Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, I

Average Error: 8.1 → 6.0

Time: 8.3s

Precision: 64

Math

\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le \frac{-191715806434571}{4835703278458516698824704}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\frac{x}{a}}{\frac{1}{y}} - \left(\frac{9}{2} \cdot \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right) \cdot \frac{z}{\sqrt[3]{a}}\\ \mathbf{elif}\;z \le \frac{-4015161094643863}{7.371020360979572953596786290992712677572 \cdot 10^{165}}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{elif}\;z \le \frac{-5872900659725985}{4.872657005699999540176691193937594155438 \cdot 10^{288}}:\\ \;\;\;\;\frac{1}{2} \cdot \left(x \cdot \frac{y}{a}\right) - \frac{9}{2} \cdot \frac{t \cdot z}{a}\\ \mathbf{elif}\;z \le \frac{8992166776767629}{4.267529237043106735411105146061604990173 \cdot 10^{242}}:\\ \;\;\;\;\frac{1}{2} \cdot \left(\frac{x}{a} \cdot y\right) - \frac{9}{2} \cdot \frac{t \cdot z}{a}\\ \mathbf{elif}\;z \le 8.760394845593749117763722532018894881612 \cdot 10^{105}:\\ \;\;\;\;\frac{1}{2} \cdot \left(x \cdot \frac{y}{a}\right) - \frac{9}{2} \cdot \frac{t \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{x}{\frac{a}{y}} - \frac{9 \cdot t}{2} \cdot \frac{z}{a}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r623255 = x;
        double r623256 = y;
        double r623257 = r623255 * r623256;
        double r623258 = z;
        double r623259 = 9.0;
        double r623260 = r623258 * r623259;
        double r623261 = t;
        double r623262 = r623260 * r623261;
        double r623263 = r623257 - r623262;
        double r623264 = a;
        double r623265 = 2.0;
        double r623266 = r623264 * r623265;
        double r623267 = r623263 / r623266;
        return r623267;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r623268 = z;
        double r623269 = -191715806434571.0;
        double r623270 = 4.835703278458517e+24;
        double r623271 = r623269 / r623270;
        bool r623272 = r623268 <= r623271;
        double r623273 = 1.0;
        double r623274 = 2.0;
        double r623275 = r623273 / r623274;
        double r623276 = x;
        double r623277 = a;
        double r623278 = r623276 / r623277;
        double r623279 = 1.0;
        double r623280 = y;
        double r623281 = r623279 / r623280;
        double r623282 = r623278 / r623281;
        double r623283 = r623275 * r623282;
        double r623284 = 9.0;
        double r623285 = r623284 / r623274;
        double r623286 = t;
        double r623287 = cbrt(r623277);
        double r623288 = r623287 * r623287;
        double r623289 = r623286 / r623288;
        double r623290 = r623285 * r623289;
        double r623291 = r623268 / r623287;
        double r623292 = r623290 * r623291;
        double r623293 = r623283 - r623292;
        double r623294 = -4015161094643863.0;
        double r623295 = 7.371020360979573e+165;
        double r623296 = r623294 / r623295;
        bool r623297 = r623268 <= r623296;
        double r623298 = r623276 * r623280;
        double r623299 = r623284 * r623286;
        double r623300 = r623268 * r623299;
        double r623301 = r623298 - r623300;
        double r623302 = r623277 * r623274;
        double r623303 = r623301 / r623302;
        double r623304 = -5872900659725985.0;
        double r623305 = 4.8726570057e+288;
        double r623306 = r623304 / r623305;
        bool r623307 = r623268 <= r623306;
        double r623308 = r623280 / r623277;
        double r623309 = r623276 * r623308;
        double r623310 = r623275 * r623309;
        double r623311 = r623286 * r623268;
        double r623312 = r623311 / r623277;
        double r623313 = r623285 * r623312;
        double r623314 = r623310 - r623313;
        double r623315 = 8992166776767629.0;
        double r623316 = 4.267529237043107e+242;
        double r623317 = r623315 / r623316;
        bool r623318 = r623268 <= r623317;
        double r623319 = r623278 * r623280;
        double r623320 = r623275 * r623319;
        double r623321 = r623320 - r623313;
        double r623322 = 8.760394845593749e+105;
        bool r623323 = r623268 <= r623322;
        double r623324 = r623277 / r623280;
        double r623325 = r623276 / r623324;
        double r623326 = r623275 * r623325;
        double r623327 = r623299 / r623274;
        double r623328 = r623268 / r623277;
        double r623329 = r623327 * r623328;
        double r623330 = r623326 - r623329;
        double r623331 = r623323 ? r623314 : r623330;
        double r623332 = r623318 ? r623321 : r623331;
        double r623333 = r623307 ? r623314 : r623332;
        double r623334 = r623297 ? r623303 : r623333;
        double r623335 = r623272 ? r623293 : r623334;
        return r623335;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	8.1
Target	5.8
Herbie	6.0

\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709043451944897028999329376 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.144030707833976090627817222818061808815 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

1. Split input into 5 regimes

2. if z < -3.964589955066153e-11

   1. Initial program 11.9

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

   2. Taylor expanded around 0 11.8

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}}\]

   3. Simplified 11.8

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{a} - \frac{9}{2} \cdot \frac{t \cdot z}{a}}\]
   4. Using strategy rm

   5. Applied associate-/l* 11.8

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{x}{\frac{a}{y}}} - \frac{9}{2} \cdot \frac{t \cdot z}{a}\]
   6. Using strategy rm

   7. Applied add-cube-cbrt 12.3

      \[\leadsto \frac{1}{2} \cdot \frac{x}{\frac{a}{y}} - \frac{9}{2} \cdot \frac{t \cdot z}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}\]

   8. Applied times-frac 6.4

      \[\leadsto \frac{1}{2} \cdot \frac{x}{\frac{a}{y}} - \frac{9}{2} \cdot \color{blue}{\left(\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}\right)}\]

   9. Applied associate-*r* 6.5

      \[\leadsto \frac{1}{2} \cdot \frac{x}{\frac{a}{y}} - \color{blue}{\left(\frac{9}{2} \cdot \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right) \cdot \frac{z}{\sqrt[3]{a}}}\]
   10. Using strategy rm

   11. Applied div-inv 6.5

       \[\leadsto \frac{1}{2} \cdot \frac{x}{\color{blue}{a \cdot \frac{1}{y}}} - \left(\frac{9}{2} \cdot \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right) \cdot \frac{z}{\sqrt[3]{a}}\]

   12. Applied associate-/r* 5.8

       \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\frac{x}{a}}{\frac{1}{y}}} - \left(\frac{9}{2} \cdot \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right) \cdot \frac{z}{\sqrt[3]{a}}\]

   if -3.964589955066153e-11 < z < -5.447225618720537e-151

   1. Initial program 3.7

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
   2. Using strategy rm

   3. Applied associate-*l* 3.7

      \[\leadsto \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2}\]

   if -5.447225618720537e-151 < z < -1.205276844410741e-273 or 2.1071131039275873e-227 < z < 8.760394845593749e+105

   1. Initial program 5.7

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

   2. Taylor expanded around 0 5.7

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}}\]

   3. Simplified 5.7

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{a} - \frac{9}{2} \cdot \frac{t \cdot z}{a}}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity 5.7

      \[\leadsto \frac{1}{2} \cdot \frac{x \cdot y}{\color{blue}{1 \cdot a}} - \frac{9}{2} \cdot \frac{t \cdot z}{a}\]

   6. Applied times-frac 6.2

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\frac{x}{1} \cdot \frac{y}{a}\right)} - \frac{9}{2} \cdot \frac{t \cdot z}{a}\]

   7. Simplified 6.2

      \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{x} \cdot \frac{y}{a}\right) - \frac{9}{2} \cdot \frac{t \cdot z}{a}\]

   if -1.205276844410741e-273 < z < 2.1071131039275873e-227

   1. Initial program 3.9

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

   2. Taylor expanded around 0 4.0

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}}\]

   3. Simplified 4.0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{a} - \frac{9}{2} \cdot \frac{t \cdot z}{a}}\]
   4. Using strategy rm

   5. Applied associate-/l* 6.8

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{x}{\frac{a}{y}}} - \frac{9}{2} \cdot \frac{t \cdot z}{a}\]
   6. Using strategy rm

   7. Applied associate-/r/ 5.5

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\frac{x}{a} \cdot y\right)} - \frac{9}{2} \cdot \frac{t \cdot z}{a}\]

   if 8.760394845593749e+105 < z

   1. Initial program 15.8

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

   2. Taylor expanded around 0 15.6

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}}\]

   3. Simplified 15.6

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{a} - \frac{9}{2} \cdot \frac{t \cdot z}{a}}\]
   4. Using strategy rm

   5. Applied associate-/l* 14.2

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{x}{\frac{a}{y}}} - \frac{9}{2} \cdot \frac{t \cdot z}{a}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 14.2

      \[\leadsto \frac{1}{2} \cdot \frac{x}{\frac{a}{y}} - \frac{9}{2} \cdot \frac{t \cdot z}{\color{blue}{1 \cdot a}}\]

   8. Applied times-frac 8.4

      \[\leadsto \frac{1}{2} \cdot \frac{x}{\frac{a}{y}} - \frac{9}{2} \cdot \color{blue}{\left(\frac{t}{1} \cdot \frac{z}{a}\right)}\]

   9. Applied associate-*r* 8.4

      \[\leadsto \frac{1}{2} \cdot \frac{x}{\frac{a}{y}} - \color{blue}{\left(\frac{9}{2} \cdot \frac{t}{1}\right) \cdot \frac{z}{a}}\]

   10. Simplified 8.4

       \[\leadsto \frac{1}{2} \cdot \frac{x}{\frac{a}{y}} - \color{blue}{\frac{9 \cdot t}{2}} \cdot \frac{z}{a}\]
3. Recombined 5 regimes into one program.

4. Final simplification 6.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le \frac{-191715806434571}{4835703278458516698824704}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\frac{x}{a}}{\frac{1}{y}} - \left(\frac{9}{2} \cdot \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right) \cdot \frac{z}{\sqrt[3]{a}}\\ \mathbf{elif}\;z \le \frac{-4015161094643863}{7.371020360979572953596786290992712677572 \cdot 10^{165}}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{elif}\;z \le \frac{-5872900659725985}{4.872657005699999540176691193937594155438 \cdot 10^{288}}:\\ \;\;\;\;\frac{1}{2} \cdot \left(x \cdot \frac{y}{a}\right) - \frac{9}{2} \cdot \frac{t \cdot z}{a}\\ \mathbf{elif}\;z \le \frac{8992166776767629}{4.267529237043106735411105146061604990173 \cdot 10^{242}}:\\ \;\;\;\;\frac{1}{2} \cdot \left(\frac{x}{a} \cdot y\right) - \frac{9}{2} \cdot \frac{t \cdot z}{a}\\ \mathbf{elif}\;z \le 8.760394845593749117763722532018894881612 \cdot 10^{105}:\\ \;\;\;\;\frac{1}{2} \cdot \left(x \cdot \frac{y}{a}\right) - \frac{9}{2} \cdot \frac{t \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{x}{\frac{a}{y}} - \frac{9 \cdot t}{2} \cdot \frac{z}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :herbie-target
  (if (< a -2.090464557976709e86) (- (* 0.5 (/ (* y x) a)) (* 4.5 (/ t (/ a z)))) (if (< a 2.14403070783397609e99) (/ (- (* x y) (* z (* 9 t))) (* a 2)) (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5)))))

  (/ (- (* x y) (* (* z 9) t)) (* a 2)))