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

Average Error: 20.6 → 9.4

Time: 16.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le 7.102196910018291020221091259853464339216 \cdot 10^{-289} \lor \neg \left(t \le 8.613546435620795812058914333288299977605 \cdot 10^{-89}\right):\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \left(\left(\frac{a}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{1}}\right) \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{c}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b} \cdot \sqrt[3]{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z} \cdot \frac{\sqrt[3]{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{c}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r516771 = x;
        double r516772 = 9.0;
        double r516773 = r516771 * r516772;
        double r516774 = y;
        double r516775 = r516773 * r516774;
        double r516776 = z;
        double r516777 = 4.0;
        double r516778 = r516776 * r516777;
        double r516779 = t;
        double r516780 = r516778 * r516779;
        double r516781 = a;
        double r516782 = r516780 * r516781;
        double r516783 = r516775 - r516782;
        double r516784 = b;
        double r516785 = r516783 + r516784;
        double r516786 = c;
        double r516787 = r516776 * r516786;
        double r516788 = r516785 / r516787;
        return r516788;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r516789 = t;
        double r516790 = 7.102196910018291e-289;
        bool r516791 = r516789 <= r516790;
        double r516792 = 8.613546435620796e-89;
        bool r516793 = r516789 <= r516792;
        double r516794 = !r516793;
        bool r516795 = r516791 || r516794;
        double r516796 = b;
        double r516797 = z;
        double r516798 = c;
        double r516799 = r516797 * r516798;
        double r516800 = r516796 / r516799;
        double r516801 = 9.0;
        double r516802 = x;
        double r516803 = y;
        double r516804 = r516799 / r516803;
        double r516805 = r516802 / r516804;
        double r516806 = r516801 * r516805;
        double r516807 = r516800 + r516806;
        double r516808 = 4.0;
        double r516809 = a;
        double r516810 = cbrt(r516798);
        double r516811 = r516810 * r516810;
        double r516812 = r516809 / r516811;
        double r516813 = cbrt(r516789);
        double r516814 = r516813 * r516813;
        double r516815 = 1.0;
        double r516816 = cbrt(r516815);
        double r516817 = r516814 / r516816;
        double r516818 = r516812 * r516817;
        double r516819 = r516813 / r516810;
        double r516820 = r516818 * r516819;
        double r516821 = r516808 * r516820;
        double r516822 = r516807 - r516821;
        double r516823 = r516802 * r516801;
        double r516824 = r516823 * r516803;
        double r516825 = r516797 * r516808;
        double r516826 = r516825 * r516789;
        double r516827 = r516826 * r516809;
        double r516828 = r516824 - r516827;
        double r516829 = r516828 + r516796;
        double r516830 = cbrt(r516829);
        double r516831 = r516830 * r516830;
        double r516832 = r516831 / r516797;
        double r516833 = r516830 / r516798;
        double r516834 = r516832 * r516833;
        double r516835 = r516795 ? r516822 : r516834;
        return r516835;
}

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	20.6
Target	14.5
Herbie	9.4

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

Derivation

1. Split input into 2 regimes

2. if t < 7.102196910018291e-289 or 8.613546435620796e-89 < t

   1. Initial program 22.4

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

   2. Taylor expanded around 0 12.3

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

   4. Applied add-cube-cbrt 12.7

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

   5. Applied times-frac 10.4

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

   7. Applied associate-/l* 9.5

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

   9. Applied *-un-lft-identity 9.5

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

   10. Applied cbrt-prod 9.5

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

   11. Applied add-cube-cbrt 9.6

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

   12. Applied times-frac 9.6

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

   13. Applied associate-*r* 9.1

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

   if 7.102196910018291e-289 < t < 8.613546435620796e-89

   1. Initial program 12.9

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

   3. Applied add-cube-cbrt 13.6

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

   4. Applied times-frac 10.8

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b} \cdot \sqrt[3]{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z} \cdot \frac{\sqrt[3]{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{c}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 9.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 7.102196910018291020221091259853464339216 \cdot 10^{-289} \lor \neg \left(t \le 8.613546435620795812058914333288299977605 \cdot 10^{-89}\right):\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \left(\left(\frac{a}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{1}}\right) \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{c}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b} \cdot \sqrt[3]{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z} \cdot \frac{\sqrt[3]{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{c}\\ \end{array}\]

Reproduce

herbie shell --seed 2019297 
(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)) -1.1001567408041049e-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)) 1.17088779117474882e-53) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 2.8768236795461372e130) (- (+ (* (* 9 (/ y c)) (/ x z)) (/ b (* c z))) (* 4 (/ (* a t) c))) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 1.3838515042456319e158) (/ (+ (- (* (* 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)))