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

Average Error: 20.8 → 3.4

Time: 20.9s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r526996 = x;
        double r526997 = 9.0;
        double r526998 = r526996 * r526997;
        double r526999 = y;
        double r527000 = r526998 * r526999;
        double r527001 = z;
        double r527002 = 4.0;
        double r527003 = r527001 * r527002;
        double r527004 = t;
        double r527005 = r527003 * r527004;
        double r527006 = a;
        double r527007 = r527005 * r527006;
        double r527008 = r527000 - r527007;
        double r527009 = b;
        double r527010 = r527008 + r527009;
        double r527011 = c;
        double r527012 = r527001 * r527011;
        double r527013 = r527010 / r527012;
        return r527013;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r527014 = x;
        double r527015 = 9.0;
        double r527016 = r527014 * r527015;
        double r527017 = y;
        double r527018 = r527016 * r527017;
        double r527019 = z;
        double r527020 = 4.0;
        double r527021 = r527019 * r527020;
        double r527022 = t;
        double r527023 = r527021 * r527022;
        double r527024 = a;
        double r527025 = r527023 * r527024;
        double r527026 = r527018 - r527025;
        double r527027 = b;
        double r527028 = r527026 + r527027;
        double r527029 = c;
        double r527030 = r527029 * r527019;
        double r527031 = r527028 / r527030;
        double r527032 = -inf.0;
        bool r527033 = r527031 <= r527032;
        double r527034 = r527027 / r527029;
        double r527035 = r527034 / r527019;
        double r527036 = r527017 / r527019;
        double r527037 = r527029 / r527036;
        double r527038 = r527016 / r527037;
        double r527039 = r527035 + r527038;
        double r527040 = r527029 / r527024;
        double r527041 = r527022 / r527040;
        double r527042 = r527041 * r527020;
        double r527043 = r527039 - r527042;
        double r527044 = -4.832158440556104e+31;
        bool r527045 = r527031 <= r527044;
        double r527046 = 0.0;
        bool r527047 = r527031 <= r527046;
        double r527048 = 1.0;
        double r527049 = r527017 * r527014;
        double r527050 = r527049 * r527015;
        double r527051 = r527027 + r527050;
        double r527052 = r527051 / r527019;
        double r527053 = r527022 * r527024;
        double r527054 = r527053 * r527020;
        double r527055 = r527052 - r527054;
        double r527056 = r527029 / r527055;
        double r527057 = r527048 / r527056;
        double r527058 = 7.4687020054859425e+289;
        bool r527059 = r527031 <= r527058;
        double r527060 = r527059 ? r527031 : r527043;
        double r527061 = r527047 ? r527057 : r527060;
        double r527062 = r527045 ? r527031 : r527061;
        double r527063 = r527033 ? r527043 : r527062;
        return r527063;
}

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)))