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

Average Error: 3.7 → 1.0

Time: 7.4s

Precision: 64

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

Math

\[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]

↓

\[\begin{array}{l} \mathbf{if}\;t \le -3.2937345159843623 \cdot 10^{-114} \lor \neg \left(t \le 1.66489209295084457 \cdot 10^{-80}\right):\\ \;\;\;\;\left(x \cdot 2 - \left(y \cdot \left(z \cdot 9\right)\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2 + \left(\left(a \cdot 27\right) \cdot b - 9 \cdot \left(\left(t \cdot z\right) \cdot y\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r906990 = x;
        double r906991 = 2.0;
        double r906992 = r906990 * r906991;
        double r906993 = y;
        double r906994 = 9.0;
        double r906995 = r906993 * r906994;
        double r906996 = z;
        double r906997 = r906995 * r906996;
        double r906998 = t;
        double r906999 = r906997 * r906998;
        double r907000 = r906992 - r906999;
        double r907001 = a;
        double r907002 = 27.0;
        double r907003 = r907001 * r907002;
        double r907004 = b;
        double r907005 = r907003 * r907004;
        double r907006 = r907000 + r907005;
        return r907006;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r907007 = t;
        double r907008 = -3.293734515984362e-114;
        bool r907009 = r907007 <= r907008;
        double r907010 = 1.6648920929508446e-80;
        bool r907011 = r907007 <= r907010;
        double r907012 = !r907011;
        bool r907013 = r907009 || r907012;
        double r907014 = x;
        double r907015 = 2.0;
        double r907016 = r907014 * r907015;
        double r907017 = y;
        double r907018 = z;
        double r907019 = 9.0;
        double r907020 = r907018 * r907019;
        double r907021 = r907017 * r907020;
        double r907022 = r907021 * r907007;
        double r907023 = r907016 - r907022;
        double r907024 = a;
        double r907025 = 27.0;
        double r907026 = b;
        double r907027 = r907025 * r907026;
        double r907028 = r907024 * r907027;
        double r907029 = r907023 + r907028;
        double r907030 = r907024 * r907025;
        double r907031 = r907030 * r907026;
        double r907032 = r907007 * r907018;
        double r907033 = r907032 * r907017;
        double r907034 = r907019 * r907033;
        double r907035 = r907031 - r907034;
        double r907036 = r907016 + r907035;
        double r907037 = r907013 ? r907029 : r907036;
        return r907037;
}

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

Target

Original	3.7
Target	2.5
Herbie	1.0

\[\begin{array}{l} \mathbf{if}\;y \lt 7.590524218811189 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if t < -3.293734515984362e-114 or 1.6648920929508446e-80 < t

   1. Initial program 1.2

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
   2. Using strategy rm

   3. Applied associate-*l* 1.2

      \[\leadsto \left(x \cdot 2 - \color{blue}{\left(y \cdot \left(9 \cdot z\right)\right)} \cdot t\right) + \left(a \cdot 27\right) \cdot b\]

   4. Simplified 1.2

      \[\leadsto \left(x \cdot 2 - \left(y \cdot \color{blue}{\left(z \cdot 9\right)}\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
   5. Using strategy rm

   6. Applied associate-*l* 1.2

      \[\leadsto \left(x \cdot 2 - \left(y \cdot \left(z \cdot 9\right)\right) \cdot t\right) + \color{blue}{a \cdot \left(27 \cdot b\right)}\]

   if -3.293734515984362e-114 < t < 1.6648920929508446e-80

   1. Initial program 7.5

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
   2. Using strategy rm

   3. Applied sub-neg 7.5

      \[\leadsto \color{blue}{\left(x \cdot 2 + \left(-\left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right)} + \left(a \cdot 27\right) \cdot b\]

   4. Applied associate-+l+ 7.5

      \[\leadsto \color{blue}{x \cdot 2 + \left(\left(-\left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\right)}\]

   5. Simplified 7.3

      \[\leadsto x \cdot 2 + \color{blue}{\left(\left(a \cdot 27\right) \cdot b - 9 \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)}\]
   6. Using strategy rm

   7. Applied associate-*r* 0.6

      \[\leadsto x \cdot 2 + \left(\left(a \cdot 27\right) \cdot b - 9 \cdot \color{blue}{\left(\left(t \cdot z\right) \cdot y\right)}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.2937345159843623 \cdot 10^{-114} \lor \neg \left(t \le 1.66489209295084457 \cdot 10^{-80}\right):\\ \;\;\;\;\left(x \cdot 2 - \left(y \cdot \left(z \cdot 9\right)\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2 + \left(\left(a \cdot 27\right) \cdot b - 9 \cdot \left(\left(t \cdot z\right) \cdot y\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< y 7.590524218811189e-161) (+ (- (* x 2) (* (* (* y 9) z) t)) (* a (* 27 b))) (+ (- (* x 2) (* 9 (* y (* t z)))) (* (* a 27) b)))

  (+ (- (* x 2) (* (* (* y 9) z) t)) (* (* a 27) b)))