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

Average Error: 7.7 → 0.9

Time: 4.9s

Precision: 64

Math

\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -1.47596775768044329 \cdot 10^{214} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 1.44327490436144402 \cdot 10^{217}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \left(x \cdot y\right) - 4.5 \cdot \left(t \cdot z\right)}{a}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r678677 = x;
        double r678678 = y;
        double r678679 = r678677 * r678678;
        double r678680 = z;
        double r678681 = 9.0;
        double r678682 = r678680 * r678681;
        double r678683 = t;
        double r678684 = r678682 * r678683;
        double r678685 = r678679 - r678684;
        double r678686 = a;
        double r678687 = 2.0;
        double r678688 = r678686 * r678687;
        double r678689 = r678685 / r678688;
        return r678689;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r678690 = x;
        double r678691 = y;
        double r678692 = r678690 * r678691;
        double r678693 = z;
        double r678694 = 9.0;
        double r678695 = r678693 * r678694;
        double r678696 = t;
        double r678697 = r678695 * r678696;
        double r678698 = r678692 - r678697;
        double r678699 = -1.4759677576804433e+214;
        bool r678700 = r678698 <= r678699;
        double r678701 = 1.443274904361444e+217;
        bool r678702 = r678698 <= r678701;
        double r678703 = !r678702;
        bool r678704 = r678700 || r678703;
        double r678705 = 0.5;
        double r678706 = a;
        double r678707 = r678691 / r678706;
        double r678708 = r678690 * r678707;
        double r678709 = r678705 * r678708;
        double r678710 = 4.5;
        double r678711 = r678706 / r678693;
        double r678712 = r678696 / r678711;
        double r678713 = r678710 * r678712;
        double r678714 = r678709 - r678713;
        double r678715 = r678705 * r678692;
        double r678716 = r678696 * r678693;
        double r678717 = r678710 * r678716;
        double r678718 = r678715 - r678717;
        double r678719 = r678718 / r678706;
        double r678720 = r678704 ? r678714 : r678719;
        return r678720;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	7.7
Target	5.6
Herbie	0.9

\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.14403070783397609 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (- (* x y) (* (* z 9.0) t)) < -1.4759677576804433e+214 or 1.443274904361444e+217 < (- (* x y) (* (* z 9.0) t))

   1. Initial program 31.2

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

   2. Taylor expanded around 0 30.5

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}}\]
   3. Using strategy rm

   4. Applied associate-/l* 16.8

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \color{blue}{\frac{t}{\frac{a}{z}}}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 16.8

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{\color{blue}{1 \cdot a}} - 4.5 \cdot \frac{t}{\frac{a}{z}}\]

   7. Applied times-frac 1.0

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{1} \cdot \frac{y}{a}\right)} - 4.5 \cdot \frac{t}{\frac{a}{z}}\]

   8. Simplified 1.0

      \[\leadsto 0.5 \cdot \left(\color{blue}{x} \cdot \frac{y}{a}\right) - 4.5 \cdot \frac{t}{\frac{a}{z}}\]

   if -1.4759677576804433e+214 < (- (* x y) (* (* z 9.0) t)) < 1.443274904361444e+217

   1. Initial program 0.8

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

   2. Taylor expanded around 0 0.8

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}}\]
   3. Using strategy rm

   4. Applied associate-*r/ 0.8

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - \color{blue}{\frac{4.5 \cdot \left(t \cdot z\right)}{a}}\]

   5. Applied associate-*r/ 0.8

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(x \cdot y\right)}{a}} - \frac{4.5 \cdot \left(t \cdot z\right)}{a}\]

   6. Applied sub-div 0.8

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(x \cdot y\right) - 4.5 \cdot \left(t \cdot z\right)}{a}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -1.47596775768044329 \cdot 10^{214} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 1.44327490436144402 \cdot 10^{217}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \left(x \cdot y\right) - 4.5 \cdot \left(t \cdot z\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 +o rules:numerics
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :herbie-target
  (if (< a -2.090464557976709e+86) (- (* 0.5 (/ (* y x) a)) (* 4.5 (/ t (/ a z)))) (if (< a 2.144030707833976e+99) (/ (- (* x y) (* z (* 9 t))) (* a 2)) (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5)))))

  (/ (- (* x y) (* (* z 9) t)) (* a 2)))