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

Average Error: 20.8 → 3.4

Time: 20.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} = -\infty:\\ \;\;\;\;\left(\frac{\frac{b}{c}}{z} + \frac{x \cdot 9}{\frac{c}{\frac{y}{z}}}\right) - \frac{t}{\frac{c}{a}} \cdot 4\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le -48321584405561035694420036419584:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 0.0:\\ \;\;\;\;\frac{1}{\frac{c}{\frac{b + \left(y \cdot x\right) \cdot 9}{z} - \left(t \cdot a\right) \cdot 4}}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 7.468702005485942468554359890686701640407 \cdot 10^{289}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{b}{c}}{z} + \frac{x \cdot 9}{\frac{c}{\frac{y}{z}}}\right) - \frac{t}{\frac{c}{a}} \cdot 4\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r621963 = x;
        double r621964 = 9.0;
        double r621965 = r621963 * r621964;
        double r621966 = y;
        double r621967 = r621965 * r621966;
        double r621968 = z;
        double r621969 = 4.0;
        double r621970 = r621968 * r621969;
        double r621971 = t;
        double r621972 = r621970 * r621971;
        double r621973 = a;
        double r621974 = r621972 * r621973;
        double r621975 = r621967 - r621974;
        double r621976 = b;
        double r621977 = r621975 + r621976;
        double r621978 = c;
        double r621979 = r621968 * r621978;
        double r621980 = r621977 / r621979;
        return r621980;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r621981 = x;
        double r621982 = 9.0;
        double r621983 = r621981 * r621982;
        double r621984 = y;
        double r621985 = r621983 * r621984;
        double r621986 = z;
        double r621987 = 4.0;
        double r621988 = r621986 * r621987;
        double r621989 = t;
        double r621990 = r621988 * r621989;
        double r621991 = a;
        double r621992 = r621990 * r621991;
        double r621993 = r621985 - r621992;
        double r621994 = b;
        double r621995 = r621993 + r621994;
        double r621996 = c;
        double r621997 = r621996 * r621986;
        double r621998 = r621995 / r621997;
        double r621999 = -inf.0;
        bool r622000 = r621998 <= r621999;
        double r622001 = r621994 / r621996;
        double r622002 = r622001 / r621986;
        double r622003 = r621984 / r621986;
        double r622004 = r621996 / r622003;
        double r622005 = r621983 / r622004;
        double r622006 = r622002 + r622005;
        double r622007 = r621996 / r621991;
        double r622008 = r621989 / r622007;
        double r622009 = r622008 * r621987;
        double r622010 = r622006 - r622009;
        double r622011 = -4.832158440556104e+31;
        bool r622012 = r621998 <= r622011;
        double r622013 = 0.0;
        bool r622014 = r621998 <= r622013;
        double r622015 = 1.0;
        double r622016 = r621984 * r621981;
        double r622017 = r622016 * r621982;
        double r622018 = r621994 + r622017;
        double r622019 = r622018 / r621986;
        double r622020 = r621989 * r621991;
        double r622021 = r622020 * r621987;
        double r622022 = r622019 - r622021;
        double r622023 = r621996 / r622022;
        double r622024 = r622015 / r622023;
        double r622025 = 7.4687020054859425e+289;
        bool r622026 = r621998 <= r622025;
        double r622027 = r622026 ? r621998 : r622010;
        double r622028 = r622014 ? r622024 : r622027;
        double r622029 = r622012 ? r621998 : r622028;
        double r622030 = r622000 ? r622010 : r622029;
        return r622030;
}

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	20.8
Target	14.8
Herbie	3.4

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

Derivation

1. Split input into 3 regimes

2. if (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -inf.0 or 7.4687020054859425e+289 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))

   1. Initial program 61.4

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

   2. Simplified 26.6

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

   4. Applied div-sub 26.6

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

   5. Simplified 30.5

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

   6. Simplified 30.4

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

   8. Applied associate-/l* 24.3

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

   9. Taylor expanded around 0 24.3

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

   10. Simplified 10.7

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

   if -inf.0 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -4.832158440556104e+31 or 0.0 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 7.4687020054859425e+289

   1. Initial program 4.4

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

   if -4.832158440556104e+31 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 0.0

   1. Initial program 11.8

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

   2. Simplified 0.7

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

   4. Applied clear-num 1.0

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

   5. Simplified 1.0

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

4. Final simplification 3.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} = -\infty:\\ \;\;\;\;\left(\frac{\frac{b}{c}}{z} + \frac{x \cdot 9}{\frac{c}{\frac{y}{z}}}\right) - \frac{t}{\frac{c}{a}} \cdot 4\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le -48321584405561035694420036419584:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 0.0:\\ \;\;\;\;\frac{1}{\frac{c}{\frac{b + \left(y \cdot x\right) \cdot 9}{z} - \left(t \cdot a\right) \cdot 4}}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 7.468702005485942468554359890686701640407 \cdot 10^{289}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{b}{c}}{z} + \frac{x \cdot 9}{\frac{c}{\frac{y}{z}}}\right) - \frac{t}{\frac{c}{a}} \cdot 4\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(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)))