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

Average Error: 25.3 → 7.2

Time: 32.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -2.8045670485597684 \cdot 10^{72}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \le 3.39666918434105311 \cdot 10^{81}:\\ \;\;\;\;x \cdot \frac{y \cdot z}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r187418 = x;
        double r187419 = y;
        double r187420 = r187418 * r187419;
        double r187421 = z;
        double r187422 = r187420 * r187421;
        double r187423 = r187421 * r187421;
        double r187424 = t;
        double r187425 = a;
        double r187426 = r187424 * r187425;
        double r187427 = r187423 - r187426;
        double r187428 = sqrt(r187427);
        double r187429 = r187422 / r187428;
        return r187429;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r187430 = z;
        double r187431 = -2.8045670485597684e+72;
        bool r187432 = r187430 <= r187431;
        double r187433 = x;
        double r187434 = y;
        double r187435 = -r187434;
        double r187436 = r187433 * r187435;
        double r187437 = 3.396669184341053e+81;
        bool r187438 = r187430 <= r187437;
        double r187439 = r187434 * r187430;
        double r187440 = r187430 * r187430;
        double r187441 = t;
        double r187442 = a;
        double r187443 = r187441 * r187442;
        double r187444 = r187440 - r187443;
        double r187445 = sqrt(r187444);
        double r187446 = r187439 / r187445;
        double r187447 = r187433 * r187446;
        double r187448 = r187434 * r187433;
        double r187449 = r187438 ? r187447 : r187448;
        double r187450 = r187432 ? r187436 : r187449;
        return r187450;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	25.3
Target	7.9
Herbie	7.2

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

Derivation

1. Split input into 3 regimes

2. if z < -2.8045670485597684e+72

   1. Initial program 41.1

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

   3. Applied add-sqr-sqrt 41.1

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

   4. Applied sqrt-prod 41.2

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

   5. Applied times-frac 39.6

      \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{\sqrt{z \cdot z - t \cdot a}}} \cdot \frac{z}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 39.6

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

   8. Applied sqrt-prod 39.6

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

   9. Applied sqrt-prod 39.6

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

   10. Applied times-frac 40.1

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

   11. Applied associate-*l* 39.1

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

   12. Taylor expanded around -inf 3.8

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

   13. Simplified 3.8

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

   if -2.8045670485597684e+72 < z < 3.396669184341053e+81

   1. Initial program 11.6

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

   3. Applied add-sqr-sqrt 11.6

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

   4. Applied sqrt-prod 11.8

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

   5. Applied times-frac 11.2

      \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{\sqrt{z \cdot z - t \cdot a}}} \cdot \frac{z}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 11.2

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

   8. Applied sqrt-prod 11.2

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

   9. Applied sqrt-prod 11.2

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

   10. Applied times-frac 11.7

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

   11. Applied associate-*l* 10.5

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

   13. Applied frac-times 11.0

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

   14. Simplified 10.8

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

   if 3.396669184341053e+81 < z

   1. Initial program 41.1

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

   3. Applied add-sqr-sqrt 41.1

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

   4. Applied sqrt-prod 41.2

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

   5. Applied times-frac 39.8

      \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{\sqrt{z \cdot z - t \cdot a}}} \cdot \frac{z}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}}\]
   6. Using strategy rm

   7. Applied associate-*l/ 39.7

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

   8. Taylor expanded around inf 2.4

      \[\leadsto \color{blue}{x \cdot y}\]

   9. Simplified 2.4

      \[\leadsto \color{blue}{y \cdot x}\]
3. Recombined 3 regimes into one program.

4. Final simplification 7.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.8045670485597684 \cdot 10^{72}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \le 3.39666918434105311 \cdot 10^{81}:\\ \;\;\;\;x \cdot \frac{y \cdot z}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 
(FPCore (x y z t a)
  :name "Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2"

  :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)))))