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

Average Error: 16.2 → 9.6

Time: 13.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y \cdot b}{t}\\ t_2 := \frac{x}{1 + \left(a + t_1\right)}\\ t_3 := \frac{z}{b} + t_2\\ \mathbf{if}\;y \leq -6.04812534866938 \cdot 10^{+167}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 3.67379042608604 \cdot 10^{+52}:\\ \;\;\;\;t_2 + \frac{y \cdot z}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(a, t, t\right)\right)}\\ \mathbf{elif}\;y \leq 1.244253742756409 \cdot 10^{+146}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{t_1 + \left(1 + a\right)}\\ \mathbf{elif}\;y \leq 5.1233805809014946 \cdot 10^{+247}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(t, a, t\right)\right)} + \frac{x}{1 + \left(a + \frac{y}{\frac{t}{b}}\right)}\\ \end{array} \]

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 (/ (* y b) t))
        (t_2 (/ x (+ 1.0 (+ a t_1))))
        (t_3 (+ (/ z b) t_2)))
   (if (<= y -6.04812534866938e+167)
     t_3
     (if (<= y 3.67379042608604e+52)
       (+ t_2 (/ (* y z) (fma y b (fma a t t))))
       (if (<= y 1.244253742756409e+146)
         (/ (+ x (/ y (/ t z))) (+ t_1 (+ 1.0 a)))
         (if (<= y 5.1233805809014946e+247)
           t_3
           (+
            (* y (/ z (fma y b (fma t a t))))
            (/ x (+ 1.0 (+ a (/ y (/ t b))))))))))))

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 = (y * b) / t;
	double t_2 = x / (1.0 + (a + t_1));
	double t_3 = (z / b) + t_2;
	double tmp;
	if (y <= -6.04812534866938e+167) {
		tmp = t_3;
	} else if (y <= 3.67379042608604e+52) {
		tmp = t_2 + ((y * z) / fma(y, b, fma(a, t, t)));
	} else if (y <= 1.244253742756409e+146) {
		tmp = (x + (y / (t / z))) / (t_1 + (1.0 + a));
	} else if (y <= 5.1233805809014946e+247) {
		tmp = t_3;
	} else {
		tmp = (y * (z / fma(y, b, fma(t, a, t)))) + (x / (1.0 + (a + (y / (t / b)))));
	}
	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	16.2
Target	13.1
Herbie	9.6

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

Derivation

1. Split input into 4 regimes

2. if y < -6.04812534866938001e167 or 1.24425374275640895e146 < y < 5.12338058090149459e247

   1. Initial program 37.9

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

   2. Taylor expanded in x around 0 36.5

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

   3. Taylor expanded in y around inf 19.6

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

   if -6.04812534866938001e167 < y < 3.6737904260860398e52

   1. Initial program 7.6

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

   2. Taylor expanded in x around 0 6.5

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

   3. Taylor expanded in z around inf 4.5

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

   4. Simplified 4.5

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

   5. Applied *-un-lft-identity_binary64 4.5

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

   6. Applied *-un-lft-identity_binary64 4.5

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

   7. Applied times-frac_binary64 4.5

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

   8. Simplified 4.5

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

   if 3.6737904260860398e52 < y < 1.24425374275640895e146

   1. Initial program 23.0

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

   2. Applied associate-/l*_binary64 19.7

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

   if 5.12338058090149459e247 < y

   1. Initial program 44.8

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

   2. Taylor expanded in x around 0 42.5

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

   3. Taylor expanded in z around inf 39.4

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

   4. Simplified 38.9

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

   5. Applied *-un-lft-identity_binary64 38.9

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

   6. Applied times-frac_binary64 32.2

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

   7. Simplified 32.2

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

   8. Simplified 32.2

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

   9. Applied associate-/l*_binary64 29.1

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

4. Final simplification 9.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.04812534866938 \cdot 10^{+167}:\\ \;\;\;\;\frac{z}{b} + \frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)}\\ \mathbf{elif}\;y \leq 3.67379042608604 \cdot 10^{+52}:\\ \;\;\;\;\frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)} + \frac{y \cdot z}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(a, t, t\right)\right)}\\ \mathbf{elif}\;y \leq 1.244253742756409 \cdot 10^{+146}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\frac{y \cdot b}{t} + \left(1 + a\right)}\\ \mathbf{elif}\;y \leq 5.1233805809014946 \cdot 10^{+247}:\\ \;\;\;\;\frac{z}{b} + \frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(t, a, t\right)\right)} + \frac{x}{1 + \left(a + \frac{y}{\frac{t}{b}}\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022005 
(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))))