Development.Shake.Progress:decay from shake-0.15.5

Average Error: 23.0 → 14.2

Time: 9.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} = -\infty:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le -1.23026288465321788 \cdot 10^{-300}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b - y, z, y\right)} \cdot \mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le 0.0:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le 4.17068802715010518 \cdot 10^{302}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b - y, z, y\right)} \cdot \mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a, double b) {
	double VAR;
	if ((((double) (((double) (((double) (x * y)) + ((double) (z * ((double) (t - a)))))) / ((double) (y + ((double) (z * ((double) (b - y)))))))) <= -inf.0)) {
		VAR = x;
	} else {
		double VAR_1;
		if ((((double) (((double) (((double) (x * y)) + ((double) (z * ((double) (t - a)))))) / ((double) (y + ((double) (z * ((double) (b - y)))))))) <= -1.2302628846532179e-300)) {
			VAR_1 = ((double) (((double) (1.0 / ((double) fma(((double) (b - y)), z, y)))) * ((double) fma(x, y, ((double) (z * ((double) (t - a))))))));
		} else {
			double VAR_2;
			if ((((double) (((double) (((double) (x * y)) + ((double) (z * ((double) (t - a)))))) / ((double) (y + ((double) (z * ((double) (b - y)))))))) <= 0.0)) {
				VAR_2 = ((double) (((double) (t / b)) - ((double) (a / b))));
			} else {
				double VAR_3;
				if ((((double) (((double) (((double) (x * y)) + ((double) (z * ((double) (t - a)))))) / ((double) (y + ((double) (z * ((double) (b - y)))))))) <= 4.170688027150105e+302)) {
					VAR_3 = ((double) (((double) (1.0 / ((double) fma(((double) (b - y)), z, y)))) * ((double) fma(x, y, ((double) (z * ((double) (t - a))))))));
				} else {
					VAR_3 = ((double) (((double) (t / b)) - ((double) (a / b))));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

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	23.0
Target	17.9
Herbie	14.2

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 64.0

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

   3. Applied clear-num 64.0

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

   4. Simplified 64.0

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

   5. Taylor expanded around 0 36.3

      \[\leadsto \color{blue}{x}\]

   if -inf.0 < (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))) < -1.2302628846532179e-300 or 0.0 < (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))) < 4.170688027150105e+302

   1. Initial program 0.3

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

   3. Applied clear-num 0.5

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

   4. Simplified 0.5

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

   6. Applied div-inv 0.6

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

   7. Applied add-cube-cbrt 0.6

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

   8. Applied times-frac 0.6

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

   9. Simplified 0.6

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

   10. Simplified 0.4

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

   if -1.2302628846532179e-300 < (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))) < 0.0 or 4.170688027150105e+302 < (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y))))

   1. Initial program 58.6

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

   3. Applied clear-num 58.7

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

   4. Simplified 58.7

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

   5. Taylor expanded around inf 36.6

      \[\leadsto \color{blue}{\frac{t}{b} - \frac{a}{b}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 14.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} = -\infty:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le -1.23026288465321788 \cdot 10^{-300}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b - y, z, y\right)} \cdot \mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le 0.0:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le 4.17068802715010518 \cdot 10^{302}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b - y, z, y\right)} \cdot \mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020120 +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)))))