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

Average Error: 16.5 → 13.2

Time: 16.2min

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -5.70151277635625122 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{y}{\frac{t}{z}} + x}{\left(a + 1\right) + b \cdot \frac{y}{t}}\\ \mathbf{elif}\;t \le 9.2919357485169675 \cdot 10^{137}:\\ \;\;\;\;\frac{x + \frac{1}{\frac{t}{y \cdot z}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{\left(a + 1\right) + \frac{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot b}{\sqrt[3]{t}}}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a, double b) {
	double VAR;
	if ((t <= -5.701512776356251e-30)) {
		VAR = (((double) ((y / (t / z)) + x)) / ((double) (((double) (a + 1.0)) + ((double) (b * (y / t))))));
	} else {
		double VAR_1;
		if ((t <= 9.291935748516968e+137)) {
			VAR_1 = (((double) (x + (1.0 / (t / ((double) (y * z)))))) / ((double) (((double) (a + 1.0)) + (((double) (y * b)) / t))));
		} else {
			VAR_1 = (((double) (x + ((double) ((y / ((double) (((double) cbrt(t)) * ((double) cbrt(t))))) * (z / ((double) cbrt(t))))))) / ((double) (((double) (a + 1.0)) + (((double) ((y / ((double) (((double) cbrt(t)) * ((double) cbrt(t))))) * b)) / ((double) cbrt(t))))));
		}
		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	12.9
Herbie	13.2

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

Derivation

1. Split input into 3 regimes

2. if t < -5.70151277635625122e-30

   1. Initial program 11.3

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

   3. Applied add-cube-cbrt 11.5

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

   4. Applied times-frac 8.5

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

   6. Applied add-cube-cbrt 8.5

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

   7. Applied times-frac 4.8

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

   9. Applied *-un-lft-identity 4.8

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

   10. Applied *-un-lft-identity 4.8

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

   11. Applied times-frac 4.8

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

   12. Simplified 4.8

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

   13. Simplified 4.9

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

   if -5.70151277635625122e-30 < t < 9.2919357485169675e137

   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 clear-num 20.0

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

   if 9.2919357485169675e137 < t

   1. Initial program 12.6

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

   3. Applied add-cube-cbrt 12.7

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

   4. Applied times-frac 8.0

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

   6. Applied add-cube-cbrt 8.0

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

   7. Applied times-frac 1.5

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

   9. Applied associate-*r/ 2.4

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

4. Final simplification 13.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.70151277635625122 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{y}{\frac{t}{z}} + x}{\left(a + 1\right) + b \cdot \frac{y}{t}}\\ \mathbf{elif}\;t \le 9.2919357485169675 \cdot 10^{137}:\\ \;\;\;\;\frac{x + \frac{1}{\frac{t}{y \cdot z}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{\left(a + 1\right) + \frac{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot b}{\sqrt[3]{t}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(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))))