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

Average Error: 16.8 → 9.5

Time: 8.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)}\\ t_2 := \frac{z}{b} + t_1\\ \mathbf{if}\;y \leq -5.981645553620746 \cdot 10^{+276}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_3 := 1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)\\ t_4 := \frac{x}{t_3}\\ \mathbf{if}\;y \leq -1.0875447241440656 \cdot 10^{+227}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{t_3}{y}}, \frac{z}{t}, t_4\right)\\ \mathbf{elif}\;y \leq -1.884688025365851 \cdot 10^{+169}:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{elif}\;y \leq 1.081035654397059 \cdot 10^{-9}:\\ \;\;\;\;t_1 + \frac{y \cdot z}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(a, t, t\right)\right)}\\ \mathbf{elif}\;y \leq 6.194405303028895 \cdot 10^{+221}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t_3}, \frac{z}{t}, t_4\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array}\\ \end{array} \]

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
 (let* ((t_1 (/ x (+ 1.0 (+ a (/ (* y b) t))))) (t_2 (+ (/ z b) t_1)))
   (if (<= y -5.981645553620746e+276)
     t_2
     (let* ((t_3 (+ 1.0 (fma b (/ y t) a))) (t_4 (/ x t_3)))
       (if (<= y -1.0875447241440656e+227)
         (fma (/ 1.0 (/ t_3 y)) (/ z t) t_4)
         (if (<= y -1.884688025365851e+169)
           (/ (+ z (/ (* x t) y)) b)
           (if (<= y 1.081035654397059e-9)
             (+ t_1 (/ (* y z) (fma y b (fma a t t))))
             (if (<= y 6.194405303028895e+221)
               (fma (/ y t_3) (/ z t) t_4)
               t_2))))))))

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 t_1 = x / (1.0 + (a + ((y * b) / t)));
	double t_2 = (z / b) + t_1;
	double tmp;
	if (y <= -5.981645553620746e+276) {
		tmp = t_2;
	} else {
		double t_3 = 1.0 + fma(b, (y / t), a);
		double t_4 = x / t_3;
		double tmp_1;
		if (y <= -1.0875447241440656e+227) {
			tmp_1 = fma((1.0 / (t_3 / y)), (z / t), t_4);
		} else if (y <= -1.884688025365851e+169) {
			tmp_1 = (z + ((x * t) / y)) / b;
		} else if (y <= 1.081035654397059e-9) {
			tmp_1 = t_1 + ((y * z) / fma(y, b, fma(a, t, t)));
		} else if (y <= 6.194405303028895e+221) {
			tmp_1 = fma((y / t_3), (z / t), t_4);
		} else {
			tmp_1 = t_2;
		}
		tmp = tmp_1;
	}
	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	16.8
Target	13.6
Herbie	9.5

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

Derivation

1. Split input into 5 regimes

2. if y < -5.9816455536207457e276 or 6.194405303028895e221 < y

   1. Initial program 43.0

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

   2. Simplified 38.2

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

   3. Taylor expanded in z around 0 40.3

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

   4. Taylor expanded in y around inf 16.0

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

   if -5.9816455536207457e276 < y < -1.0875447241440656e227

   1. Initial program 42.2

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

   2. Simplified 36.2

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

   3. Taylor expanded in z around 0 40.0

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

   4. Simplified 31.5

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

   5. Applied clear-num_binary64 31.5

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

   if -1.0875447241440656e227 < y < -1.884688025365851e169

   1. Initial program 37.4

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

   2. Simplified 28.7

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

   3. Taylor expanded in z around 0 35.4

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

   4. Simplified 24.2

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

   5. Taylor expanded in b around inf 23.2

      \[\leadsto \color{blue}{\frac{\frac{t \cdot x}{y} + z}{b}} \]

   if -1.884688025365851e169 < y < 1.081035654397059e-9

   1. Initial program 7.5

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

   2. Simplified 10.7

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

   3. Taylor expanded in z around 0 6.1

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

   4. Taylor expanded in z around inf 4.1

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

   5. Simplified 4.0

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

   if 1.081035654397059e-9 < y < 6.194405303028895e221

   1. Initial program 25.5

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

   2. Simplified 20.4

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

   3. Taylor expanded in z around 0 22.8

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

   4. Simplified 17.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{z}{t}, \frac{x}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 9.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.981645553620746 \cdot 10^{+276}:\\ \;\;\;\;\frac{z}{b} + \frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)}\\ \mathbf{elif}\;y \leq -1.0875447241440656 \cdot 10^{+227}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}{y}}, \frac{z}{t}, \frac{x}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}\right)\\ \mathbf{elif}\;y \leq -1.884688025365851 \cdot 10^{+169}:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{elif}\;y \leq 1.081035654397059 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)} + \frac{y \cdot z}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(a, t, t\right)\right)}\\ \mathbf{elif}\;y \leq 6.194405303028895 \cdot 10^{+221}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{z}{t}, \frac{x}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b} + \frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021280 
(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))))