Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A

Average Error: 10.5 → 5.2

Time: 6.1s

Precision: binary64

Math

\[\frac{x - y \cdot z}{t - a \cdot z} \]

↓

\[\begin{array}{l} t_1 := x - y \cdot z\\ t_2 := \frac{t_1}{t - z \cdot a}\\ t_3 := \frac{t_1}{\mathsf{fma}\left(-a, z, t\right)}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;t_2 \leq -1.20736653492792 \cdot 10^{-309}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;t_2 \leq 1.9822646599559802 \cdot 10^{+292}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (/ (- x (* y z)) (- t (* a z))))

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (* y z)))
        (t_2 (/ t_1 (- t (* z a))))
        (t_3 (/ t_1 (fma (- a) z t))))
   (if (<= t_2 (- INFINITY))
     (/ y a)
     (if (<= t_2 -1.20736653492792e-309)
       t_3
       (if (<= t_2 0.0)
         (/ y a)
         (if (<= t_2 1.9822646599559802e+292) t_3 (/ y a)))))))

C

double code(double x, double y, double z, double t, double a) {
	return (x - (y * z)) / (t - (a * z));
}

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (y * z);
	double t_2 = t_1 / (t - (z * a));
	double t_3 = t_1 / fma(-a, z, t);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = y / a;
	} else if (t_2 <= -1.20736653492792e-309) {
		tmp = t_3;
	} else if (t_2 <= 0.0) {
		tmp = y / a;
	} else if (t_2 <= 1.9822646599559802e+292) {
		tmp = t_3;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	10.5
Target	1.7
Herbie	5.2

\[\begin{array}{l} \mathbf{if}\;z < -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z < 3.5139522372978296 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -inf.0 or -1.207366534927921e-309 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -0.0 or 1.98226465995598025e292 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 41.4

      \[\frac{x - y \cdot z}{t - a \cdot z} \]

   2. Taylor expanded in z around inf 20.1

      \[\leadsto \color{blue}{\frac{y}{a}} \]

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.207366534927921e-309 or -0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 1.98226465995598025e292

   1. Initial program 0.2

      \[\frac{x - y \cdot z}{t - a \cdot z} \]

   2. Applied egg-rr 0.2

      \[\leadsto \frac{x - y \cdot z}{\color{blue}{\mathsf{fma}\left(-a, z, t\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 5.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -1.20736653492792 \cdot 10^{-309}:\\ \;\;\;\;\frac{x - y \cdot z}{\mathsf{fma}\left(-a, z, t\right)}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 1.9822646599559802 \cdot 10^{+292}:\\ \;\;\;\;\frac{x - y \cdot z}{\mathsf{fma}\left(-a, z, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -32113435955957344.0) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1.0 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

  (/ (- x (* y z)) (- t (* a z))))