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

Average Error: 16.6 → 13.1

Time: 6.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -5.7447987803197 \cdot 10^{-311}:\\ \;\;\;\;\frac{x + \frac{z \cdot \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -0:\\ \;\;\;\;\left(x + y \cdot \frac{z}{t}\right) \cdot \frac{1}{a + \left(1 + y \cdot \frac{b}{t}\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 5.525389761919301 \cdot 10^{+251}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{a + \left(1 + y \cdot \frac{b}{t}\right)}\\ \end{array}\]

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 (((((double) (x + (((double) (y * z)) / t))) / ((double) (((double) (a + 1.0)) + (((double) (y * b)) / t)))) <= -5.7447987803197e-311)) {
		VAR = (((double) (x + (((double) (z * (y / ((double) (((double) cbrt(t)) * ((double) cbrt(t))))))) / ((double) cbrt(t))))) / ((double) (((double) (a + 1.0)) + (((double) (y * b)) / t))));
	} else {
		double VAR_1;
		if (((((double) (x + (((double) (y * z)) / t))) / ((double) (((double) (a + 1.0)) + (((double) (y * b)) / t)))) <= -0.0)) {
			VAR_1 = ((double) (((double) (x + ((double) (y * (z / t))))) * (1.0 / ((double) (a + ((double) (1.0 + ((double) (y * (b / t))))))))));
		} else {
			double VAR_2;
			if (((((double) (x + (((double) (y * z)) / t))) / ((double) (((double) (a + 1.0)) + (((double) (y * b)) / t)))) <= 5.525389761919301e+251)) {
				VAR_2 = (((double) (x + (((double) (y * z)) / t))) / ((double) (((double) (a + 1.0)) + (((double) (y * b)) / t))));
			} else {
				VAR_2 = (((double) (x + ((double) ((y / ((double) (((double) cbrt(t)) * ((double) cbrt(t))))) * (z / ((double) cbrt(t))))))) / ((double) (a + ((double) (1.0 + ((double) (y * (b / t))))))));
			}
			VAR_1 = VAR_2;
		}
		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	16.6
Target	13.4
Herbie	13.1

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

Derivation

1. Split input into 4 regimes

2. if (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))) < -5.74479878031971e-311

   1. Initial program 6.7

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

   3. Applied add-cube-cbrt 7.1

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

   4. Applied associate-/r* 7.1

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

   5. Simplified 5.6

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

   if -5.74479878031971e-311 < (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))) < -0.0

   1. Initial program 29.2

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

   2. Simplified 20.0

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

   4. Applied div-inv 20.1

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

   if -0.0 < (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))) < 5.5253897619193015e251

   1. Initial program 0.2

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

   if 5.5253897619193015e251 < (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))

   1. Initial program 58.1

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

   2. Simplified 48.0

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

   4. Applied add-cube-cbrt 48.2

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

   5. Applied *-un-lft-identity 48.2

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

   6. Applied times-frac 48.2

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

   7. Applied associate-*r* 47.9

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

   8. Simplified 47.9

      \[\leadsto \frac{x + \color{blue}{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \frac{z}{\sqrt[3]{t}}}{a + \left(1 + y \cdot \frac{b}{t}\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 13.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -5.7447987803197 \cdot 10^{-311}:\\ \;\;\;\;\frac{x + \frac{z \cdot \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -0:\\ \;\;\;\;\left(x + y \cdot \frac{z}{t}\right) \cdot \frac{1}{a + \left(1 + y \cdot \frac{b}{t}\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 5.525389761919301 \cdot 10^{+251}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{a + \left(1 + y \cdot \frac{b}{t}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020198 
(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))))