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

Average Error: 20.3 → 3.1

Time: 20.1s

Precision: 64

Math

\[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} = -\infty:\\ \;\;\;\;\frac{\frac{\sqrt[3]{b} \cdot \sqrt[3]{b}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}{\frac{c}{\sqrt[3]{b}} \cdot \sqrt[3]{z}} + \left(\left(\frac{9}{c} \cdot \frac{x}{z}\right) \cdot y - \frac{t}{\frac{c}{a}} \cdot 4\right)\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le -1.143656966902138628653369801663880934825 \cdot 10^{-208}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 0.0:\\ \;\;\;\;\left(\frac{\frac{x}{\frac{c}{y \cdot 9}}}{z} - \frac{t}{\frac{c}{a}} \cdot 4\right) + \frac{\frac{b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 1.646526307585928545464016302227471938453 \cdot 10^{294}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{b} \cdot \sqrt[3]{b}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}{\frac{c}{\sqrt[3]{b}} \cdot \sqrt[3]{z}} + \left(\left(\frac{9}{c} \cdot \frac{x}{z}\right) \cdot y - \frac{t}{\frac{c}{a}} \cdot 4\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r456441 = x;
        double r456442 = 9.0;
        double r456443 = r456441 * r456442;
        double r456444 = y;
        double r456445 = r456443 * r456444;
        double r456446 = z;
        double r456447 = 4.0;
        double r456448 = r456446 * r456447;
        double r456449 = t;
        double r456450 = r456448 * r456449;
        double r456451 = a;
        double r456452 = r456450 * r456451;
        double r456453 = r456445 - r456452;
        double r456454 = b;
        double r456455 = r456453 + r456454;
        double r456456 = c;
        double r456457 = r456446 * r456456;
        double r456458 = r456455 / r456457;
        return r456458;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r456459 = x;
        double r456460 = 9.0;
        double r456461 = r456459 * r456460;
        double r456462 = y;
        double r456463 = r456461 * r456462;
        double r456464 = z;
        double r456465 = 4.0;
        double r456466 = r456464 * r456465;
        double r456467 = t;
        double r456468 = r456466 * r456467;
        double r456469 = a;
        double r456470 = r456468 * r456469;
        double r456471 = r456463 - r456470;
        double r456472 = b;
        double r456473 = r456471 + r456472;
        double r456474 = c;
        double r456475 = r456474 * r456464;
        double r456476 = r456473 / r456475;
        double r456477 = -inf.0;
        bool r456478 = r456476 <= r456477;
        double r456479 = cbrt(r456472);
        double r456480 = r456479 * r456479;
        double r456481 = cbrt(r456464);
        double r456482 = r456481 * r456481;
        double r456483 = r456480 / r456482;
        double r456484 = r456474 / r456479;
        double r456485 = r456484 * r456481;
        double r456486 = r456483 / r456485;
        double r456487 = r456460 / r456474;
        double r456488 = r456459 / r456464;
        double r456489 = r456487 * r456488;
        double r456490 = r456489 * r456462;
        double r456491 = r456474 / r456469;
        double r456492 = r456467 / r456491;
        double r456493 = r456492 * r456465;
        double r456494 = r456490 - r456493;
        double r456495 = r456486 + r456494;
        double r456496 = -1.1436569669021386e-208;
        bool r456497 = r456476 <= r456496;
        double r456498 = 0.0;
        bool r456499 = r456476 <= r456498;
        double r456500 = r456462 * r456460;
        double r456501 = r456474 / r456500;
        double r456502 = r456459 / r456501;
        double r456503 = r456502 / r456464;
        double r456504 = r456503 - r456493;
        double r456505 = r456472 / r456464;
        double r456506 = r456505 / r456474;
        double r456507 = r456504 + r456506;
        double r456508 = 1.6465263075859285e+294;
        bool r456509 = r456476 <= r456508;
        double r456510 = r456509 ? r456476 : r456495;
        double r456511 = r456499 ? r456507 : r456510;
        double r456512 = r456497 ? r456476 : r456511;
        double r456513 = r456478 ? r456495 : r456512;
        return r456513;
}

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	20.3
Target	14.3
Herbie	3.1

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.100156740804104887233830094663413900721 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.170887791174748819600820354912645756062 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.876823679546137226963937101710277849382 \cdot 10^{130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.383851504245631860711731716196098366993 \cdot 10^{158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -inf.0 or 1.6465263075859285e+294 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))

   1. Initial program 61.8

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

   2. Simplified 26.9

      \[\leadsto \color{blue}{\frac{\frac{\left(x \cdot 9\right) \cdot y + b}{z} - \left(4 \cdot a\right) \cdot t}{c}}\]

   3. Taylor expanded around 0 30.6

      \[\leadsto \color{blue}{\left(9 \cdot \frac{x \cdot y}{z \cdot c} + \frac{b}{z \cdot c}\right) - 4 \cdot \frac{t \cdot a}{c}}\]

   4. Simplified 17.8

      \[\leadsto \color{blue}{\frac{\frac{b}{z}}{c} + \left(\left(\frac{x}{z} \cdot y\right) \cdot \frac{9}{c} - \frac{t \cdot a}{c} \cdot 4\right)}\]
   5. Using strategy rm

   6. Applied associate-/l* 13.4

      \[\leadsto \frac{\frac{b}{z}}{c} + \left(\left(\frac{x}{z} \cdot y\right) \cdot \frac{9}{c} - \color{blue}{\frac{t}{\frac{c}{a}}} \cdot 4\right)\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 13.5

      \[\leadsto \frac{\frac{b}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}}{c} + \left(\left(\frac{x}{z} \cdot y\right) \cdot \frac{9}{c} - \frac{t}{\frac{c}{a}} \cdot 4\right)\]

   9. Applied add-cube-cbrt 13.5

      \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \sqrt[3]{b}}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}{c} + \left(\left(\frac{x}{z} \cdot y\right) \cdot \frac{9}{c} - \frac{t}{\frac{c}{a}} \cdot 4\right)\]

   10. Applied times-frac 13.5

       \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{b} \cdot \sqrt[3]{b}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\sqrt[3]{b}}{\sqrt[3]{z}}}}{c} + \left(\left(\frac{x}{z} \cdot y\right) \cdot \frac{9}{c} - \frac{t}{\frac{c}{a}} \cdot 4\right)\]

   11. Applied associate-/l* 12.8

       \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{b} \cdot \sqrt[3]{b}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}{\frac{c}{\frac{\sqrt[3]{b}}{\sqrt[3]{z}}}}} + \left(\left(\frac{x}{z} \cdot y\right) \cdot \frac{9}{c} - \frac{t}{\frac{c}{a}} \cdot 4\right)\]

   12. Simplified 13.0

       \[\leadsto \frac{\frac{\sqrt[3]{b} \cdot \sqrt[3]{b}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}{\color{blue}{\frac{c}{\sqrt[3]{b}} \cdot \sqrt[3]{z}}} + \left(\left(\frac{x}{z} \cdot y\right) \cdot \frac{9}{c} - \frac{t}{\frac{c}{a}} \cdot 4\right)\]
   13. Using strategy rm

   14. Applied pow1 13.0

       \[\leadsto \frac{\frac{\sqrt[3]{b} \cdot \sqrt[3]{b}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}{\frac{c}{\sqrt[3]{b}} \cdot \sqrt[3]{z}} + \left(\left(\frac{x}{z} \cdot y\right) \cdot \color{blue}{{\left(\frac{9}{c}\right)}^{1}} - \frac{t}{\frac{c}{a}} \cdot 4\right)\]

   15. Applied pow1 13.0

       \[\leadsto \frac{\frac{\sqrt[3]{b} \cdot \sqrt[3]{b}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}{\frac{c}{\sqrt[3]{b}} \cdot \sqrt[3]{z}} + \left(\left(\frac{x}{z} \cdot \color{blue}{{y}^{1}}\right) \cdot {\left(\frac{9}{c}\right)}^{1} - \frac{t}{\frac{c}{a}} \cdot 4\right)\]

   16. Applied pow1 13.0

       \[\leadsto \frac{\frac{\sqrt[3]{b} \cdot \sqrt[3]{b}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}{\frac{c}{\sqrt[3]{b}} \cdot \sqrt[3]{z}} + \left(\left(\color{blue}{{\left(\frac{x}{z}\right)}^{1}} \cdot {y}^{1}\right) \cdot {\left(\frac{9}{c}\right)}^{1} - \frac{t}{\frac{c}{a}} \cdot 4\right)\]

   17. Applied pow-prod-down 13.0

       \[\leadsto \frac{\frac{\sqrt[3]{b} \cdot \sqrt[3]{b}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}{\frac{c}{\sqrt[3]{b}} \cdot \sqrt[3]{z}} + \left(\color{blue}{{\left(\frac{x}{z} \cdot y\right)}^{1}} \cdot {\left(\frac{9}{c}\right)}^{1} - \frac{t}{\frac{c}{a}} \cdot 4\right)\]

   18. Applied pow-prod-down 13.0

       \[\leadsto \frac{\frac{\sqrt[3]{b} \cdot \sqrt[3]{b}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}{\frac{c}{\sqrt[3]{b}} \cdot \sqrt[3]{z}} + \left(\color{blue}{{\left(\left(\frac{x}{z} \cdot y\right) \cdot \frac{9}{c}\right)}^{1}} - \frac{t}{\frac{c}{a}} \cdot 4\right)\]

   19. Simplified 9.2

       \[\leadsto \frac{\frac{\sqrt[3]{b} \cdot \sqrt[3]{b}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}{\frac{c}{\sqrt[3]{b}} \cdot \sqrt[3]{z}} + \left({\color{blue}{\left(\left(\frac{x}{z} \cdot \frac{9}{c}\right) \cdot y\right)}}^{1} - \frac{t}{\frac{c}{a}} \cdot 4\right)\]

   if -inf.0 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -1.1436569669021386e-208 or 0.0 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 1.6465263075859285e+294

   1. Initial program 3.5

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

   if -1.1436569669021386e-208 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 0.0

   1. Initial program 30.3

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

   2. Simplified 0.3

      \[\leadsto \color{blue}{\frac{\frac{\left(x \cdot 9\right) \cdot y + b}{z} - \left(4 \cdot a\right) \cdot t}{c}}\]

   3. Taylor expanded around 0 19.5

      \[\leadsto \color{blue}{\left(9 \cdot \frac{x \cdot y}{z \cdot c} + \frac{b}{z \cdot c}\right) - 4 \cdot \frac{t \cdot a}{c}}\]

   4. Simplified 1.5

      \[\leadsto \color{blue}{\frac{\frac{b}{z}}{c} + \left(\left(\frac{x}{z} \cdot y\right) \cdot \frac{9}{c} - \frac{t \cdot a}{c} \cdot 4\right)}\]
   5. Using strategy rm

   6. Applied associate-/l* 3.4

      \[\leadsto \frac{\frac{b}{z}}{c} + \left(\left(\frac{x}{z} \cdot y\right) \cdot \frac{9}{c} - \color{blue}{\frac{t}{\frac{c}{a}}} \cdot 4\right)\]
   7. Using strategy rm

   8. Applied pow1 3.4

      \[\leadsto \frac{\frac{b}{z}}{c} + \left(\left(\frac{x}{z} \cdot y\right) \cdot \color{blue}{{\left(\frac{9}{c}\right)}^{1}} - \frac{t}{\frac{c}{a}} \cdot 4\right)\]

   9. Applied pow1 3.4

      \[\leadsto \frac{\frac{b}{z}}{c} + \left(\left(\frac{x}{z} \cdot \color{blue}{{y}^{1}}\right) \cdot {\left(\frac{9}{c}\right)}^{1} - \frac{t}{\frac{c}{a}} \cdot 4\right)\]

   10. Applied pow1 3.4

       \[\leadsto \frac{\frac{b}{z}}{c} + \left(\left(\color{blue}{{\left(\frac{x}{z}\right)}^{1}} \cdot {y}^{1}\right) \cdot {\left(\frac{9}{c}\right)}^{1} - \frac{t}{\frac{c}{a}} \cdot 4\right)\]

   11. Applied pow-prod-down 3.4

       \[\leadsto \frac{\frac{b}{z}}{c} + \left(\color{blue}{{\left(\frac{x}{z} \cdot y\right)}^{1}} \cdot {\left(\frac{9}{c}\right)}^{1} - \frac{t}{\frac{c}{a}} \cdot 4\right)\]

   12. Applied pow-prod-down 3.4

       \[\leadsto \frac{\frac{b}{z}}{c} + \left(\color{blue}{{\left(\left(\frac{x}{z} \cdot y\right) \cdot \frac{9}{c}\right)}^{1}} - \frac{t}{\frac{c}{a}} \cdot 4\right)\]

   13. Simplified 4.4

       \[\leadsto \frac{\frac{b}{z}}{c} + \left({\color{blue}{\left(\frac{\frac{x}{\frac{c}{y \cdot 9}}}{z}\right)}}^{1} - \frac{t}{\frac{c}{a}} \cdot 4\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 3.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} = -\infty:\\ \;\;\;\;\frac{\frac{\sqrt[3]{b} \cdot \sqrt[3]{b}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}{\frac{c}{\sqrt[3]{b}} \cdot \sqrt[3]{z}} + \left(\left(\frac{9}{c} \cdot \frac{x}{z}\right) \cdot y - \frac{t}{\frac{c}{a}} \cdot 4\right)\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le -1.143656966902138628653369801663880934825 \cdot 10^{-208}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 0.0:\\ \;\;\;\;\left(\frac{\frac{x}{\frac{c}{y \cdot 9}}}{z} - \frac{t}{\frac{c}{a}} \cdot 4\right) + \frac{\frac{b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 1.646526307585928545464016302227471938453 \cdot 10^{294}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{b} \cdot \sqrt[3]{b}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}{\frac{c}{\sqrt[3]{b}} \cdot \sqrt[3]{z}} + \left(\left(\frac{9}{c} \cdot \frac{x}{z}\right) \cdot y - \frac{t}{\frac{c}{a}} \cdot 4\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(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)))