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

Average Error: 20.2 → 8.3

Time: 24.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le -9.88584605604387315427172643975722969846 \cdot 10^{302}:\\ \;\;\;\;\frac{\frac{1}{\frac{z}{\left(x \cdot 9\right) \cdot y + b}}}{c} - \frac{t \cdot \left(a \cdot 4\right)}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le -6.165302790627336852530598431195247451468 \cdot 10^{136}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 1.325458288189529837322656396878629664913 \cdot 10^{70}:\\ \;\;\;\;\frac{\frac{y \cdot x}{z} \cdot 9 + \frac{b}{z}}{c} - \frac{t \cdot \left(a \cdot 4\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{c} \cdot \left(\left(x \cdot 9\right) \cdot y + b\right) - \frac{a \cdot 4}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{t}{\sqrt[3]{c}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r32335596 = x;
        double r32335597 = 9.0;
        double r32335598 = r32335596 * r32335597;
        double r32335599 = y;
        double r32335600 = r32335598 * r32335599;
        double r32335601 = z;
        double r32335602 = 4.0;
        double r32335603 = r32335601 * r32335602;
        double r32335604 = t;
        double r32335605 = r32335603 * r32335604;
        double r32335606 = a;
        double r32335607 = r32335605 * r32335606;
        double r32335608 = r32335600 - r32335607;
        double r32335609 = b;
        double r32335610 = r32335608 + r32335609;
        double r32335611 = c;
        double r32335612 = r32335601 * r32335611;
        double r32335613 = r32335610 / r32335612;
        return r32335613;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r32335614 = x;
        double r32335615 = 9.0;
        double r32335616 = r32335614 * r32335615;
        double r32335617 = y;
        double r32335618 = r32335616 * r32335617;
        double r32335619 = z;
        double r32335620 = 4.0;
        double r32335621 = r32335619 * r32335620;
        double r32335622 = t;
        double r32335623 = r32335621 * r32335622;
        double r32335624 = a;
        double r32335625 = r32335623 * r32335624;
        double r32335626 = r32335618 - r32335625;
        double r32335627 = b;
        double r32335628 = r32335626 + r32335627;
        double r32335629 = c;
        double r32335630 = r32335629 * r32335619;
        double r32335631 = r32335628 / r32335630;
        double r32335632 = -9.885846056043873e+302;
        bool r32335633 = r32335631 <= r32335632;
        double r32335634 = 1.0;
        double r32335635 = r32335618 + r32335627;
        double r32335636 = r32335619 / r32335635;
        double r32335637 = r32335634 / r32335636;
        double r32335638 = r32335637 / r32335629;
        double r32335639 = r32335624 * r32335620;
        double r32335640 = r32335622 * r32335639;
        double r32335641 = r32335640 / r32335629;
        double r32335642 = r32335638 - r32335641;
        double r32335643 = -6.165302790627337e+136;
        bool r32335644 = r32335631 <= r32335643;
        double r32335645 = 1.3254582881895298e+70;
        bool r32335646 = r32335631 <= r32335645;
        double r32335647 = r32335617 * r32335614;
        double r32335648 = r32335647 / r32335619;
        double r32335649 = r32335648 * r32335615;
        double r32335650 = r32335627 / r32335619;
        double r32335651 = r32335649 + r32335650;
        double r32335652 = r32335651 / r32335629;
        double r32335653 = r32335652 - r32335641;
        double r32335654 = r32335634 / r32335619;
        double r32335655 = r32335654 / r32335629;
        double r32335656 = r32335655 * r32335635;
        double r32335657 = cbrt(r32335629);
        double r32335658 = r32335657 * r32335657;
        double r32335659 = r32335639 / r32335658;
        double r32335660 = r32335622 / r32335657;
        double r32335661 = r32335659 * r32335660;
        double r32335662 = r32335656 - r32335661;
        double r32335663 = r32335646 ? r32335653 : r32335662;
        double r32335664 = r32335644 ? r32335631 : r32335663;
        double r32335665 = r32335633 ? r32335642 : r32335664;
        return r32335665;
}

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	20.2
Target	14.7
Herbie	8.3

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

Derivation

1. Split input into 4 regimes

2. if (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -9.885846056043873e+302

   1. Initial program 61.9

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

   2. Simplified 22.6

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

   4. Applied div-sub 22.6

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

   6. Applied clear-num 22.6

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

   if -9.885846056043873e+302 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -6.165302790627337e+136

   1. Initial program 0.4

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

   if -6.165302790627337e+136 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 1.3254582881895298e+70

   1. Initial program 9.7

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

   2. Simplified 2.9

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

   4. Applied div-sub 2.9

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

   5. Taylor expanded around 0 2.9

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

   if 1.3254582881895298e+70 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))

   1. Initial program 31.8

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

   2. Simplified 21.8

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

   4. Applied div-sub 21.8

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

   6. Applied add-cube-cbrt 22.1

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

   7. Applied times-frac 20.3

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

   9. Applied *-un-lft-identity 20.3

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

   10. Applied div-inv 20.3

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

   11. Applied times-frac 15.0

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

   12. Simplified 15.0

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

4. Final simplification 8.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le -9.88584605604387315427172643975722969846 \cdot 10^{302}:\\ \;\;\;\;\frac{\frac{1}{\frac{z}{\left(x \cdot 9\right) \cdot y + b}}}{c} - \frac{t \cdot \left(a \cdot 4\right)}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le -6.165302790627336852530598431195247451468 \cdot 10^{136}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 1.325458288189529837322656396878629664913 \cdot 10^{70}:\\ \;\;\;\;\frac{\frac{y \cdot x}{z} \cdot 9 + \frac{b}{z}}{c} - \frac{t \cdot \left(a \cdot 4\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{c} \cdot \left(\left(x \cdot 9\right) \cdot y + b\right) - \frac{a \cdot 4}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{t}{\sqrt[3]{c}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(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)))