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

Average Error: 16.9 → 4.5

Time: 39.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -2.4972769510363455 \cdot 10^{+80}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;t \leq 2.3404855533204465 \cdot 10^{+32}:\\ \;\;\;\;\frac{z}{\frac{t \cdot a}{y} + \left(b + \frac{t}{y}\right)} + \frac{x}{a + \left(1 + \frac{y \cdot b}{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a + \left(1 + \frac{y \cdot b}{t}\right)} + \frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\frac{\sqrt[3]{t}}{\frac{y}{a + \left(1 + \frac{b}{\frac{t}{y}}\right)}}}\\ \end{array}\]

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 (<= t -2.4972769510363455e+80)
   (/ (+ x (* y (/ z t))) (+ (+ a 1.0) (/ (* y b) t)))
   (if (<= t 2.3404855533204465e+32)
     (+
      (/ z (+ (/ (* t a) y) (+ b (/ t y))))
      (/ x (+ a (+ 1.0 (/ (* y b) t)))))
     (+
      (/ x (+ a (+ 1.0 (/ (* y b) t))))
      (*
       (/ 1.0 (* (cbrt t) (cbrt t)))
       (/ z (/ (cbrt t) (/ y (+ a (+ 1.0 (/ b (/ t y))))))))))))

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 (t <= -2.4972769510363455e+80) {
		tmp = (x + (y * (z / t))) / ((a + 1.0) + ((y * b) / t));
	} else if (t <= 2.3404855533204465e+32) {
		tmp = (z / (((t * a) / y) + (b + (t / y)))) + (x / (a + (1.0 + ((y * b) / t))));
	} else {
		tmp = (x / (a + (1.0 + ((y * b) / t)))) + ((1.0 / (cbrt(t) * cbrt(t))) * (z / (cbrt(t) / (y / (a + (1.0 + (b / (t / y))))))));
	}
	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	16.9
Target	13.2
Herbie	4.5

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

Derivation

1. Split input into 3 regimes

2. if t < -2.49727695103634555e80

   1. Initial program 12.1

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

   3. Applied *-un-lft-identity_binary64_16105 12.1

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

   4. Applied times-frac_binary64_16111 6.9

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

   5. Simplified 6.9

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

   if -2.49727695103634555e80 < t < 2.34048555332044649e32

   1. Initial program 20.8

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

   2. Taylor expanded around 0 16.5

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

   3. Simplified 16.5

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

   5. Applied associate-/l*_binary64_16050 14.8

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

   6. Simplified 17.8

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

   7. Taylor expanded around 0 2.9

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

   if 2.34048555332044649e32 < t

   1. Initial program 11.2

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

   2. Taylor expanded around 0 12.6

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

   3. Simplified 12.6

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

   5. Applied associate-/l*_binary64_16050 10.5

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

   6. Simplified 8.8

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

   8. Applied *-un-lft-identity_binary64_16105 8.8

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

   9. Applied *-un-lft-identity_binary64_16105 8.8

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

   10. Applied times-frac_binary64_16111 8.8

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

   11. Applied add-cube-cbrt_binary64_16140 9.0

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

   12. Applied times-frac_binary64_16111 9.0

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

   13. Applied *-un-lft-identity_binary64_16105 9.0

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

   14. Applied times-frac_binary64_16111 8.7

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

   15. Simplified 8.7

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

   16. Simplified 6.6

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

4. Final simplification 4.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.4972769510363455 \cdot 10^{+80}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;t \leq 2.3404855533204465 \cdot 10^{+32}:\\ \;\;\;\;\frac{z}{\frac{t \cdot a}{y} + \left(b + \frac{t}{y}\right)} + \frac{x}{a + \left(1 + \frac{y \cdot b}{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a + \left(1 + \frac{y \cdot b}{t}\right)} + \frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\frac{\sqrt[3]{t}}{\frac{y}{a + \left(1 + \frac{b}{\frac{t}{y}}\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021023 
(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))))