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

Average Error: 21.4 → 4.3

Time: 17.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} = -\infty:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \left(t \cdot \frac{a}{c}\right)\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -2.044945372715348494311419335429311721877 \cdot 10^{-258}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 0.0:\\ \;\;\;\;\frac{\sqrt[3]{\frac{b + \left(x \cdot 9\right) \cdot y}{z} - \left(a \cdot 4\right) \cdot t} \cdot \sqrt[3]{\frac{b + \left(x \cdot 9\right) \cdot y}{z} - \left(a \cdot 4\right) \cdot t}}{\frac{c}{\sqrt[3]{\frac{b + \left(x \cdot 9\right) \cdot y}{z} - \left(a \cdot 4\right) \cdot t}}}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 5.226249926821728616154563669432090090445 \cdot 10^{297}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\left(z \cdot c\right) \cdot \frac{1}{y}}\right) - 4 \cdot \frac{t}{\frac{c}{a}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r488671 = x;
        double r488672 = 9.0;
        double r488673 = r488671 * r488672;
        double r488674 = y;
        double r488675 = r488673 * r488674;
        double r488676 = z;
        double r488677 = 4.0;
        double r488678 = r488676 * r488677;
        double r488679 = t;
        double r488680 = r488678 * r488679;
        double r488681 = a;
        double r488682 = r488680 * r488681;
        double r488683 = r488675 - r488682;
        double r488684 = b;
        double r488685 = r488683 + r488684;
        double r488686 = c;
        double r488687 = r488676 * r488686;
        double r488688 = r488685 / r488687;
        return r488688;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r488689 = x;
        double r488690 = 9.0;
        double r488691 = r488689 * r488690;
        double r488692 = y;
        double r488693 = r488691 * r488692;
        double r488694 = z;
        double r488695 = 4.0;
        double r488696 = r488694 * r488695;
        double r488697 = t;
        double r488698 = r488696 * r488697;
        double r488699 = a;
        double r488700 = r488698 * r488699;
        double r488701 = r488693 - r488700;
        double r488702 = b;
        double r488703 = r488701 + r488702;
        double r488704 = c;
        double r488705 = r488694 * r488704;
        double r488706 = r488703 / r488705;
        double r488707 = -inf.0;
        bool r488708 = r488706 <= r488707;
        double r488709 = r488702 / r488705;
        double r488710 = r488705 / r488692;
        double r488711 = r488689 / r488710;
        double r488712 = r488690 * r488711;
        double r488713 = r488709 + r488712;
        double r488714 = r488699 / r488704;
        double r488715 = r488697 * r488714;
        double r488716 = r488695 * r488715;
        double r488717 = r488713 - r488716;
        double r488718 = -2.0449453727153485e-258;
        bool r488719 = r488706 <= r488718;
        double r488720 = 0.0;
        bool r488721 = r488706 <= r488720;
        double r488722 = r488702 + r488693;
        double r488723 = r488722 / r488694;
        double r488724 = r488699 * r488695;
        double r488725 = r488724 * r488697;
        double r488726 = r488723 - r488725;
        double r488727 = cbrt(r488726);
        double r488728 = r488727 * r488727;
        double r488729 = r488704 / r488727;
        double r488730 = r488728 / r488729;
        double r488731 = 5.2262499268217286e+297;
        bool r488732 = r488706 <= r488731;
        double r488733 = 1.0;
        double r488734 = r488733 / r488692;
        double r488735 = r488705 * r488734;
        double r488736 = r488689 / r488735;
        double r488737 = r488690 * r488736;
        double r488738 = r488709 + r488737;
        double r488739 = r488704 / r488699;
        double r488740 = r488697 / r488739;
        double r488741 = r488695 * r488740;
        double r488742 = r488738 - r488741;
        double r488743 = r488732 ? r488706 : r488742;
        double r488744 = r488721 ? r488730 : r488743;
        double r488745 = r488719 ? r488706 : r488744;
        double r488746 = r488708 ? r488717 : r488745;
        return r488746;
}

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	21.4
Target	14.7
Herbie	4.3

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

Derivation

1. Split input into 4 regimes

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

   1. Initial program 64.0

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

   2. Simplified 23.4

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

   3. Taylor expanded around 0 28.4

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

   5. Applied associate-/l* 12.9

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

   7. Applied *-un-lft-identity 12.9

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

   8. Applied times-frac 8.7

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

   9. Simplified 8.7

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

   if -inf.0 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -2.0449453727153485e-258 or 0.0 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 5.2262499268217286e+297

   1. Initial program 3.7

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

   if -2.0449453727153485e-258 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 0.0

   1. Initial program 35.0

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

   2. Simplified 0.9

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

   4. Applied add-cube-cbrt 1.6

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

   5. Applied associate-/l* 1.6

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

   if 5.2262499268217286e+297 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))

   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 27.3

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

   3. Taylor expanded around 0 30.4

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

   5. Applied associate-/l* 22.4

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

   7. Applied associate-/l* 15.8

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

   9. Applied div-inv 15.8

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

4. Final simplification 4.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} = -\infty:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \left(t \cdot \frac{a}{c}\right)\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -2.044945372715348494311419335429311721877 \cdot 10^{-258}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 0.0:\\ \;\;\;\;\frac{\sqrt[3]{\frac{b + \left(x \cdot 9\right) \cdot y}{z} - \left(a \cdot 4\right) \cdot t} \cdot \sqrt[3]{\frac{b + \left(x \cdot 9\right) \cdot y}{z} - \left(a \cdot 4\right) \cdot t}}{\frac{c}{\sqrt[3]{\frac{b + \left(x \cdot 9\right) \cdot y}{z} - \left(a \cdot 4\right) \cdot t}}}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 5.226249926821728616154563669432090090445 \cdot 10^{297}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\left(z \cdot c\right) \cdot \frac{1}{y}}\right) - 4 \cdot \frac{t}{\frac{c}{a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, J"
  :precision binary64

  :herbie-target
  (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) -1.100156740804105e-171) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) -0.0) (/ (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) z) c) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 1.1708877911747488e-53) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 2.876823679546137e+130) (- (+ (* (* 9 (/ y c)) (/ x z)) (/ b (* c z))) (* 4 (/ (* a t) c))) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 1.3838515042456319e+158) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (- (+ (* 9 (* (/ y (* c z)) x)) (/ b (* c z))) (* 4 (/ (* a t) c))))))))

  (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)))