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

Average Error: 10.2 → 2.6

Time: 6.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := x - y \cdot z\\ t_2 := t - z \cdot a\\ t_3 := \frac{t_1}{t_2}\\ t_4 := \frac{x}{t_2}\\ \mathbf{if}\;t_3 \leq -1.6490978098404618 \cdot 10^{-72}:\\ \;\;\;\;t_4 - \frac{y}{\frac{t_2}{z}}\\ \mathbf{elif}\;t_3 \leq -1.91994533465 \cdot 10^{-313}:\\ \;\;\;\;t_1 \cdot \frac{1}{t_2}\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;t_3 \leq 1.4642476401666246 \cdot 10^{+25}:\\ \;\;\;\;t_4 - \frac{y \cdot z}{t_2}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;t_4 - y \cdot \frac{z}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} - \frac{x}{z \cdot a}\\ \end{array} \]

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 (* z a)))
        (t_3 (/ t_1 t_2))
        (t_4 (/ x t_2)))
   (if (<= t_3 -1.6490978098404618e-72)
     (- t_4 (/ y (/ t_2 z)))
     (if (<= t_3 -1.91994533465e-313)
       (* t_1 (/ 1.0 t_2))
       (if (<= t_3 0.0)
         (/ (- y (/ x z)) a)
         (if (<= t_3 1.4642476401666246e+25)
           (- t_4 (/ (* y z) t_2))
           (if (<= t_3 INFINITY)
             (- t_4 (* y (/ z t_2)))
             (- (/ y a) (/ x (* z 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 - (z * a);
	double t_3 = t_1 / t_2;
	double t_4 = x / t_2;
	double tmp;
	if (t_3 <= -1.6490978098404618e-72) {
		tmp = t_4 - (y / (t_2 / z));
	} else if (t_3 <= -1.91994533465e-313) {
		tmp = t_1 * (1.0 / t_2);
	} else if (t_3 <= 0.0) {
		tmp = (y - (x / z)) / a;
	} else if (t_3 <= 1.4642476401666246e+25) {
		tmp = t_4 - ((y * z) / t_2);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4 - (y * (z / t_2));
	} else {
		tmp = (y / a) - (x / (z * 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	10.2
Target	1.8
Herbie	2.6

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

Derivation

1. Split input into 6 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.64909780984046183e-72

   1. Initial program 7.0

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Using strategy rm

   3. Applied div-sub_binary64 7.0

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

   4. Simplified 7.0

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

   5. Simplified 7.0

      \[\leadsto \frac{x}{t - z \cdot a} - \color{blue}{\frac{y \cdot z}{t - z \cdot a}} \]
   6. Using strategy rm

   7. Applied associate-/l*_binary64 0.3

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

   if -1.64909780984046183e-72 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.91994533465e-313

   1. Initial program 0.2

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Using strategy rm

   3. Applied div-inv_binary64 0.4

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

   4. Simplified 0.4

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

   if -1.91994533465e-313 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0

   1. Initial program 25.3

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Using strategy rm

   3. Applied div-sub_binary64 25.3

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

   4. Simplified 25.3

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

   5. Simplified 25.3

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

   6. Taylor expanded in a around inf 16.5

      \[\leadsto \color{blue}{\frac{y - \frac{x}{z}}{a}} \]

   if 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 1.4642476401666246e25

   1. Initial program 0.2

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Using strategy rm

   3. Applied div-sub_binary64 0.2

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

   4. Simplified 0.2

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

   5. Simplified 0.2

      \[\leadsto \frac{x}{t - z \cdot a} - \color{blue}{\frac{y \cdot z}{t - z \cdot a}} \]
   6. Using strategy rm

   7. Applied cancel-sign-sub-inv_binary64 0.2

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

   if 1.4642476401666246e25 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 9.6

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Using strategy rm

   3. Applied div-sub_binary64 9.6

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

   4. Simplified 9.6

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

   5. Simplified 9.6

      \[\leadsto \frac{x}{t - z \cdot a} - \color{blue}{\frac{y \cdot z}{t - z \cdot a}} \]
   6. Using strategy rm

   7. Applied *-un-lft-identity_binary64 9.6

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

   8. Applied times-frac_binary64 0.2

      \[\leadsto \frac{x}{t - z \cdot a} - \color{blue}{\frac{y}{1} \cdot \frac{z}{t - z \cdot a}} \]

   9. Simplified 0.2

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

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

   1. Initial program 64.0

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Using strategy rm

   3. Applied div-sub_binary64 64.0

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

   4. Simplified 64.0

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

   5. Simplified 64.0

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

   6. Taylor expanded in a around inf 0

      \[\leadsto \color{blue}{\frac{y - \frac{x}{z}}{a}} \]

   7. Simplified 0.1

      \[\leadsto \color{blue}{\frac{y}{a} - \frac{x}{z \cdot a}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 2.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -1.6490978098404618 \cdot 10^{-72}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y}{\frac{t - z \cdot a}{z}}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -1.91994533465 \cdot 10^{-313}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 1.4642476401666246 \cdot 10^{+25}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq \infty:\\ \;\;\;\;\frac{x}{t - z \cdot a} - y \cdot \frac{z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} - \frac{x}{z \cdot a}\\ \end{array} \]

Reproduce

herbie shell --seed 2021206 
(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))))