Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B

Average Error: 16.3 → 16.6

Time: 16.9s

Precision: 64

Math

\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.68308155764658379047185600858342380105 \cdot 10^{215}:\\ \;\;\;\;\frac{x + \frac{1}{t} \cdot \left(y \cdot z\right)}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ \mathbf{elif}\;z \le 3.794251549758276033940358680162827177832 \cdot 10^{146}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y}{\frac{t}{b}} + \left(1 + a\right)}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r34211032 = x;
        double r34211033 = y;
        double r34211034 = z;
        double r34211035 = r34211033 * r34211034;
        double r34211036 = t;
        double r34211037 = r34211035 / r34211036;
        double r34211038 = r34211032 + r34211037;
        double r34211039 = a;
        double r34211040 = 1.0;
        double r34211041 = r34211039 + r34211040;
        double r34211042 = b;
        double r34211043 = r34211033 * r34211042;
        double r34211044 = r34211043 / r34211036;
        double r34211045 = r34211041 + r34211044;
        double r34211046 = r34211038 / r34211045;
        return r34211046;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r34211047 = z;
        double r34211048 = -1.6830815576465838e+215;
        bool r34211049 = r34211047 <= r34211048;
        double r34211050 = x;
        double r34211051 = 1.0;
        double r34211052 = t;
        double r34211053 = r34211051 / r34211052;
        double r34211054 = y;
        double r34211055 = r34211054 * r34211047;
        double r34211056 = r34211053 * r34211055;
        double r34211057 = r34211050 + r34211056;
        double r34211058 = 1.0;
        double r34211059 = a;
        double r34211060 = r34211058 + r34211059;
        double r34211061 = b;
        double r34211062 = r34211061 * r34211054;
        double r34211063 = r34211062 / r34211052;
        double r34211064 = r34211060 + r34211063;
        double r34211065 = r34211057 / r34211064;
        double r34211066 = 3.794251549758276e+146;
        bool r34211067 = r34211047 <= r34211066;
        double r34211068 = r34211052 / r34211047;
        double r34211069 = r34211054 / r34211068;
        double r34211070 = r34211050 + r34211069;
        double r34211071 = r34211070 / r34211064;
        double r34211072 = r34211055 / r34211052;
        double r34211073 = r34211050 + r34211072;
        double r34211074 = r34211052 / r34211061;
        double r34211075 = r34211054 / r34211074;
        double r34211076 = r34211075 + r34211060;
        double r34211077 = r34211073 / r34211076;
        double r34211078 = r34211067 ? r34211071 : r34211077;
        double r34211079 = r34211049 ? r34211065 : r34211078;
        return r34211079;
}

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	16.3
Target	13.0
Herbie	16.6

\[\begin{array}{l} \mathbf{if}\;t \lt -1.365908536631008841640163147697088508132 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.036967103737245906066829435890093573122 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if z < -1.6830815576465838e+215

   1. Initial program 30.5

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
   2. Using strategy rm

   3. Applied div-inv 30.6

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

   if -1.6830815576465838e+215 < z < 3.794251549758276e+146

   1. Initial program 13.2

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
   2. Using strategy rm

   3. Applied associate-/l* 13.3

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

   if 3.794251549758276e+146 < z

   1. Initial program 29.1

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
   2. Using strategy rm

   3. Applied associate-/l* 31.0

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

4. Final simplification 16.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.68308155764658379047185600858342380105 \cdot 10^{215}:\\ \;\;\;\;\frac{x + \frac{1}{t} \cdot \left(y \cdot z\right)}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ \mathbf{elif}\;z \le 3.794251549758276033940358680162827177832 \cdot 10^{146}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y}{\frac{t}{b}} + \left(1 + a\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B"

  :herbie-target
  (if (< t -1.3659085366310088e-271) (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b))))) (if (< t 3.036967103737246e-130) (/ z b) (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b)))))))

  (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))