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

Average Error: 16.5 → 13.9

Time: 5.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -5.01008916636701195 \cdot 10^{61}:\\ \;\;\;\;\frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{\left(a + 1\right) + \left(\sqrt[3]{y \cdot \frac{b}{t}} \cdot \sqrt[3]{y \cdot \frac{b}{t}}\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{\frac{b}{t}}\right)}\\ \mathbf{elif}\;t \le 1.3709026813582713 \cdot 10^{237}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \left(y \cdot b\right) \cdot \frac{1}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \end{array}\]

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 <= -5.010089166367012e+61)) {
		VAR = ((x + ((y / (cbrt(t) * cbrt(t))) * (z / cbrt(t)))) / ((a + 1.0) + ((cbrt((y * (b / t))) * cbrt((y * (b / t)))) * (cbrt(y) * cbrt((b / t))))));
	} else {
		double VAR_1;
		if ((t <= 1.3709026813582713e+237)) {
			VAR_1 = ((x + ((y * z) / t)) / ((a + 1.0) + ((y * b) * (1.0 / t))));
		} else {
			VAR_1 = ((x + (y / (t / z))) / ((a + 1.0) + (y * (b / 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	13.5
Herbie	13.9

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

Derivation

1. Split input into 3 regimes

2. if t < -5.010089166367012e+61

   1. Initial program 11.6

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

   3. Applied *-un-lft-identity 11.6

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

   4. Applied times-frac 8.1

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

   5. Simplified 8.1

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

   7. Applied add-cube-cbrt 8.3

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

   8. Applied times-frac 3.1

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

   10. Applied add-cube-cbrt 3.2

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

   12. Applied cbrt-prod 3.2

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

   if -5.010089166367012e+61 < t < 1.3709026813582713e+237

   1. Initial program 18.1

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

   3. Applied div-inv 18.1

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

   if 1.3709026813582713e+237 < t

   1. Initial program 14.0

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

   3. Applied *-un-lft-identity 14.0

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

   4. Applied times-frac 9.1

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

   5. Simplified 9.1

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

   7. Applied associate-/l* 0.5

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

4. Final simplification 13.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.01008916636701195 \cdot 10^{61}:\\ \;\;\;\;\frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{\left(a + 1\right) + \left(\sqrt[3]{y \cdot \frac{b}{t}} \cdot \sqrt[3]{y \cdot \frac{b}{t}}\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{\frac{b}{t}}\right)}\\ \mathbf{elif}\;t \le 1.3709026813582713 \cdot 10^{237}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \left(y \cdot b\right) \cdot \frac{1}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020075 
(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))))