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

Average Error: 19.6 → 2.7

Time: 17.5s

Precision: 64

Math

\[\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} = -\infty:\\ \;\;\;\;\left(\frac{\frac{b}{z}}{c} + \frac{x}{\frac{c \cdot z}{y}} \cdot 9.0\right) - \frac{t}{\frac{c}{a}} \cdot 4.0\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le -3.4143397402486294 \cdot 10^{-292}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 2.350132438968108 \cdot 10^{-119}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \left(b + \left(y \cdot x\right) \cdot 9.0\right) - a \cdot \left(4.0 \cdot t\right)}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 7.263326730960966 \cdot 10^{+307}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{c \cdot z} + \frac{x}{\frac{c}{\sqrt[3]{y}} \cdot \frac{z}{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot 9.0\right) - \frac{t}{\frac{c}{a}} \cdot 4.0\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r30848298 = x;
        double r30848299 = 9.0;
        double r30848300 = r30848298 * r30848299;
        double r30848301 = y;
        double r30848302 = r30848300 * r30848301;
        double r30848303 = z;
        double r30848304 = 4.0;
        double r30848305 = r30848303 * r30848304;
        double r30848306 = t;
        double r30848307 = r30848305 * r30848306;
        double r30848308 = a;
        double r30848309 = r30848307 * r30848308;
        double r30848310 = r30848302 - r30848309;
        double r30848311 = b;
        double r30848312 = r30848310 + r30848311;
        double r30848313 = c;
        double r30848314 = r30848303 * r30848313;
        double r30848315 = r30848312 / r30848314;
        return r30848315;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r30848316 = x;
        double r30848317 = 9.0;
        double r30848318 = r30848316 * r30848317;
        double r30848319 = y;
        double r30848320 = r30848318 * r30848319;
        double r30848321 = z;
        double r30848322 = 4.0;
        double r30848323 = r30848321 * r30848322;
        double r30848324 = t;
        double r30848325 = r30848323 * r30848324;
        double r30848326 = a;
        double r30848327 = r30848325 * r30848326;
        double r30848328 = r30848320 - r30848327;
        double r30848329 = b;
        double r30848330 = r30848328 + r30848329;
        double r30848331 = c;
        double r30848332 = r30848331 * r30848321;
        double r30848333 = r30848330 / r30848332;
        double r30848334 = -inf.0;
        bool r30848335 = r30848333 <= r30848334;
        double r30848336 = r30848329 / r30848321;
        double r30848337 = r30848336 / r30848331;
        double r30848338 = r30848332 / r30848319;
        double r30848339 = r30848316 / r30848338;
        double r30848340 = r30848339 * r30848317;
        double r30848341 = r30848337 + r30848340;
        double r30848342 = r30848331 / r30848326;
        double r30848343 = r30848324 / r30848342;
        double r30848344 = r30848343 * r30848322;
        double r30848345 = r30848341 - r30848344;
        double r30848346 = -3.4143397402486294e-292;
        bool r30848347 = r30848333 <= r30848346;
        double r30848348 = 2.350132438968108e-119;
        bool r30848349 = r30848333 <= r30848348;
        double r30848350 = 1.0;
        double r30848351 = r30848350 / r30848321;
        double r30848352 = r30848319 * r30848316;
        double r30848353 = r30848352 * r30848317;
        double r30848354 = r30848329 + r30848353;
        double r30848355 = r30848351 * r30848354;
        double r30848356 = r30848322 * r30848324;
        double r30848357 = r30848326 * r30848356;
        double r30848358 = r30848355 - r30848357;
        double r30848359 = r30848358 / r30848331;
        double r30848360 = 7.263326730960966e+307;
        bool r30848361 = r30848333 <= r30848360;
        double r30848362 = r30848329 / r30848332;
        double r30848363 = cbrt(r30848319);
        double r30848364 = r30848331 / r30848363;
        double r30848365 = r30848363 * r30848363;
        double r30848366 = r30848321 / r30848365;
        double r30848367 = r30848364 * r30848366;
        double r30848368 = r30848316 / r30848367;
        double r30848369 = r30848368 * r30848317;
        double r30848370 = r30848362 + r30848369;
        double r30848371 = r30848370 - r30848344;
        double r30848372 = r30848361 ? r30848333 : r30848371;
        double r30848373 = r30848349 ? r30848359 : r30848372;
        double r30848374 = r30848347 ? r30848333 : r30848373;
        double r30848375 = r30848335 ? r30848345 : r30848374;
        return r30848375;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	19.6
Target	13.7
Herbie	2.7

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.100156740804105 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(z \cdot 4.0\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.1708877911747488 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(z \cdot 4.0\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.876823679546137 \cdot 10^{+130}:\\ \;\;\;\;\left(\left(9.0 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4.0 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.3838515042456319 \cdot 10^{+158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(z \cdot 4.0\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9.0 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4.0 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -inf.0

   1. Initial program 60.2

      \[\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

   2. Simplified 22.1

      \[\leadsto \color{blue}{\frac{\frac{9.0 \cdot \left(y \cdot x\right) + b}{z} - a \cdot \left(4.0 \cdot t\right)}{c}}\]

   3. Taylor expanded around 0 26.2

      \[\leadsto \color{blue}{\left(9.0 \cdot \frac{x \cdot y}{z \cdot c} + \frac{b}{z \cdot c}\right) - 4.0 \cdot \frac{t \cdot a}{c}}\]
   4. Using strategy rm

   5. Applied associate-/l* 13.0

      \[\leadsto \left(9.0 \cdot \color{blue}{\frac{x}{\frac{z \cdot c}{y}}} + \frac{b}{z \cdot c}\right) - 4.0 \cdot \frac{t \cdot a}{c}\]
   6. Using strategy rm

   7. Applied associate-/l* 6.0

      \[\leadsto \left(9.0 \cdot \frac{x}{\frac{z \cdot c}{y}} + \frac{b}{z \cdot c}\right) - 4.0 \cdot \color{blue}{\frac{t}{\frac{c}{a}}}\]
   8. Using strategy rm

   9. Applied associate-/r* 6.7

      \[\leadsto \left(9.0 \cdot \frac{x}{\frac{z \cdot c}{y}} + \color{blue}{\frac{\frac{b}{z}}{c}}\right) - 4.0 \cdot \frac{t}{\frac{c}{a}}\]

   if -inf.0 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -3.4143397402486294e-292 or 2.350132438968108e-119 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 7.263326730960966e+307

   1. Initial program 0.6

      \[\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

   if -3.4143397402486294e-292 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 2.350132438968108e-119

   1. Initial program 26.4

      \[\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

   2. Simplified 0.8

      \[\leadsto \color{blue}{\frac{\frac{9.0 \cdot \left(y \cdot x\right) + b}{z} - a \cdot \left(4.0 \cdot t\right)}{c}}\]
   3. Using strategy rm

   4. Applied div-inv 0.8

      \[\leadsto \frac{\color{blue}{\left(9.0 \cdot \left(y \cdot x\right) + b\right) \cdot \frac{1}{z}} - a \cdot \left(4.0 \cdot t\right)}{c}\]

   if 7.263326730960966e+307 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))

   1. Initial program 61.5

      \[\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

   2. Simplified 26.7

      \[\leadsto \color{blue}{\frac{\frac{9.0 \cdot \left(y \cdot x\right) + b}{z} - a \cdot \left(4.0 \cdot t\right)}{c}}\]

   3. Taylor expanded around 0 30.4

      \[\leadsto \color{blue}{\left(9.0 \cdot \frac{x \cdot y}{z \cdot c} + \frac{b}{z \cdot c}\right) - 4.0 \cdot \frac{t \cdot a}{c}}\]
   4. Using strategy rm

   5. Applied associate-/l* 21.8

      \[\leadsto \left(9.0 \cdot \color{blue}{\frac{x}{\frac{z \cdot c}{y}}} + \frac{b}{z \cdot c}\right) - 4.0 \cdot \frac{t \cdot a}{c}\]
   6. Using strategy rm

   7. Applied associate-/l* 14.9

      \[\leadsto \left(9.0 \cdot \frac{x}{\frac{z \cdot c}{y}} + \frac{b}{z \cdot c}\right) - 4.0 \cdot \color{blue}{\frac{t}{\frac{c}{a}}}\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 15.0

      \[\leadsto \left(9.0 \cdot \frac{x}{\frac{z \cdot c}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}} + \frac{b}{z \cdot c}\right) - 4.0 \cdot \frac{t}{\frac{c}{a}}\]

   10. Applied times-frac 10.0

       \[\leadsto \left(9.0 \cdot \frac{x}{\color{blue}{\frac{z}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{c}{\sqrt[3]{y}}}} + \frac{b}{z \cdot c}\right) - 4.0 \cdot \frac{t}{\frac{c}{a}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 2.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} = -\infty:\\ \;\;\;\;\left(\frac{\frac{b}{z}}{c} + \frac{x}{\frac{c \cdot z}{y}} \cdot 9.0\right) - \frac{t}{\frac{c}{a}} \cdot 4.0\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le -3.4143397402486294 \cdot 10^{-292}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 2.350132438968108 \cdot 10^{-119}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \left(b + \left(y \cdot x\right) \cdot 9.0\right) - a \cdot \left(4.0 \cdot t\right)}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 7.263326730960966 \cdot 10^{+307}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{c \cdot z} + \frac{x}{\frac{c}{\sqrt[3]{y}} \cdot \frac{z}{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot 9.0\right) - \frac{t}{\frac{c}{a}} \cdot 4.0\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, J"

  :herbie-target
  (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) -1.100156740804105e-171) (/ (+ (- (* (* x 9.0) y) (* (* z 4.0) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) -0.0) (/ (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) z) c) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 1.1708877911747488e-53) (/ (+ (- (* (* x 9.0) y) (* (* z 4.0) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 2.876823679546137e+130) (- (+ (* (* 9.0 (/ y c)) (/ x z)) (/ b (* c z))) (* 4.0 (/ (* a t) c))) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 1.3838515042456319e+158) (/ (+ (- (* (* x 9.0) y) (* (* z 4.0) (* t a))) b) (* z c)) (- (+ (* 9.0 (* (/ y (* c z)) x)) (/ b (* c z))) (* 4.0 (/ (* a t) c))))))))

  (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)))