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

Average Error: 16.6 → 8.5

Time: 8.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t_1 \leq -4.152181355139608 \cdot 10^{-284}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{1 + \left(a + \frac{y}{\frac{t}{b}}\right)}, \frac{z}{t}, \frac{x}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y}, \frac{x}{b}, \frac{z}{b}\right) - \mathsf{fma}\left(\frac{t}{y}, \frac{z}{b \cdot b}, \frac{a \cdot \left(z \cdot t\right)}{y \cdot \left(b \cdot b\right)}\right)\\ \mathbf{elif}\;t_1 \leq 4.064482167495813 \cdot 10^{+299}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

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 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_1 -4.152181355139608e-284)
     (fma
      (/ y (+ 1.0 (+ a (/ y (/ t b)))))
      (/ z t)
      (/ x (+ 1.0 (fma b (/ y t) a))))
     (if (<= t_1 0.0)
       (-
        (fma (/ t y) (/ x b) (/ z b))
        (fma (/ t y) (/ z (* b b)) (/ (* a (* z t)) (* y (* b b)))))
       (if (<= t_1 4.064482167495813e+299) t_1 (/ 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 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double tmp;
	if (t_1 <= -4.152181355139608e-284) {
		tmp = fma((y / (1.0 + (a + (y / (t / b))))), (z / t), (x / (1.0 + fma(b, (y / t), a))));
	} else if (t_1 <= 0.0) {
		tmp = fma((t / y), (x / b), (z / b)) - fma((t / y), (z / (b * b)), ((a * (z * t)) / (y * (b * b))));
	} else if (t_1 <= 4.064482167495813e+299) {
		tmp = t_1;
	} else {
		tmp = 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	16.6
Target	12.9
Herbie	8.5

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -4.1521813551396081e-284

   1. Initial program 7.3

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

   2. Simplified 9.8

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

   3. Taylor expanded in z around 0 5.3

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

   4. Simplified 9.6

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

   5. Applied fma-udef_binary64 9.6

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

   6. Simplified 7.7

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

   7. Applied associate-/l*_binary64 7.7

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

   if -4.1521813551396081e-284 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 28.1

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

   2. Simplified 18.7

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

   3. Taylor expanded in y around inf 27.7

      \[\leadsto \color{blue}{\left(\frac{z}{b} + \frac{t \cdot x}{y \cdot b}\right) - \left(\frac{t \cdot z}{y \cdot {b}^{2}} + \frac{a \cdot \left(t \cdot z\right)}{y \cdot {b}^{2}}\right)} \]

   4. Simplified 22.1

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

   if -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 4.0644821674958132e299

   1. Initial program 0.4

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

   if 4.0644821674958132e299 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t)))

   1. Initial program 63.2

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

   2. Simplified 51.9

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

   3. Taylor expanded in y around inf 12.9

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 8.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -4.152181355139608 \cdot 10^{-284}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{1 + \left(a + \frac{y}{\frac{t}{b}}\right)}, \frac{z}{t}, \frac{x}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y}, \frac{x}{b}, \frac{z}{b}\right) - \mathsf{fma}\left(\frac{t}{y}, \frac{z}{b \cdot b}, \frac{a \cdot \left(z \cdot t\right)}{y \cdot \left(b \cdot b\right)}\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 4.064482167495813 \cdot 10^{+299}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(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))))