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

Average Error: 7.8 → 0.9

Time: 5.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -1.56341946069433755 \cdot 10^{290} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 7.94655575938796826 \cdot 10^{301}\right):\\ \;\;\;\;0.5 \cdot \left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}\right) - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{2}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r687026 = x;
        double r687027 = y;
        double r687028 = r687026 * r687027;
        double r687029 = z;
        double r687030 = 9.0;
        double r687031 = r687029 * r687030;
        double r687032 = t;
        double r687033 = r687031 * r687032;
        double r687034 = r687028 - r687033;
        double r687035 = a;
        double r687036 = 2.0;
        double r687037 = r687035 * r687036;
        double r687038 = r687034 / r687037;
        return r687038;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r687039 = x;
        double r687040 = y;
        double r687041 = r687039 * r687040;
        double r687042 = z;
        double r687043 = 9.0;
        double r687044 = r687042 * r687043;
        double r687045 = t;
        double r687046 = r687044 * r687045;
        double r687047 = r687041 - r687046;
        double r687048 = -1.5634194606943375e+290;
        bool r687049 = r687047 <= r687048;
        double r687050 = 7.946555759387968e+301;
        bool r687051 = r687047 <= r687050;
        double r687052 = !r687051;
        bool r687053 = r687049 || r687052;
        double r687054 = 0.5;
        double r687055 = a;
        double r687056 = cbrt(r687055);
        double r687057 = r687056 * r687056;
        double r687058 = r687039 / r687057;
        double r687059 = r687040 / r687056;
        double r687060 = r687058 * r687059;
        double r687061 = r687054 * r687060;
        double r687062 = 4.5;
        double r687063 = r687055 / r687042;
        double r687064 = r687045 / r687063;
        double r687065 = r687062 * r687064;
        double r687066 = r687061 - r687065;
        double r687067 = 1.0;
        double r687068 = r687067 / r687055;
        double r687069 = 2.0;
        double r687070 = r687047 / r687069;
        double r687071 = r687068 * r687070;
        double r687072 = r687053 ? r687066 : r687071;
        return r687072;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	7.8
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)) < -1.5634194606943375e+290 or 7.946555759387968e+301 < (- (* x y) (* (* z 9.0) t))

   1. Initial program 57.7

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

   2. Taylor expanded around 0 57.0

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

   4. Applied add-cube-cbrt 57.0

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}} - 4.5 \cdot \frac{t \cdot z}{a}\]

   5. Applied times-frac 28.5

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}\right)} - 4.5 \cdot \frac{t \cdot z}{a}\]
   6. Using strategy rm

   7. Applied associate-/l* 0.9

      \[\leadsto 0.5 \cdot \left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}\right) - 4.5 \cdot \color{blue}{\frac{t}{\frac{a}{z}}}\]

   if -1.5634194606943375e+290 < (- (* x y) (* (* z 9.0) t)) < 7.946555759387968e+301

   1. Initial program 0.8

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

   3. Applied *-un-lft-identity 0.8

      \[\leadsto \frac{\color{blue}{1 \cdot \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)}}{a \cdot 2}\]

   4. Applied times-frac 0.9

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{2}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -1.56341946069433755 \cdot 10^{290} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 7.94655575938796826 \cdot 10^{301}\right):\\ \;\;\;\;0.5 \cdot \left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}\right) - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(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)))