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

Average Error: 10.7 → 3.7

Time: 7.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := t - z \cdot a\\ t_2 := \frac{x - y \cdot z}{t_1}\\ t_3 := \frac{x}{t_1}\\ \mathbf{if}\;t_2 \leq -5.15120624 \cdot 10^{-316}:\\ \;\;\;\;\begin{array}{l} t_4 := \sqrt[3]{t_1}\\ t_3 - \frac{y}{t_4 \cdot t_4} \cdot \frac{z}{t_4} \end{array}\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;t_2 \leq 2.5937576388216173 \cdot 10^{+162}:\\ \;\;\;\;t_3 - \frac{y \cdot z}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_3 - \frac{y}{\frac{t}{z} - a}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* z a))) (t_2 (/ (- x (* y z)) t_1)) (t_3 (/ x t_1)))
   (if (<= t_2 -5.15120624e-316)
     (let* ((t_4 (cbrt t_1))) (- t_3 (* (/ y (* t_4 t_4)) (/ z t_4))))
     (if (<= t_2 0.0)
       (/ (- y (/ x z)) a)
       (if (<= t_2 2.5937576388216173e+162)
         (- t_3 (/ (* y z) t_1))
         (- t_3 (/ 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 t_1 = t - (z * a);
	double t_2 = (x - (y * z)) / t_1;
	double t_3 = x / t_1;
	double tmp;
	if (t_2 <= -5.15120624e-316) {
		double t_4_1 = cbrt(t_1);
		tmp = t_3 - ((y / (t_4_1 * t_4_1)) * (z / t_4_1));
	} else if (t_2 <= 0.0) {
		tmp = (y - (x / z)) / a;
	} else if (t_2 <= 2.5937576388216173e+162) {
		tmp = t_3 - ((y * z) / t_1);
	} else {
		tmp = t_3 - (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	1.9
Herbie	3.7

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -5.151206239e-316

   1. Initial program 4.5

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

   2. Taylor expanded in x around 0 4.5

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

   3. Applied add-cube-cbrt_binary64 4.9

      \[\leadsto \frac{x}{t - a \cdot z} - \frac{y \cdot z}{\color{blue}{\left(\sqrt[3]{t - a \cdot z} \cdot \sqrt[3]{t - a \cdot z}\right) \cdot \sqrt[3]{t - a \cdot z}}} \]

   4. Applied times-frac_binary64 3.1

      \[\leadsto \frac{x}{t - a \cdot z} - \color{blue}{\frac{y}{\sqrt[3]{t - a \cdot z} \cdot \sqrt[3]{t - a \cdot z}} \cdot \frac{z}{\sqrt[3]{t - a \cdot z}}} \]

   if -5.151206239e-316 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0

   1. Initial program 26.3

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

   2. Taylor expanded in x around 0 26.3

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

   3. Taylor expanded in t around 0 27.2

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

   4. Simplified 16.6

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

   if 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 2.5937576388216173e162

   1. Initial program 0.2

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

   2. Taylor expanded in x around 0 0.2

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

   if 2.5937576388216173e162 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 34.8

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

   2. Taylor expanded in x around 0 34.8

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

   3. Applied *-un-lft-identity_binary64 34.8

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

   4. Applied times-frac_binary64 21.0

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

   5. Simplified 21.0

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

   6. Simplified 21.0

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

   7. Taylor expanded in y around 0 34.8

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

   8. Simplified 0.1

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

4. Final simplification 3.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -5.15120624 \cdot 10^{-316}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y}{\sqrt[3]{t - z \cdot a} \cdot \sqrt[3]{t - z \cdot a}} \cdot \frac{z}{\sqrt[3]{t - z \cdot a}}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 2.5937576388216173 \cdot 10^{+162}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y \cdot z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y}{\frac{t}{z} - a}\\ \end{array} \]

Reproduce

herbie shell --seed 2022019 
(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))))