Development.Shake.Progress:decay from shake-0.15.5

Average Error: 22.4 → 19.2

Time: 7.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -3.79681007453336803 \cdot 10^{235} \lor \neg \left(z \le 9.3702509275758488 \cdot 10^{111}\right):\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, t - a, -x \cdot y\right)}{-\mathsf{fma}\left(b - y, z, y\right)}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a, double b) {
	double temp;
	if (((z <= -3.796810074533368e+235) || !(z <= 9.370250927575849e+111))) {
		temp = ((t / b) - (a / b));
	} else {
		temp = (fma(-z, (t - a), -(x * y)) / -fma((b - y), z, y));
	}
	return temp;
}

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	22.4
Target	17.3
Herbie	19.2

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

Derivation

1. Split input into 2 regimes

2. if z < -3.796810074533368e+235 or 9.370250927575849e+111 < z

   1. Initial program 49.2

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
   2. Using strategy rm

   3. Applied clear-num 49.2

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

   4. Simplified 49.2

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

   5. Taylor expanded around inf 34.0

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

   if -3.796810074533368e+235 < z < 9.370250927575849e+111

   1. Initial program 15.3

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
   2. Using strategy rm

   3. Applied div-inv 15.4

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

   4. Simplified 15.4

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

   6. Applied add-cube-cbrt 15.8

      \[\leadsto \left(x \cdot y + z \cdot \color{blue}{\left(\left(\sqrt[3]{t - a} \cdot \sqrt[3]{t - a}\right) \cdot \sqrt[3]{t - a}\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(b - y, z, y\right)}\]

   7. Applied associate-*r* 15.8

      \[\leadsto \left(x \cdot y + \color{blue}{\left(z \cdot \left(\sqrt[3]{t - a} \cdot \sqrt[3]{t - a}\right)\right) \cdot \sqrt[3]{t - a}}\right) \cdot \frac{1}{\mathsf{fma}\left(b - y, z, y\right)}\]
   8. Using strategy rm

   9. Applied frac-2neg 15.8

      \[\leadsto \left(x \cdot y + \left(z \cdot \left(\sqrt[3]{t - a} \cdot \sqrt[3]{t - a}\right)\right) \cdot \sqrt[3]{t - a}\right) \cdot \color{blue}{\frac{-1}{-\mathsf{fma}\left(b - y, z, y\right)}}\]

   10. Applied associate-*r/ 15.8

       \[\leadsto \color{blue}{\frac{\left(x \cdot y + \left(z \cdot \left(\sqrt[3]{t - a} \cdot \sqrt[3]{t - a}\right)\right) \cdot \sqrt[3]{t - a}\right) \cdot \left(-1\right)}{-\mathsf{fma}\left(b - y, z, y\right)}}\]

   11. Simplified 15.3

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

4. Final simplification 19.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.79681007453336803 \cdot 10^{235} \lor \neg \left(z \le 9.3702509275758488 \cdot 10^{111}\right):\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, t - a, -x \cdot y\right)}{-\mathsf{fma}\left(b - y, z, y\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(FPCore (x y z t a b)
  :name "Development.Shake.Progress:decay from shake-0.15.5"
  :precision binary64

  :herbie-target
  (- (/ (+ (* z t) (* y x)) (+ y (* z (- b y)))) (/ a (+ (- b y) (/ y z))))

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