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

Average Error: 16.8 → 8.8

Time: 7.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := 1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)\\ \mathbf{if}\;t \leq -5.099390865643201 \cdot 10^{-35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_2 := \frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)}\\ \mathbf{if}\;t \leq -5.019015735760609 \cdot 10^{-197}:\\ \;\;\;\;\frac{y}{\frac{\mathsf{fma}\left(y, b, \mathsf{fma}\left(t, a, t\right)\right)}{z}} + t_2\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_3 := t_2 + \frac{z}{b}\\ \mathbf{if}\;t \leq 1.8963796502357436 \cdot 10^{-175}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 9975149583244896:\\ \;\;\;\;t_2 + \frac{y \cdot z}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(a, t, t\right)\right)}\\ \mathbf{elif}\;t \leq 2.0973695853553908 \cdot 10^{+40}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t_1}, \frac{z}{t}, \frac{x}{t_1}\right)\\ \end{array}\\ \end{array}\\ \end{array} \]

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 (+ 1.0 (fma b (/ y t) a))))
   (if (<= t -5.099390865643201e-35)
     (/ (fma y (/ z t) x) t_1)
     (let* ((t_2 (/ x (+ 1.0 (+ a (/ (* y b) t))))))
       (if (<= t -5.019015735760609e-197)
         (+ (/ y (/ (fma y b (fma t a t)) z)) t_2)
         (let* ((t_3 (+ t_2 (/ z b))))
           (if (<= t 1.8963796502357436e-175)
             t_3
             (if (<= t 9975149583244896.0)
               (+ t_2 (/ (* y z) (fma y b (fma a t t))))
               (if (<= t 2.0973695853553908e+40)
                 t_3
                 (fma (/ y t_1) (/ z t) (/ x t_1)))))))))))

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 = 1.0 + fma(b, (y / t), a);
	double tmp;
	if (t <= -5.099390865643201e-35) {
		tmp = fma(y, (z / t), x) / t_1;
	} else {
		double t_2 = x / (1.0 + (a + ((y * b) / t)));
		double tmp_1;
		if (t <= -5.019015735760609e-197) {
			tmp_1 = (y / (fma(y, b, fma(t, a, t)) / z)) + t_2;
		} else {
			double t_3 = t_2 + (z / b);
			double tmp_2;
			if (t <= 1.8963796502357436e-175) {
				tmp_2 = t_3;
			} else if (t <= 9975149583244896.0) {
				tmp_2 = t_2 + ((y * z) / fma(y, b, fma(a, t, t)));
			} else if (t <= 2.0973695853553908e+40) {
				tmp_2 = t_3;
			} else {
				tmp_2 = fma((y / t_1), (z / t), (x / t_1));
			}
			tmp_1 = tmp_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	13.4
Herbie	8.8

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

Derivation

1. Split input into 5 regimes

2. if t < -5.09939086564320129e-35

   1. Initial program 11.3

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

   2. Simplified 4.7

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

   if -5.09939086564320129e-35 < t < -5.01901573576060879e-197

   1. Initial program 18.2

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

   2. Simplified 25.4

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

   3. Taylor expanded in z around 0 14.7

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

   4. Taylor expanded in z around inf 11.2

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

   5. Simplified 11.2

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

   6. Applied associate-/l*_binary64 11.8

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

   7. Simplified 11.8

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

   if -5.01901573576060879e-197 < t < 1.89637965023574361e-175 or 9975149583244896 < t < 2.09736958535539077e40

   1. Initial program 28.4

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

   2. Simplified 35.3

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

   3. Taylor expanded in z around 0 21.7

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

   4. Taylor expanded in y around inf 17.9

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

   if 1.89637965023574361e-175 < t < 9975149583244896

   1. Initial program 15.9

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

   2. Simplified 19.3

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

   3. Taylor expanded in z around 0 13.2

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

   4. Taylor expanded in z around inf 10.6

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

   5. Simplified 10.6

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

   if 2.09736958535539077e40 < t

   1. Initial program 12.3

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

   2. Simplified 3.2

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

   3. Taylor expanded in z around 0 14.3

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

   4. Simplified 2.4

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

4. Final simplification 8.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.099390865643201 \cdot 10^{-35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}\\ \mathbf{elif}\;t \leq -5.019015735760609 \cdot 10^{-197}:\\ \;\;\;\;\frac{y}{\frac{\mathsf{fma}\left(y, b, \mathsf{fma}\left(t, a, t\right)\right)}{z}} + \frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)}\\ \mathbf{elif}\;t \leq 1.8963796502357436 \cdot 10^{-175}:\\ \;\;\;\;\frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)} + \frac{z}{b}\\ \mathbf{elif}\;t \leq 9975149583244896:\\ \;\;\;\;\frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)} + \frac{y \cdot z}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(a, t, t\right)\right)}\\ \mathbf{elif}\;t \leq 2.0973695853553908 \cdot 10^{+40}:\\ \;\;\;\;\frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)} + \frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{z}{t}, \frac{x}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022104 
(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))))