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

Average Error: 16.3 → 13.5

Time: 17.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -4.147142268992736217334744595000847977693 \cdot 10^{-47}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(1 + a\right) + \frac{y}{\frac{t}{b}}}\\ \mathbf{elif}\;t \le 1.159687065919828568064040293359474030002 \cdot 10^{-206}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{y \cdot z}{t} + x} \cdot \sqrt[3]{\frac{y \cdot z}{t} + x}}{\frac{\left(1 + a\right) + \frac{y \cdot b}{t}}{\sqrt[3]{\frac{y \cdot z}{t} + x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(1 + a\right) + \frac{y}{\frac{t}{b}}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r36706385 = x;
        double r36706386 = y;
        double r36706387 = z;
        double r36706388 = r36706386 * r36706387;
        double r36706389 = t;
        double r36706390 = r36706388 / r36706389;
        double r36706391 = r36706385 + r36706390;
        double r36706392 = a;
        double r36706393 = 1.0;
        double r36706394 = r36706392 + r36706393;
        double r36706395 = b;
        double r36706396 = r36706386 * r36706395;
        double r36706397 = r36706396 / r36706389;
        double r36706398 = r36706394 + r36706397;
        double r36706399 = r36706391 / r36706398;
        return r36706399;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r36706400 = t;
        double r36706401 = -4.147142268992736e-47;
        bool r36706402 = r36706400 <= r36706401;
        double r36706403 = x;
        double r36706404 = y;
        double r36706405 = z;
        double r36706406 = r36706400 / r36706405;
        double r36706407 = r36706404 / r36706406;
        double r36706408 = r36706403 + r36706407;
        double r36706409 = 1.0;
        double r36706410 = a;
        double r36706411 = r36706409 + r36706410;
        double r36706412 = b;
        double r36706413 = r36706400 / r36706412;
        double r36706414 = r36706404 / r36706413;
        double r36706415 = r36706411 + r36706414;
        double r36706416 = r36706408 / r36706415;
        double r36706417 = 1.1596870659198286e-206;
        bool r36706418 = r36706400 <= r36706417;
        double r36706419 = r36706404 * r36706405;
        double r36706420 = r36706419 / r36706400;
        double r36706421 = r36706420 + r36706403;
        double r36706422 = cbrt(r36706421);
        double r36706423 = r36706422 * r36706422;
        double r36706424 = r36706404 * r36706412;
        double r36706425 = r36706424 / r36706400;
        double r36706426 = r36706411 + r36706425;
        double r36706427 = r36706426 / r36706422;
        double r36706428 = r36706423 / r36706427;
        double r36706429 = r36706418 ? r36706428 : r36706416;
        double r36706430 = r36706402 ? r36706416 : r36706429;
        return r36706430;
}

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	12.8
Herbie	13.5

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

Derivation

1. Split input into 2 regimes

2. if t < -4.147142268992736e-47 or 1.1596870659198286e-206 < t

   1. Initial program 12.6

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

   3. Applied associate-/l* 11.2

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

   5. Applied associate-/l* 8.5

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

   if -4.147142268992736e-47 < t < 1.1596870659198286e-206

   1. Initial program 25.3

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

   3. Applied add-cube-cbrt 25.8

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

   4. Applied associate-/l* 25.8

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

4. Final simplification 13.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.147142268992736217334744595000847977693 \cdot 10^{-47}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(1 + a\right) + \frac{y}{\frac{t}{b}}}\\ \mathbf{elif}\;t \le 1.159687065919828568064040293359474030002 \cdot 10^{-206}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{y \cdot z}{t} + x} \cdot \sqrt[3]{\frac{y \cdot z}{t} + x}}{\frac{\left(1 + a\right) + \frac{y \cdot b}{t}}{\sqrt[3]{\frac{y \cdot z}{t} + x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(1 + a\right) + \frac{y}{\frac{t}{b}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(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))))