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

Average Error: 24.5 → 5.8

Time: 4.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.0932223177976295 \cdot 10^{154}:\\ \;\;\;\;\left(x \cdot y\right) \cdot -1\\ \mathbf{elif}\;z \le 3.43469549411492992 \cdot 10^{83}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r274455 = x;
        double r274456 = y;
        double r274457 = r274455 * r274456;
        double r274458 = z;
        double r274459 = r274457 * r274458;
        double r274460 = r274458 * r274458;
        double r274461 = t;
        double r274462 = a;
        double r274463 = r274461 * r274462;
        double r274464 = r274460 - r274463;
        double r274465 = sqrt(r274464);
        double r274466 = r274459 / r274465;
        return r274466;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r274467 = z;
        double r274468 = -1.0932223177976295e+154;
        bool r274469 = r274467 <= r274468;
        double r274470 = x;
        double r274471 = y;
        double r274472 = r274470 * r274471;
        double r274473 = -1.0;
        double r274474 = r274472 * r274473;
        double r274475 = 3.43469549411493e+83;
        bool r274476 = r274467 <= r274475;
        double r274477 = r274467 * r274467;
        double r274478 = t;
        double r274479 = a;
        double r274480 = r274478 * r274479;
        double r274481 = r274477 - r274480;
        double r274482 = sqrt(r274481);
        double r274483 = r274467 / r274482;
        double r274484 = r274471 * r274483;
        double r274485 = r274470 * r274484;
        double r274486 = r274476 ? r274485 : r274472;
        double r274487 = r274469 ? r274474 : r274486;
        return r274487;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.5
Target	7.3
Herbie	5.8

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

Derivation

1. Split input into 3 regimes

2. if z < -1.0932223177976295e+154

   1. Initial program 54.4

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

   3. Applied *-un-lft-identity 54.4

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

   4. Applied sqrt-prod 54.4

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

   5. Applied times-frac 54.0

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

   6. Simplified 54.0

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

   7. Taylor expanded around -inf 1.5

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

   if -1.0932223177976295e+154 < z < 3.43469549411493e+83

   1. Initial program 10.6

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

   3. Applied *-un-lft-identity 10.6

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

   4. Applied sqrt-prod 10.6

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

   5. Applied times-frac 8.3

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

   6. Simplified 8.3

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

   8. Applied associate-*l* 8.1

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

   if 3.43469549411493e+83 < z

   1. Initial program 42.2

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

   3. Applied *-un-lft-identity 42.2

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

   4. Applied sqrt-prod 42.2

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

   5. Applied times-frac 39.7

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

   6. Simplified 39.7

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

   8. Applied associate-*l* 39.7

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

   9. Taylor expanded around inf 2.6

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

4. Final simplification 5.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.0932223177976295 \cdot 10^{154}:\\ \;\;\;\;\left(x \cdot y\right) \cdot -1\\ \mathbf{elif}\;z \le 3.43469549411492992 \cdot 10^{83}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(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)))))