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

Average Error: 16.5 → 14.0

Time: 5.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.54215107921783331 \cdot 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right) \cdot \frac{1}{a + \mathsf{fma}\left(\frac{y}{t}, b, 1\right)}\\ \mathbf{elif}\;t \le 1.03638931227158246 \cdot 10^{242}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{1}{\frac{t}{y \cdot b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a + \mathsf{fma}\left(\frac{y}{t}, b, 1\right)}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}}\\ \end{array}\]

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 VAR;
	if ((t <= -1.5421510792178333e-37)) {
		VAR = (fma((y / t), z, x) * (1.0 / (a + fma((y / t), b, 1.0))));
	} else {
		double VAR_1;
		if ((t <= 1.0363893122715825e+242)) {
			VAR_1 = ((x + ((y * z) / t)) / ((a + 1.0) + (1.0 / (t / (y * b)))));
		} else {
			VAR_1 = (1.0 / ((a + fma((y / t), b, 1.0)) / fma((y / t), z, x)));
		}
		VAR = VAR_1;
	}
	return VAR;
}

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	13.5
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 3 regimes

2. if t < -1.5421510792178333e-37

   1. Initial program 11.5

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

   3. Applied associate-+l+ 11.5

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

   4. Simplified 8.8

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

   6. Applied *-un-lft-identity 8.8

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

   7. Applied associate-/r* 8.8

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

   8. Simplified 5.1

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

   10. Applied div-inv 5.2

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

   if -1.5421510792178333e-37 < t < 1.0363893122715825e+242

   1. Initial program 18.9

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

   3. Applied clear-num 18.9

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

   if 1.0363893122715825e+242 < t

   1. Initial program 14.1

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

   3. Applied associate-+l+ 14.1

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

   4. Simplified 9.1

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

   6. Applied *-un-lft-identity 9.1

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

   7. Applied associate-/r* 9.1

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

   8. Simplified 0.8

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

   10. Applied clear-num 1.3

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

4. Final simplification 14.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.54215107921783331 \cdot 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right) \cdot \frac{1}{a + \mathsf{fma}\left(\frac{y}{t}, b, 1\right)}\\ \mathbf{elif}\;t \le 1.03638931227158246 \cdot 10^{242}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{1}{\frac{t}{y \cdot b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a + \mathsf{fma}\left(\frac{y}{t}, b, 1\right)}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020075 +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))))