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

Average Error: 23.8 → 5.4

Time: 16.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.8913959868564195 \cdot 10^{+154}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \le 1.1848486164183457 \cdot 10^{+114}:\\ \;\;\;\;\left(x \cdot \left(y \cdot \left(\frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}}}\right)\right)\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r15198488 = x;
        double r15198489 = y;
        double r15198490 = r15198488 * r15198489;
        double r15198491 = z;
        double r15198492 = r15198490 * r15198491;
        double r15198493 = r15198491 * r15198491;
        double r15198494 = t;
        double r15198495 = a;
        double r15198496 = r15198494 * r15198495;
        double r15198497 = r15198493 - r15198496;
        double r15198498 = sqrt(r15198497);
        double r15198499 = r15198492 / r15198498;
        return r15198499;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r15198500 = z;
        double r15198501 = -1.8913959868564195e+154;
        bool r15198502 = r15198500 <= r15198501;
        double r15198503 = x;
        double r15198504 = y;
        double r15198505 = -r15198504;
        double r15198506 = r15198503 * r15198505;
        double r15198507 = 1.1848486164183457e+114;
        bool r15198508 = r15198500 <= r15198507;
        double r15198509 = cbrt(r15198500);
        double r15198510 = r15198500 * r15198500;
        double r15198511 = t;
        double r15198512 = a;
        double r15198513 = r15198511 * r15198512;
        double r15198514 = r15198510 - r15198513;
        double r15198515 = sqrt(r15198514);
        double r15198516 = cbrt(r15198515);
        double r15198517 = r15198509 / r15198516;
        double r15198518 = r15198517 * r15198517;
        double r15198519 = r15198504 * r15198518;
        double r15198520 = r15198503 * r15198519;
        double r15198521 = r15198520 * r15198517;
        double r15198522 = r15198503 * r15198504;
        double r15198523 = r15198508 ? r15198521 : r15198522;
        double r15198524 = r15198502 ? r15198506 : r15198523;
        return r15198524;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	23.8
Target	7.6
Herbie	5.4

\[\begin{array}{l} \mathbf{if}\;z \lt -3.1921305903852764 \cdot 10^{+46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.976268120920894 \cdot 10^{+90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if z < -1.8913959868564195e+154

   1. Initial program 53.3

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

   3. Applied *-un-lft-identity 53.3

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

   4. Applied sqrt-prod 53.3

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

   5. Applied times-frac 53.4

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

   6. Simplified 53.4

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

   7. Taylor expanded around -inf 1.3

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

   8. Simplified 1.3

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

   if -1.8913959868564195e+154 < z < 1.1848486164183457e+114

   1. Initial program 10.2

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

   3. Applied *-un-lft-identity 10.2

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

   4. Applied sqrt-prod 10.2

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

   5. Applied times-frac 8.2

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

   6. Simplified 8.2

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

   8. Applied add-cube-cbrt 8.9

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

   9. Applied add-cube-cbrt 8.5

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

   10. Applied times-frac 8.5

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

   11. Applied associate-*r* 7.9

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

   12. Simplified 7.5

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

   if 1.1848486164183457e+114 < z

   1. Initial program 45.5

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

   3. Applied *-un-lft-identity 45.5

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

   4. Applied sqrt-prod 45.5

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

   5. Applied times-frac 44.0

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

   6. Simplified 44.0

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

   8. Applied associate-*l* 44.0

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

   9. Taylor expanded around inf 1.6

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

4. Final simplification 5.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.8913959868564195 \cdot 10^{+154}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \le 1.1848486164183457 \cdot 10^{+114}:\\ \;\;\;\;\left(x \cdot \left(y \cdot \left(\frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}}}\right)\right)\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(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)))))