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

Average Error: 16.3 → 5.1

Time: 7.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y \cdot b}{t}\\ t_2 := 1 + \left(a + t_1\right)\\ t_3 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + t_1}\\ t_4 := \frac{x}{t_2}\\ \mathbf{if}\;t_3 \leq -8.818781100815444 \cdot 10^{-35}:\\ \;\;\;\;z \cdot \frac{y}{t \cdot t_2} + t_4\\ \mathbf{elif}\;t_3 \leq 1.2837857584797247 \cdot 10^{-59}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;t_4 + z \cdot \frac{y}{y \cdot b + \left(t + t \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;t_4 + \frac{z}{b}\\ \end{array} \]

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 (+ 1.0 (+ a t_1)))
        (t_3 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) t_1)))
        (t_4 (/ x t_2)))
   (if (<= t_3 -8.818781100815444e-35)
     (+ (* z (/ y (* t t_2))) t_4)
     (if (<= t_3 1.2837857584797247e-59)
       (/ (fma y (/ z t) x) (+ 1.0 (fma b (/ y t) a)))
       (if (<= t_3 INFINITY)
         (+ t_4 (* z (/ y (+ (* y b) (+ t (* t a))))))
         (+ t_4 (/ z 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 = 1.0 + (a + t_1);
	double t_3 = (x + ((y * z) / t)) / ((a + 1.0) + t_1);
	double t_4 = x / t_2;
	double tmp;
	if (t_3 <= -8.818781100815444e-35) {
		tmp = (z * (y / (t * t_2))) + t_4;
	} else if (t_3 <= 1.2837857584797247e-59) {
		tmp = fma(y, (z / t), x) / (1.0 + fma(b, (y / t), a));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4 + (z * (y / ((y * b) + (t + (t * a)))));
	} else {
		tmp = t_4 + (z / 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	16.3
Target	13.4
Herbie	5.1

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -8.8187811008154439e-35

   1. Initial program 10.3

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

   2. Taylor expanded in x around 0 6.1

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

   3. Applied associate-/l*_binary64 3.4

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

   4. Applied associate-/r/_binary64 1.0

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

   if -8.8187811008154439e-35 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 1.28378575847972467e-59

   1. Initial program 11.5

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

   2. Simplified 9.7

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

   if 1.28378575847972467e-59 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 11.5

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

   2. Taylor expanded in x around 0 7.0

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

   3. Applied associate-/l*_binary64 4.0

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

   4. Applied associate-/r/_binary64 1.6

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

   5. Taylor expanded in a around 0 1.6

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

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t)))

   1. Initial program 64.0

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

   2. Taylor expanded in x around 0 60.9

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

   3. Taylor expanded in y around inf 2.6

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

4. Final simplification 5.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -8.818781100815444 \cdot 10^{-35}:\\ \;\;\;\;z \cdot \frac{y}{t \cdot \left(1 + \left(a + \frac{y \cdot b}{t}\right)\right)} + \frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 1.2837857584797247 \cdot 10^{-59}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq \infty:\\ \;\;\;\;\frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)} + z \cdot \frac{y}{y \cdot b + \left(t + t \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)} + \frac{z}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2022061 
(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))))