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

Average Error: 17.0 → 13.4

Time: 7.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} = -inf.0 \lor \neg \left(\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \le -6.98981573682074936 \cdot 10^{-167} \lor \neg \left(\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \le -0.0 \lor \neg \left(\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \le 1.68643473026138206 \cdot 10^{157}\right)\right)\right):\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \left(y \cdot b\right) \cdot \frac{1}{t}}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a, double b) {
	double VAR;
	if (((((double) (((double) (x + ((double) (((double) (y * z)) / t)))) / ((double) (((double) (a + 1.0)) + ((double) (((double) (y * b)) / t)))))) <= -inf.0) || !((((double) (((double) (x + ((double) (((double) (y * z)) / t)))) / ((double) (((double) (a + 1.0)) + ((double) (((double) (y * b)) / t)))))) <= -6.98981573682075e-167) || !((((double) (((double) (x + ((double) (((double) (y * z)) / t)))) / ((double) (((double) (a + 1.0)) + ((double) (((double) (y * b)) / t)))))) <= -0.0) || !(((double) (((double) (x + ((double) (((double) (y * z)) / t)))) / ((double) (((double) (a + 1.0)) + ((double) (((double) (y * b)) / t)))))) <= 1.686434730261382e+157))))) {
		VAR = ((double) (((double) (x + ((double) (y * ((double) (z / t)))))) / ((double) (((double) (a + 1.0)) + ((double) (y * ((double) (b / t))))))));
	} else {
		VAR = ((double) (((double) (x + ((double) (((double) (y * z)) / t)))) / ((double) (((double) (a + 1.0)) + ((double) (((double) (y * b)) * ((double) (1.0 / t))))))));
	}
	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	17.0
Target	13.2
Herbie	13.4

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

Derivation

1. Split input into 2 regimes

2. if (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))) < -inf.0 or -6.98981573682074936e-167 < (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))) < -0.0 or 1.68643473026138206e157 < (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))

   1. Initial program 36.4

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

   3. Applied associate-/l* 32.5

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

   5. Applied *-un-lft-identity 32.5

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

   6. Applied times-frac 28.5

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

   7. Simplified 28.5

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

   9. Applied div-inv 28.6

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

   10. Simplified 28.5

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

   if -inf.0 < (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))) < -6.98981573682074936e-167 or -0.0 < (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))) < 1.68643473026138206e157

   1. Initial program 0.4

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

   3. Applied div-inv 0.4

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

4. Final simplification 13.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} = -inf.0 \lor \neg \left(\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \le -6.98981573682074936 \cdot 10^{-167} \lor \neg \left(\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \le -0.0 \lor \neg \left(\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \le 1.68643473026138206 \cdot 10^{157}\right)\right)\right):\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \left(y \cdot b\right) \cdot \frac{1}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020173 
(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))))