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

Average Error: 19.9 → 4.9

Time: 10.9s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} = -\infty:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{c \cdot \frac{z}{y}}\right) - 4 \cdot \frac{t}{\frac{c}{a}}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -4.5747116168721846 \cdot 10^{107}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 1.557205577643591 \cdot 10^{-75}:\\ \;\;\;\;\left(\frac{b + \left(x \cdot 9\right) \cdot y}{z} - a \cdot \left(4 \cdot t\right)\right) \cdot \frac{1}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 7.62562086493039623 \cdot 10^{282}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + \frac{9 \cdot \left(x \cdot \frac{y}{c}\right)}{z}\right) - 4 \cdot \frac{t}{\frac{c}{a}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r631947 = x;
        double r631948 = 9.0;
        double r631949 = r631947 * r631948;
        double r631950 = y;
        double r631951 = r631949 * r631950;
        double r631952 = z;
        double r631953 = 4.0;
        double r631954 = r631952 * r631953;
        double r631955 = t;
        double r631956 = r631954 * r631955;
        double r631957 = a;
        double r631958 = r631956 * r631957;
        double r631959 = r631951 - r631958;
        double r631960 = b;
        double r631961 = r631959 + r631960;
        double r631962 = c;
        double r631963 = r631952 * r631962;
        double r631964 = r631961 / r631963;
        return r631964;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r631965 = x;
        double r631966 = 9.0;
        double r631967 = r631965 * r631966;
        double r631968 = y;
        double r631969 = r631967 * r631968;
        double r631970 = z;
        double r631971 = 4.0;
        double r631972 = r631970 * r631971;
        double r631973 = t;
        double r631974 = r631972 * r631973;
        double r631975 = a;
        double r631976 = r631974 * r631975;
        double r631977 = r631969 - r631976;
        double r631978 = b;
        double r631979 = r631977 + r631978;
        double r631980 = c;
        double r631981 = r631970 * r631980;
        double r631982 = r631979 / r631981;
        double r631983 = -inf.0;
        bool r631984 = r631982 <= r631983;
        double r631985 = r631978 / r631981;
        double r631986 = r631970 / r631968;
        double r631987 = r631980 * r631986;
        double r631988 = r631965 / r631987;
        double r631989 = r631966 * r631988;
        double r631990 = r631985 + r631989;
        double r631991 = r631980 / r631975;
        double r631992 = r631973 / r631991;
        double r631993 = r631971 * r631992;
        double r631994 = r631990 - r631993;
        double r631995 = -4.5747116168721846e+107;
        bool r631996 = r631982 <= r631995;
        double r631997 = 1.557205577643591e-75;
        bool r631998 = r631982 <= r631997;
        double r631999 = r631978 + r631969;
        double r632000 = r631999 / r631970;
        double r632001 = r631971 * r631973;
        double r632002 = r631975 * r632001;
        double r632003 = r632000 - r632002;
        double r632004 = 1.0;
        double r632005 = r632004 / r631980;
        double r632006 = r632003 * r632005;
        double r632007 = 7.625620864930396e+282;
        bool r632008 = r631982 <= r632007;
        double r632009 = r631968 / r631980;
        double r632010 = r631965 * r632009;
        double r632011 = r631966 * r632010;
        double r632012 = r632011 / r631970;
        double r632013 = r631985 + r632012;
        double r632014 = r632013 - r631993;
        double r632015 = r632008 ? r631982 : r632014;
        double r632016 = r631998 ? r632006 : r632015;
        double r632017 = r631996 ? r631982 : r632016;
        double r632018 = r631984 ? r631994 : r632017;
        return r632018;
}

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	19.9
Target	14.4
Herbie	4.9

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.10015674080410512 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.17088779117474882 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.8768236795461372 \cdot 10^{130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.3838515042456319 \cdot 10^{158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

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 24.3

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

   3. Taylor expanded around 0 30.8

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

   5. Applied associate-/l* 25.8

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

   7. Applied associate-/l* 9.7

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

   8. Simplified 12.0

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

   if -inf.0 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -4.5747116168721846e+107 or 1.557205577643591e-75 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 7.625620864930396e+282

   1. Initial program 0.7

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

   if -4.5747116168721846e+107 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 1.557205577643591e-75

   1. Initial program 12.2

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

   2. Simplified 2.3

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

   4. Applied div-inv 2.4

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

   if 7.625620864930396e+282 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))

   1. Initial program 58.2

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

   2. Simplified 27.1

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

   3. Taylor expanded around 0 28.1

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

   5. Applied associate-/l* 26.1

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

   7. Applied times-frac 12.9

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

   9. Applied associate-*l/ 15.8

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

   10. Applied associate-*r/ 15.8

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

4. Final simplification 4.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} = -\infty:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{c \cdot \frac{z}{y}}\right) - 4 \cdot \frac{t}{\frac{c}{a}}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -4.5747116168721846 \cdot 10^{107}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 1.557205577643591 \cdot 10^{-75}:\\ \;\;\;\;\left(\frac{b + \left(x \cdot 9\right) \cdot y}{z} - a \cdot \left(4 \cdot t\right)\right) \cdot \frac{1}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 7.62562086493039623 \cdot 10^{282}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + \frac{9 \cdot \left(x \cdot \frac{y}{c}\right)}{z}\right) - 4 \cdot \frac{t}{\frac{c}{a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(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)) -1.1001567408041051e-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)))