Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2

Average Error: 25.5 → 5.7

Time: 6.5s

Precision: 64

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

Math

\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -3.30855613287552178 \cdot 10^{104}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \le 6.3074422229433126 \cdot 10^{112}:\\ \;\;\;\;\left(x \cdot \left(y \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\left|\sqrt[3]{z \cdot z - t \cdot a}\right|}\right)\right) \cdot \frac{\sqrt[3]{z}}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r385454 = x;
        double r385455 = y;
        double r385456 = r385454 * r385455;
        double r385457 = z;
        double r385458 = r385456 * r385457;
        double r385459 = r385457 * r385457;
        double r385460 = t;
        double r385461 = a;
        double r385462 = r385460 * r385461;
        double r385463 = r385459 - r385462;
        double r385464 = sqrt(r385463);
        double r385465 = r385458 / r385464;
        return r385465;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r385466 = z;
        double r385467 = -3.308556132875522e+104;
        bool r385468 = r385466 <= r385467;
        double r385469 = x;
        double r385470 = y;
        double r385471 = r385469 * r385470;
        double r385472 = -r385471;
        double r385473 = 6.307442222943313e+112;
        bool r385474 = r385466 <= r385473;
        double r385475 = cbrt(r385466);
        double r385476 = r385475 * r385475;
        double r385477 = r385466 * r385466;
        double r385478 = t;
        double r385479 = a;
        double r385480 = r385478 * r385479;
        double r385481 = r385477 - r385480;
        double r385482 = cbrt(r385481);
        double r385483 = fabs(r385482);
        double r385484 = r385476 / r385483;
        double r385485 = r385470 * r385484;
        double r385486 = r385469 * r385485;
        double r385487 = sqrt(r385482);
        double r385488 = r385475 / r385487;
        double r385489 = r385486 * r385488;
        double r385490 = r385474 ? r385489 : r385471;
        double r385491 = r385468 ? r385472 : r385490;
        return r385491;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	25.5
Target	7.6
Herbie	5.7

\[\begin{array}{l} \mathbf{if}\;z \lt -3.1921305903852764 \cdot 10^{46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.9762681209208942 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if z < -3.308556132875522e+104

   1. Initial program 44.2

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 44.2

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\color{blue}{1 \cdot \left(z \cdot z - t \cdot a\right)}}}\]

   4. Applied sqrt-prod 44.2

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{\sqrt{1} \cdot \sqrt{z \cdot z - t \cdot a}}}\]

   5. Applied times-frac 41.4

      \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{1}} \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}}\]

   6. Simplified 41.4

      \[\leadsto \color{blue}{\left(x \cdot y\right)} \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\]

   7. Taylor expanded around -inf 2.6

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

   if -3.308556132875522e+104 < z < 6.307442222943313e+112

   1. Initial program 11.2

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 11.2

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\color{blue}{1 \cdot \left(z \cdot z - t \cdot a\right)}}}\]

   4. Applied sqrt-prod 11.2

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{\sqrt{1} \cdot \sqrt{z \cdot z - t \cdot a}}}\]

   5. Applied times-frac 9.3

      \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{1}} \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}}\]

   6. Simplified 9.3

      \[\leadsto \color{blue}{\left(x \cdot y\right)} \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 9.8

      \[\leadsto \left(x \cdot y\right) \cdot \frac{z}{\sqrt{\color{blue}{\left(\sqrt[3]{z \cdot z - t \cdot a} \cdot \sqrt[3]{z \cdot z - t \cdot a}\right) \cdot \sqrt[3]{z \cdot z - t \cdot a}}}}\]

   9. Applied sqrt-prod 9.8

      \[\leadsto \left(x \cdot y\right) \cdot \frac{z}{\color{blue}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a} \cdot \sqrt[3]{z \cdot z - t \cdot a}} \cdot \sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}}\]

   10. Applied add-cube-cbrt 10.0

       \[\leadsto \left(x \cdot y\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a} \cdot \sqrt[3]{z \cdot z - t \cdot a}} \cdot \sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}\]

   11. Applied times-frac 10.1

       \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a} \cdot \sqrt[3]{z \cdot z - t \cdot a}}} \cdot \frac{\sqrt[3]{z}}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}\right)}\]

   12. Applied associate-*r* 9.3

       \[\leadsto \color{blue}{\left(\left(x \cdot y\right) \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a} \cdot \sqrt[3]{z \cdot z - t \cdot a}}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}}\]

   13. Simplified 8.2

       \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\left|\sqrt[3]{z \cdot z - t \cdot a}\right|}\right)\right)} \cdot \frac{\sqrt[3]{z}}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}\]

   if 6.307442222943313e+112 < z

   1. Initial program 46.4

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 46.4

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\color{blue}{1 \cdot \left(z \cdot z - t \cdot a\right)}}}\]

   4. Applied sqrt-prod 46.4

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{\sqrt{1} \cdot \sqrt{z \cdot z - t \cdot a}}}\]

   5. Applied times-frac 44.3

      \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{1}} \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}}\]

   6. Simplified 44.3

      \[\leadsto \color{blue}{\left(x \cdot y\right)} \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\]

   7. Taylor expanded around inf 2.0

      \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{1}\]
3. Recombined 3 regimes into one program.

4. Final simplification 5.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.30855613287552178 \cdot 10^{104}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \le 6.3074422229433126 \cdot 10^{112}:\\ \;\;\;\;\left(x \cdot \left(y \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\left|\sqrt[3]{z \cdot z - t \cdot a}\right|}\right)\right) \cdot \frac{\sqrt[3]{z}}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (x y z t a)
  :name "Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z -3.1921305903852764e+46) (- (* y x)) (if (< z 5.976268120920894e+90) (/ (* x z) (/ (sqrt (- (* z z) (* a t))) y)) (* y x)))

  (/ (* (* x y) z) (sqrt (- (* z z) (* t a)))))