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

Average Error: 10.7 → 1.7

Time: 8.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -1.945477398004434 \cdot 10^{+304}:\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -1.3715219134961312 \cdot 10^{-297}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{-1}{\frac{\frac{t}{z} - a}{y}}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 3.300473813311166 \cdot 10^{+303}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y \cdot z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \end{array}\]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (if (<= (/ (- x (* y z)) (- t (* z a))) -1.945477398004434e+304)
   (/ (- y) (- (/ t z) a))
   (if (<= (/ (- x (* y z)) (- t (* z a))) -1.3715219134961312e-297)
     (/ (- x (* y z)) (- t (* z a)))
     (if (<= (/ (- x (* y z)) (- t (* z a))) 0.0)
       (/ -1.0 (/ (- (/ t z) a) y))
       (if (<= (/ (- x (* y z)) (- t (* z a))) 3.300473813311166e+303)
         (- (/ x (- t (* z a))) (/ (* y z) (- t (* z a))))
         (/ (- y) (- (/ t 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 tmp;
	if (((x - (y * z)) / (t - (z * a))) <= -1.945477398004434e+304) {
		tmp = -y / ((t / z) - a);
	} else if (((x - (y * z)) / (t - (z * a))) <= -1.3715219134961312e-297) {
		tmp = (x - (y * z)) / (t - (z * a));
	} else if (((x - (y * z)) / (t - (z * a))) <= 0.0) {
		tmp = -1.0 / (((t / z) - a) / y);
	} else if (((x - (y * z)) / (t - (z * a))) <= 3.300473813311166e+303) {
		tmp = (x / (t - (z * a))) - ((y * z) / (t - (z * a)));
	} else {
		tmp = -y / ((t / 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	10.7
Target	1.5
Herbie	1.7

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.9454773980044341e304 or 3.300473813311166e303 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 62.1

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

   3. Applied div-inv_binary64_14738 62.1

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

   4. Simplified 62.1

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

   5. Taylor expanded around 0 63.3

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

   6. Simplified 1.3

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

   if -1.9454773980044341e304 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.3715219134961312e-297

   1. Initial program 0.2

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

   if -1.3715219134961312e-297 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0

   1. Initial program 24.0

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

   3. Applied div-inv_binary64_14738 24.0

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

   4. Simplified 24.0

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

   5. Taylor expanded around 0 24.7

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

   6. Simplified 9.4

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

   8. Applied neg-mul-1_binary64_14737 9.4

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

   9. Applied associate-/l*_binary64_14686 10.1

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

   if 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 3.300473813311166e303

   1. Initial program 0.2

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

   3. Applied div-sub_binary64_14746 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

      \[\leadsto \frac{x}{t - z \cdot a} - \color{blue}{\frac{z \cdot y}{t - z \cdot a}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 1.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -1.945477398004434 \cdot 10^{+304}:\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -1.3715219134961312 \cdot 10^{-297}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{-1}{\frac{\frac{t}{z} - a}{y}}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 3.300473813311166 \cdot 10^{+303}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y \cdot z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \end{array}\]

Reproduce

herbie shell --seed 2021064 
(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))))