 * r491103;
        double r491105 = r491102 * r491104;
        double r491106 = r491098 * r491105;
        double r491107 = 4.450802574869413e+153;
        bool r491108 = r491087 <= r491107;
        double r491109 = fma(r491087, r491087, r491099);
        double r491110 = r491109 * r491103;
        double r491111 = r491096 - r491093;
        double r491112 = r491087 + r491111;
        double r491113 = sqrt(r491112);
        double r491114 = r491113 * r491105;
        double r491115 = r491108 ? r491110 : r491114;
        double r491116 = r491089 ? r491106 : r491115;
        return r491116;
}

Error

Bits error versus x

Bits error versus y

Target

Original	5.8
Target	0.1
Herbie	0.1

\[x + \left(x \cdot y\right) \cdot y\]

Derivation

1. Split input into 3 regimes

2. if y < -6.685152828313653e+28

   1. Initial program 21.1

      \[x \cdot \left(1 + y \cdot y\right)\]

   2. Simplified 21.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, y, 1\right) \cdot x}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt 21.1

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(y, y, 1\right)} \cdot \sqrt{\mathsf{fma}\left(y, y, 1\right)}\right)} \cdot x\]

   5. Applied associate-*l* 21.1

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(y, y, 1\right)} \cdot \left(\sqrt{\mathsf{fma}\left(y, y, 1\right)} \cdot x\right)}\]
   6. Using strategy rm

   7. Applied add-sqr-sqrt 21.1

      \[\leadsto \sqrt{\color{blue}{\sqrt{\mathsf{fma}\left(y, y, 1\right)} \cdot \sqrt{\mathsf{fma}\left(y, y, 1\right)}}} \cdot \left(\sqrt{\mathsf{fma}\left(y, y, 1\right)} \cdot x\right)\]

   8. Applied sqrt-prod 21.3

      \[\leadsto \color{blue}{\left(\sqrt{\sqrt{\mathsf{fma}\left(y, y, 1\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(y, y, 1\right)}}\right)} \cdot \left(\sqrt{\mathsf{fma}\left(y, y, 1\right)} \cdot x\right)\]

   9. Applied associate-*l* 21.3

      \[\leadsto \color{blue}{\sqrt{\sqrt{\mathsf{fma}\left(y, y, 1\right)}} \cdot \left(\sqrt{\sqrt{\mathsf{fma}\left(y, y, 1\right)}} \cdot \left(\sqrt{\mathsf{fma}\left(y, y, 1\right)} \cdot x\right)\right)}\]

   10. Simplified 21.3

       \[\leadsto \sqrt{\sqrt{\mathsf{fma}\left(y, y, 1\right)}} \cdot \color{blue}{\left(\sqrt{\mathsf{hypot}\left(y, \sqrt{1}\right)} \cdot \left(\mathsf{hypot}\left(y, \sqrt{1}\right) \cdot x\right)\right)}\]

   11. Taylor expanded around -inf 0.6

       \[\leadsto \sqrt{\color{blue}{0.125 \cdot \frac{1}{{y}^{3}} - \left(y + 0.5 \cdot \frac{1}{y}\right)}} \cdot \left(\sqrt{\mathsf{hypot}\left(y, \sqrt{1}\right)} \cdot \left(\mathsf{hypot}\left(y, \sqrt{1}\right) \cdot x\right)\right)\]

   12. Simplified 0.6

       \[\leadsto \sqrt{\color{blue}{\left(\frac{0.125}{{y}^{3}} - y\right) - \frac{0.5}{y}}} \cdot \left(\sqrt{\mathsf{hypot}\left(y, \sqrt{1}\right)} \cdot \left(\mathsf{hypot}\left(y, \sqrt{1}\right) \cdot x\right)\right)\]

   if -6.685152828313653e+28 < y < 4.450802574869413e+153

   1. Initial program 0.1

      \[x \cdot \left(1 + y \cdot y\right)\]

   2. Simplified 0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, y, 1\right) \cdot x}\]

   if 4.450802574869413e+153 < y

   1. Initial program 63.8

      \[x \cdot \left(1 + y \cdot y\right)\]

   2. Simplified 63.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, y, 1\right) \cdot x}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt 63.8

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(y, y, 1\right)} \cdot \sqrt{\mathsf{fma}\left(y, y, 1\right)}\right)} \cdot x\]

   5. Applied associate-*l* 63.8

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(y, y, 1\right)} \cdot \left(\sqrt{\mathsf{fma}\left(y, y, 1\right)} \cdot x\right)}\]
   6. Using strategy rm

   7. Applied add-sqr-sqrt 63.8

      \[\leadsto \sqrt{\color{blue}{\sqrt{\mathsf{fma}\left(y, y, 1\right)} \cdot \sqrt{\mathsf{fma}\left(y, y, 1\right)}}} \cdot \left(\sqrt{\mathsf{fma}\left(y, y, 1\right)} \cdot x\right)\]

   8. Applied sqrt-prod 63.8

      \[\leadsto \color{blue}{\left(\sqrt{\sqrt{\mathsf{fma}\left(y, y, 1\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(y, y, 1\right)}}\right)} \cdot \left(\sqrt{\mathsf{fma}\left(y, y, 1\right)} \cdot x\right)\]

   9. Applied associate-*l* 63.8

      \[\leadsto \color{blue}{\sqrt{\sqrt{\mathsf{fma}\left(y, y, 1\right)}} \cdot \left(\sqrt{\sqrt{\mathsf{fma}\left(y, y, 1\right)}} \cdot \left(\sqrt{\mathsf{fma}\left(y, y, 1\right)} \cdot x\right)\right)}\]

   10. Simplified 63.8

       \[\leadsto \sqrt{\sqrt{\mathsf{fma}\left(y, y, 1\right)}} \cdot \color{blue}{\left(\sqrt{\mathsf{hypot}\left(y, \sqrt{1}\right)} \cdot \left(\mathsf{hypot}\left(y, \sqrt{1}\right) \cdot x\right)\right)}\]

   11. Taylor expanded around inf 0.6

       \[\leadsto \sqrt{\color{blue}{\left(y + 0.5 \cdot \frac{1}{y}\right) - 0.125 \cdot \frac{1}{{y}^{3}}}} \cdot \left(\sqrt{\mathsf{hypot}\left(y, \sqrt{1}\right)} \cdot \left(\mathsf{hypot}\left(y, \sqrt{1}\right) \cdot x\right)\right)\]

   12. Simplified 0.6

       \[\leadsto \sqrt{\color{blue}{y + \left(\frac{0.5}{y} - \frac{0.125}{{y}^{3}}\right)}} \cdot \left(\sqrt{\mathsf{hypot}\left(y, \sqrt{1}\right)} \cdot \left(\mathsf{hypot}\left(y, \sqrt{1}\right) \cdot x\right)\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -6.68515282831365304 \cdot 10^{28}:\\ \;\;\;\;\sqrt{\left(\frac{0.125}{{y}^{3}} - y\right) - \frac{0.5}{y}} \cdot \left(\sqrt{\mathsf{hypot}\left(y, \sqrt{1}\right)} \cdot \left(\mathsf{hypot}\left(y, \sqrt{1}\right) \cdot x\right)\right)\\ \mathbf{elif}\;y \le 4.4508025748694133 \cdot 10^{153}:\\ \;\;\;\;\mathsf{fma}\left(y, y, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y + \left(\frac{0.5}{y} - \frac{0.125}{{y}^{3}}\right)} \cdot \left(\sqrt{\mathsf{hypot}\left(y, \sqrt{1}\right)} \cdot \left(\mathsf{hypot}\left(y, \sqrt{1}\right) \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x y)
  :name "Numeric.Integration.TanhSinh:everywhere from integration-0.2.1"
  :precision binary64

  :herbie-target
  (+ x (* (* x y) y))

  (* x (+ 1 (* y y))))