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

Average Error: 16.2 → 13.3

Time: 9.1s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -6.3581812405522004 \cdot 10^{-131} \lor \neg \left(y \le 7.59364648446620877 \cdot 10^{-92}\right):\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{\frac{y \cdot b}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{t}}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r617105 = x;
        double r617106 = y;
        double r617107 = z;
        double r617108 = r617106 * r617107;
        double r617109 = t;
        double r617110 = r617108 / r617109;
        double r617111 = r617105 + r617110;
        double r617112 = a;
        double r617113 = 1.0;
        double r617114 = r617112 + r617113;
        double r617115 = b;
        double r617116 = r617106 * r617115;
        double r617117 = r617116 / r617109;
        double r617118 = r617114 + r617117;
        double r617119 = r617111 / r617118;
        return r617119;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r617120 = y;
        double r617121 = -6.3581812405522e-131;
        bool r617122 = r617120 <= r617121;
        double r617123 = 7.593646484466209e-92;
        bool r617124 = r617120 <= r617123;
        double r617125 = !r617124;
        bool r617126 = r617122 || r617125;
        double r617127 = x;
        double r617128 = z;
        double r617129 = t;
        double r617130 = r617128 / r617129;
        double r617131 = r617120 * r617130;
        double r617132 = r617127 + r617131;
        double r617133 = a;
        double r617134 = 1.0;
        double r617135 = r617133 + r617134;
        double r617136 = b;
        double r617137 = r617136 / r617129;
        double r617138 = r617120 * r617137;
        double r617139 = r617135 + r617138;
        double r617140 = r617132 / r617139;
        double r617141 = r617120 * r617128;
        double r617142 = r617141 / r617129;
        double r617143 = r617127 + r617142;
        double r617144 = r617120 * r617136;
        double r617145 = cbrt(r617129);
        double r617146 = r617145 * r617145;
        double r617147 = r617144 / r617146;
        double r617148 = r617147 / r617145;
        double r617149 = r617135 + r617148;
        double r617150 = r617143 / r617149;
        double r617151 = r617126 ? r617140 : r617150;
        return r617151;
}

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	16.2
Target	13.2
Herbie	13.3

\[\begin{array}{l} \mathbf{if}\;t \lt -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.0369671037372459 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < -6.3581812405522e-131 or 7.593646484466209e-92 < y

   1. Initial program 23.2

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

   3. Applied *-un-lft-identity 23.2

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

   4. Applied times-frac 21.7

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

   5. Simplified 21.7

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

   7. Applied *-un-lft-identity 21.7

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

   8. Applied times-frac 18.8

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

   9. Simplified 18.8

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

   if -6.3581812405522e-131 < y < 7.593646484466209e-92

   1. Initial program 2.1

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

   3. Applied add-cube-cbrt 2.2

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}}\]

   4. Applied associate-/r* 2.2

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

4. Final simplification 13.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -6.3581812405522004 \cdot 10^{-131} \lor \neg \left(y \le 7.59364648446620877 \cdot 10^{-92}\right):\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{\frac{y \cdot b}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{t}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (if (< t -1.3659085366310088e-271) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b))))) (if (< t 3.036967103737246e-130) (/ z b) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b)))))))

  (/ (+ x (/ (* y z) t)) (+ (+ a 1) (/ (* y b) t))))