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

Average Error: 16.5 → 14.0

Time: 6.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -3.5472664252508322 \cdot 10^{-9} \lor \neg \left(z \le 2.45143159650892174 \cdot 10^{-200}\right):\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{b}{\frac{t}{y}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(\frac{b}{t}, y, a + 1\right)}\\ \end{array}\]

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 temp;
	if (((z <= -3.547266425250832e-09) || !(z <= 2.4514315965089217e-200))) {
		temp = ((x + (z / (t / y))) / ((a + 1.0) + (b / (t / y))));
	} else {
		temp = (1.0 * (fma((z / t), y, x) / fma((b / t), y, (a + 1.0))));
	}
	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	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 z < -3.547266425250832e-09 or 2.4514315965089217e-200 < z

   1. Initial program 20.4

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

   3. Applied *-un-lft-identity 20.4

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

   4. Applied *-commutative 20.4

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

   5. Applied times-frac 19.7

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

   6. Simplified 19.7

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

   8. Applied *-commutative 19.7

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

   9. Applied associate-/l* 16.4

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

   11. Applied clear-num 16.4

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

   12. Applied un-div-inv 16.4

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

   if -3.547266425250832e-09 < z < 2.4514315965089217e-200

   1. Initial program 9.4

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

   3. Applied *-un-lft-identity 9.4

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

   4. Applied *-commutative 9.4

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

   5. Applied times-frac 9.9

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

   6. Simplified 9.9

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

   8. Applied *-commutative 9.9

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

   9. Applied associate-/l* 11.0

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

   11. Applied clear-num 11.0

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

   12. Applied un-div-inv 10.7

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

   14. Applied *-un-lft-identity 10.7

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

   15. Applied *-un-lft-identity 10.7

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

   16. Applied times-frac 10.7

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

   17. Simplified 10.7

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

   18. Simplified 9.5

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

4. Final simplification 14.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.5472664252508322 \cdot 10^{-9} \lor \neg \left(z \le 2.45143159650892174 \cdot 10^{-200}\right):\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{b}{\frac{t}{y}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(\frac{b}{t}, y, a + 1\right)}\\ \end{array}\]

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))))