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

Average Error: 19.9 → 17.6

Time: 25.6s

Precision: 64

Math

\[\left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3.0}\right) - \frac{a}{b \cdot 3.0}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t = -\infty:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \frac{-1}{2} + 1\right) \cdot \left(\sqrt{x} \cdot 2.0\right) - \frac{a}{3.0 \cdot b}\\ \mathbf{elif}\;z \cdot t \le 4.529717789787369 \cdot 10^{+261}:\\ \;\;\;\;\left(\left(\sqrt{x} \cdot 2.0\right) \cdot \left(\cos \left(0.3333333333333333 \cdot \left(z \cdot t\right)\right) \cdot \cos y\right) + \left(\sqrt{x} \cdot 2.0\right) \cdot \left(\sin y \cdot \sin \left(0.3333333333333333 \cdot \left(z \cdot t\right)\right)\right)\right) - \frac{a}{3.0 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \frac{-1}{2} + 1\right) \cdot \left(\sqrt{x} \cdot 2.0\right) - \frac{a}{3.0 \cdot b}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r14892959 = 2.0;
        double r14892960 = x;
        double r14892961 = sqrt(r14892960);
        double r14892962 = r14892959 * r14892961;
        double r14892963 = y;
        double r14892964 = z;
        double r14892965 = t;
        double r14892966 = r14892964 * r14892965;
        double r14892967 = 3.0;
        double r14892968 = r14892966 / r14892967;
        double r14892969 = r14892963 - r14892968;
        double r14892970 = cos(r14892969);
        double r14892971 = r14892962 * r14892970;
        double r14892972 = a;
        double r14892973 = b;
        double r14892974 = r14892973 * r14892967;
        double r14892975 = r14892972 / r14892974;
        double r14892976 = r14892971 - r14892975;
        return r14892976;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r14892977 = z;
        double r14892978 = t;
        double r14892979 = r14892977 * r14892978;
        double r14892980 = -inf.0;
        bool r14892981 = r14892979 <= r14892980;
        double r14892982 = y;
        double r14892983 = r14892982 * r14892982;
        double r14892984 = -0.5;
        double r14892985 = r14892983 * r14892984;
        double r14892986 = 1.0;
        double r14892987 = r14892985 + r14892986;
        double r14892988 = x;
        double r14892989 = sqrt(r14892988);
        double r14892990 = 2.0;
        double r14892991 = r14892989 * r14892990;
        double r14892992 = r14892987 * r14892991;
        double r14892993 = a;
        double r14892994 = 3.0;
        double r14892995 = b;
        double r14892996 = r14892994 * r14892995;
        double r14892997 = r14892993 / r14892996;
        double r14892998 = r14892992 - r14892997;
        double r14892999 = 4.529717789787369e+261;
        bool r14893000 = r14892979 <= r14892999;
        double r14893001 = 0.3333333333333333;
        double r14893002 = r14893001 * r14892979;
        double r14893003 = cos(r14893002);
        double r14893004 = cos(r14892982);
        double r14893005 = r14893003 * r14893004;
        double r14893006 = r14892991 * r14893005;
        double r14893007 = sin(r14892982);
        double r14893008 = sin(r14893002);
        double r14893009 = r14893007 * r14893008;
        double r14893010 = r14892991 * r14893009;
        double r14893011 = r14893006 + r14893010;
        double r14893012 = r14893011 - r14892997;
        double r14893013 = r14893000 ? r14893012 : r14892998;
        double r14893014 = r14892981 ? r14892998 : r14893013;
        return r14893014;
}

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	19.9
Target	18.3
Herbie	17.6

\[\begin{array}{l} \mathbf{if}\;z \lt -1.3793337487235141 \cdot 10^{+129}:\\ \;\;\;\;\left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333}{z}}{t}\right) - \frac{\frac{a}{3.0}}{b}\\ \mathbf{elif}\;z \lt 3.516290613555987 \cdot 10^{+106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2.0\right) \cdot \cos \left(y - \frac{t}{3.0} \cdot z\right) - \frac{\frac{a}{3.0}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333}{z}}{t}\right) \cdot \left(2.0 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3.0}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (* z t) < -inf.0 or 4.529717789787369e+261 < (* z t)

   1. Initial program 58.5

      \[\left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3.0}\right) - \frac{a}{b \cdot 3.0}\]

   2. Taylor expanded around 0 44.8

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(1 - \frac{1}{2} \cdot {y}^{2}\right)} - \frac{a}{b \cdot 3.0}\]

   3. Simplified 44.8

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot \left(y \cdot y\right)\right)} - \frac{a}{b \cdot 3.0}\]

   if -inf.0 < (* z t) < 4.529717789787369e+261

   1. Initial program 13.4

      \[\left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3.0}\right) - \frac{a}{b \cdot 3.0}\]
   2. Using strategy rm

   3. Applied cos-diff 13.0

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\cos y \cdot \cos \left(\frac{z \cdot t}{3.0}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3.0}\right)\right)} - \frac{a}{b \cdot 3.0}\]

   4. Applied distribute-lft-in 13.0

      \[\leadsto \color{blue}{\left(\left(2.0 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3.0}\right)\right) + \left(2.0 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3.0}\right)\right)\right)} - \frac{a}{b \cdot 3.0}\]

   5. Taylor expanded around inf 13.0

      \[\leadsto \left(\left(2.0 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3.0}\right)\right) + \left(2.0 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \color{blue}{\sin \left(0.3333333333333333 \cdot \left(t \cdot z\right)\right)}\right)\right) - \frac{a}{b \cdot 3.0}\]

   6. Taylor expanded around inf 13.0

      \[\leadsto \left(\left(2.0 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \color{blue}{\cos \left(0.3333333333333333 \cdot \left(t \cdot z\right)\right)}\right) + \left(2.0 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.3333333333333333 \cdot \left(t \cdot z\right)\right)\right)\right) - \frac{a}{b \cdot 3.0}\]
   7. Using strategy rm

   8. Applied *-commutative 13.0

      \[\leadsto \left(\left(2.0 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.3333333333333333 \cdot \left(t \cdot z\right)\right)\right) + \color{blue}{\left(\sin y \cdot \sin \left(0.3333333333333333 \cdot \left(t \cdot z\right)\right)\right) \cdot \left(2.0 \cdot \sqrt{x}\right)}\right) - \frac{a}{b \cdot 3.0}\]
3. Recombined 2 regimes into one program.

4. Final simplification 17.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t = -\infty:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \frac{-1}{2} + 1\right) \cdot \left(\sqrt{x} \cdot 2.0\right) - \frac{a}{3.0 \cdot b}\\ \mathbf{elif}\;z \cdot t \le 4.529717789787369 \cdot 10^{+261}:\\ \;\;\;\;\left(\left(\sqrt{x} \cdot 2.0\right) \cdot \left(\cos \left(0.3333333333333333 \cdot \left(z \cdot t\right)\right) \cdot \cos y\right) + \left(\sqrt{x} \cdot 2.0\right) \cdot \left(\sin y \cdot \sin \left(0.3333333333333333 \cdot \left(z \cdot t\right)\right)\right)\right) - \frac{a}{3.0 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \frac{-1}{2} + 1\right) \cdot \left(\sqrt{x} \cdot 2.0\right) - \frac{a}{3.0 \cdot b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"

  :herbie-target
  (if (< z -1.3793337487235141e+129) (- (* (* 2.0 (sqrt x)) (cos (- (/ 1 y) (/ (/ 0.3333333333333333 z) t)))) (/ (/ a 3.0) b)) (if (< z 3.516290613555987e+106) (- (* (* (sqrt x) 2.0) (cos (- y (* (/ t 3.0) z)))) (/ (/ a 3.0) b)) (- (* (cos (- y (/ (/ 0.3333333333333333 z) t))) (* 2.0 (sqrt x))) (/ (/ a b) 3.0))))

  (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))