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

Average Error: 17.1 → 3.7

Time: 8.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1.2246087210784099 \cdot 10^{+43}:\\ \;\;\;\;\frac{z}{b + \frac{t}{y} \cdot \left(a + 1\right)} + \frac{x}{a + \left(1 + \frac{y}{\frac{t}{b}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b + \frac{t \cdot \left(a + 1\right)}{y}} + \frac{x}{a + \left(1 + \frac{y \cdot b}{t}\right)}\\ \end{array}\]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))

↓

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.2246087210784099e+43)
   (+ (/ z (+ b (* (/ t y) (+ a 1.0)))) (/ x (+ a (+ 1.0 (/ y (/ t b))))))
   (+ (/ z (+ b (/ (* t (+ a 1.0)) y))) (/ x (+ a (+ 1.0 (/ (* y b) t)))))))

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 tmp;
	if (y <= -1.2246087210784099e+43) {
		tmp = (z / (b + ((t / y) * (a + 1.0)))) + (x / (a + (1.0 + (y / (t / b)))));
	} else {
		tmp = (z / (b + ((t * (a + 1.0)) / y))) + (x / (a + (1.0 + ((y * b) / t))));
	}
	return tmp;
}

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	17.1
Target	13.3
Herbie	3.7

\[\begin{array}{l} \mathbf{if}\;t < -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t < 3.036967103737246 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < -1.22460872107840986e43

   1. Initial program 33.3

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

   2. Taylor expanded around 0 30.5

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

   3. Simplified 30.5

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

   5. Applied associate-/l*_binary64 24.6

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

   6. Simplified 22.8

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

   7. Taylor expanded around 0 8.0

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

   8. Simplified 6.0

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

   10. Applied associate-/l*_binary64 2.9

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

   if -1.22460872107840986e43 < y

   1. Initial program 12.7

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

   2. Taylor expanded around 0 11.1

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

   3. Simplified 11.1

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

   5. Applied associate-/l*_binary64 9.8

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

   6. Simplified 11.4

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

   7. Taylor expanded around 0 7.3

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

   8. Simplified 3.5

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

   10. Applied associate-*l/_binary64 4.0

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

4. Final simplification 3.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2246087210784099 \cdot 10^{+43}:\\ \;\;\;\;\frac{z}{b + \frac{t}{y} \cdot \left(a + 1\right)} + \frac{x}{a + \left(1 + \frac{y}{\frac{t}{b}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b + \frac{t \cdot \left(a + 1\right)}{y}} + \frac{x}{a + \left(1 + \frac{y \cdot b}{t}\right)}\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021126 
(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) (* 1.0 (* (+ 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))))