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

Average Error: 16.9 → 14.0

Time: 6.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -2.0726898936729369 \cdot 10^{-88} \lor \neg \left(t \le 2.45368231681103658 \cdot 10^{-133}\right):\\ \;\;\;\;\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x + \frac{y \cdot z}{t}} \cdot \sqrt[3]{x + \frac{y \cdot z}{t}}}{\frac{\left(a + 1\right) + \frac{y \cdot b}{t}}{\sqrt[3]{x + \frac{y \cdot z}{t}}}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r667247 = x;
        double r667248 = y;
        double r667249 = z;
        double r667250 = r667248 * r667249;
        double r667251 = t;
        double r667252 = r667250 / r667251;
        double r667253 = r667247 + r667252;
        double r667254 = a;
        double r667255 = 1.0;
        double r667256 = r667254 + r667255;
        double r667257 = b;
        double r667258 = r667248 * r667257;
        double r667259 = r667258 / r667251;
        double r667260 = r667256 + r667259;
        double r667261 = r667253 / r667260;
        return r667261;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r667262 = t;
        double r667263 = -2.072689893672937e-88;
        bool r667264 = r667262 <= r667263;
        double r667265 = 2.4536823168110366e-133;
        bool r667266 = r667262 <= r667265;
        double r667267 = !r667266;
        bool r667268 = r667264 || r667267;
        double r667269 = x;
        double r667270 = y;
        double r667271 = r667270 / r667262;
        double r667272 = z;
        double r667273 = r667271 * r667272;
        double r667274 = r667269 + r667273;
        double r667275 = 1.0;
        double r667276 = a;
        double r667277 = 1.0;
        double r667278 = r667276 + r667277;
        double r667279 = b;
        double r667280 = r667262 / r667279;
        double r667281 = r667270 / r667280;
        double r667282 = r667278 + r667281;
        double r667283 = r667275 / r667282;
        double r667284 = r667274 * r667283;
        double r667285 = r667270 * r667272;
        double r667286 = r667285 / r667262;
        double r667287 = r667269 + r667286;
        double r667288 = cbrt(r667287);
        double r667289 = r667288 * r667288;
        double r667290 = r667270 * r667279;
        double r667291 = r667290 / r667262;
        double r667292 = r667278 + r667291;
        double r667293 = r667292 / r667288;
        double r667294 = r667289 / r667293;
        double r667295 = r667268 ? r667284 : r667294;
        return r667295;
}

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	16.9
Target	12.8
Herbie	14.0

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

Derivation

1. Split input into 2 regimes

2. if t < -2.072689893672937e-88 or 2.4536823168110366e-133 < t

   1. Initial program 12.1

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

   3. Applied associate-/l* 9.9

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

   5. Applied associate-/l* 7.4

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

   7. Applied div-inv 7.5

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

   9. Applied associate-/r/ 7.7

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

   if -2.072689893672937e-88 < t < 2.4536823168110366e-133

   1. Initial program 28.5

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

   3. Applied add-cube-cbrt 29.0

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

   4. Applied associate-/l* 29.0

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

4. Final simplification 14.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.0726898936729369 \cdot 10^{-88} \lor \neg \left(t \le 2.45368231681103658 \cdot 10^{-133}\right):\\ \;\;\;\;\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x + \frac{y \cdot z}{t}} \cdot \sqrt[3]{x + \frac{y \cdot z}{t}}}{\frac{\left(a + 1\right) + \frac{y \cdot b}{t}}{\sqrt[3]{x + \frac{y \cdot z}{t}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 
(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))))