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

Average Error: 16.5 → 13.8

Time: 5.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.98981247134042296 \cdot 10^{-78} \lor \neg \left(z \le 4.61495656583718254 \cdot 10^{-191}\right):\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{1}{\frac{t}{y \cdot b}}}\\ \end{array}\]

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t)));
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	double temp;
	if (((z <= -1.989812471340423e-78) || !(z <= 4.6149565658371825e-191))) {
		temp = pow((fma((y / t), z, x) / fma((y / t), b, (a + 1.0))), 1.0);
	} else {
		temp = ((x + ((y * z) / t)) / ((a + 1.0) + (1.0 / (t / (y * b)))));
	}
	return temp;
}

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	16.5
Target	13.2
Herbie	13.8

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

Derivation

1. Split input into 2 regimes

2. if z < -1.989812471340423e-78 or 4.6149565658371825e-191 < z

   1. Initial program 20.0

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

   3. Applied *-un-lft-identity 20.0

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

   4. Applied associate-/r* 20.0

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

   5. Simplified 18.3

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

   7. Applied div-inv 18.3

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

   9. Applied pow1 18.3

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

   10. Applied pow1 18.3

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

   11. Applied pow-prod-down 18.3

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

   12. Simplified 16.1

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

   if -1.989812471340423e-78 < z < 4.6149565658371825e-191

   1. Initial program 8.4

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

   3. Applied clear-num 8.4

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

4. Final simplification 13.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.98981247134042296 \cdot 10^{-78} \lor \neg \left(z \le 4.61495656583718254 \cdot 10^{-191}\right):\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{1}{\frac{t}{y \cdot b}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(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))))