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

Average Error: 25.0 → 6.2

Time: 14.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -4.610508646786433287550784748309774160666 \cdot 10^{104}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \le 9.925985792422320279557825298855191033448 \cdot 10^{124}:\\ \;\;\;\;x \cdot \frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r263307 = x;
        double r263308 = y;
        double r263309 = r263307 * r263308;
        double r263310 = z;
        double r263311 = r263309 * r263310;
        double r263312 = r263310 * r263310;
        double r263313 = t;
        double r263314 = a;
        double r263315 = r263313 * r263314;
        double r263316 = r263312 - r263315;
        double r263317 = sqrt(r263316);
        double r263318 = r263311 / r263317;
        return r263318;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r263319 = z;
        double r263320 = -4.610508646786433e+104;
        bool r263321 = r263319 <= r263320;
        double r263322 = x;
        double r263323 = y;
        double r263324 = r263322 * r263323;
        double r263325 = -r263324;
        double r263326 = 9.92598579242232e+124;
        bool r263327 = r263319 <= r263326;
        double r263328 = r263319 * r263319;
        double r263329 = t;
        double r263330 = a;
        double r263331 = r263329 * r263330;
        double r263332 = r263328 - r263331;
        double r263333 = sqrt(r263332);
        double r263334 = r263333 / r263319;
        double r263335 = r263323 / r263334;
        double r263336 = r263322 * r263335;
        double r263337 = r263327 ? r263336 : r263324;
        double r263338 = r263321 ? r263325 : r263337;
        return r263338;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	25.0
Target	7.7
Herbie	6.2

\[\begin{array}{l} \mathbf{if}\;z \lt -3.192130590385276419686361646843883646209 \cdot 10^{46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.976268120920894210257945708950453212935 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if z < -4.610508646786433e+104

   1. Initial program 45.4

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

   3. Applied associate-/l* 43.4

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

   5. Applied div-inv 43.4

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

   6. Simplified 43.4

      \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{\frac{z}{\sqrt{{z}^{2} - a \cdot t}}}\]

   7. Taylor expanded around -inf 2.1

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

   if -4.610508646786433e+104 < z < 9.92598579242232e+124

   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 associate-/l* 9.2

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

   5. Applied *-un-lft-identity 9.2

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

   6. Applied *-un-lft-identity 9.2

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

   7. Applied sqrt-prod 9.2

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

   8. Applied times-frac 9.2

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

   9. Applied times-frac 9.0

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

   10. Simplified 9.0

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

   if 9.92598579242232e+124 < z

   1. Initial program 47.3

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

   3. Applied associate-/l* 45.8

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

   5. Applied div-inv 45.8

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

   6. Simplified 45.8

      \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{\frac{z}{\sqrt{{z}^{2} - a \cdot t}}}\]

   7. Taylor expanded around inf 1.8

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

4. Final simplification 6.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.610508646786433287550784748309774160666 \cdot 10^{104}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \le 9.925985792422320279557825298855191033448 \cdot 10^{124}:\\ \;\;\;\;x \cdot \frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(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)))))