Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, J

Average Error: 20.1 → 9.1

Time: 8.2s

Precision: binary64

Math

\[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

↓

\[\begin{array}{l} \mathbf{if}\;t \le -8.34880938412968186 \cdot 10^{-171}:\\ \;\;\;\;\frac{b}{z \cdot c} + \left(x \cdot \frac{9 \cdot y}{z \cdot c} - 4 \cdot \left(\frac{t}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{a}{\sqrt[3]{c}}\right)\right)\\ \mathbf{elif}\;t \le 7.27959378407285776 \cdot 10^{-268}:\\ \;\;\;\;\frac{\left(x \cdot \frac{9 \cdot y}{z} + \frac{b}{z}\right) - 4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{elif}\;t \le 3.14191028065253483 \cdot 10^{-240}:\\ \;\;\;\;\frac{b}{z \cdot c} + \left(\left(x \cdot \frac{9}{z}\right) \cdot \frac{y}{c} - 4 \cdot \left(t \cdot \frac{a}{c}\right)\right)\\ \mathbf{elif}\;t \le 0.06196880829952622:\\ \;\;\;\;\frac{\left(x \cdot \frac{9 \cdot y}{z} + \frac{b}{z}\right) - 4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{z \cdot c} + \left(x \cdot \left(\left(9 \cdot y\right) \cdot \frac{1}{z \cdot c}\right) - 4 \cdot \left(t \cdot \frac{a}{c}\right)\right)\\ \end{array}\]

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return ((double) (((double) (((double) (((double) (((double) (x * 9.0)) * y)) - ((double) (((double) (((double) (z * 4.0)) * t)) * a)))) + b)) / ((double) (z * c))));
}

↓

double code(double x, double y, double z, double t, double a, double b, double c) {
	double VAR;
	if ((t <= -8.348809384129682e-171)) {
		VAR = ((double) (((double) (b / ((double) (z * c)))) + ((double) (((double) (x * ((double) (((double) (9.0 * y)) / ((double) (z * c)))))) - ((double) (4.0 * ((double) (((double) (t / ((double) (((double) cbrt(c)) * ((double) cbrt(c)))))) * ((double) (a / ((double) cbrt(c))))))))))));
	} else {
		double VAR_1;
		if ((t <= 7.279593784072858e-268)) {
			VAR_1 = ((double) (((double) (((double) (((double) (x * ((double) (((double) (9.0 * y)) / z)))) + ((double) (b / z)))) - ((double) (4.0 * ((double) (t * a)))))) / c));
		} else {
			double VAR_2;
			if ((t <= 3.141910280652535e-240)) {
				VAR_2 = ((double) (((double) (b / ((double) (z * c)))) + ((double) (((double) (((double) (x * ((double) (9.0 / z)))) * ((double) (y / c)))) - ((double) (4.0 * ((double) (t * ((double) (a / c))))))))));
			} else {
				double VAR_3;
				if ((t <= 0.061968808299526223)) {
					VAR_3 = ((double) (((double) (((double) (((double) (x * ((double) (((double) (9.0 * y)) / z)))) + ((double) (b / z)))) - ((double) (4.0 * ((double) (t * a)))))) / c));
				} else {
					VAR_3 = ((double) (((double) (b / ((double) (z * c)))) + ((double) (((double) (x * ((double) (((double) (9.0 * y)) * ((double) (1.0 / ((double) (z * c)))))))) - ((double) (4.0 * ((double) (t * ((double) (a / c))))))))));
				}
				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

Bits error versus c

Target

Original	20.1
Target	14.5
Herbie	9.1

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.10015674080410512 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.17088779117474882 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.8768236795461372 \cdot 10^{130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.3838515042456319 \cdot 10^{158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if t < -8.34880938412968186e-171

   1. Initial program 22.6

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

   2. Simplified 14.4

      \[\leadsto \color{blue}{\frac{\frac{x \cdot \left(9 \cdot y\right) + b}{z} - 4 \cdot \left(t \cdot a\right)}{c}}\]

   3. Taylor expanded around 0 12.0

      \[\leadsto \color{blue}{\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a \cdot t}{c}}\]

   4. Simplified 9.0

      \[\leadsto \color{blue}{\frac{b}{z \cdot c} + \left(x \cdot \frac{9 \cdot y}{z \cdot c} - 4 \cdot \left(t \cdot \frac{a}{c}\right)\right)}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 9.4

      \[\leadsto \frac{b}{z \cdot c} + \left(x \cdot \frac{9 \cdot y}{z \cdot c} - 4 \cdot \left(t \cdot \frac{a}{\color{blue}{\left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right) \cdot \sqrt[3]{c}}}\right)\right)\]

   7. Applied *-un-lft-identity 9.4

      \[\leadsto \frac{b}{z \cdot c} + \left(x \cdot \frac{9 \cdot y}{z \cdot c} - 4 \cdot \left(t \cdot \frac{\color{blue}{1 \cdot a}}{\left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right) \cdot \sqrt[3]{c}}\right)\right)\]

   8. Applied times-frac 9.4

      \[\leadsto \frac{b}{z \cdot c} + \left(x \cdot \frac{9 \cdot y}{z \cdot c} - 4 \cdot \left(t \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{a}{\sqrt[3]{c}}\right)}\right)\right)\]

   9. Applied associate-*r* 9.5

      \[\leadsto \frac{b}{z \cdot c} + \left(x \cdot \frac{9 \cdot y}{z \cdot c} - 4 \cdot \color{blue}{\left(\left(t \cdot \frac{1}{\sqrt[3]{c} \cdot \sqrt[3]{c}}\right) \cdot \frac{a}{\sqrt[3]{c}}\right)}\right)\]

   10. Simplified 9.5

       \[\leadsto \frac{b}{z \cdot c} + \left(x \cdot \frac{9 \cdot y}{z \cdot c} - 4 \cdot \left(\color{blue}{\frac{t}{\sqrt[3]{c} \cdot \sqrt[3]{c}}} \cdot \frac{a}{\sqrt[3]{c}}\right)\right)\]

   if -8.34880938412968186e-171 < t < 7.27959378407285776e-268 or 3.14191028065253483e-240 < t < 0.06196880829952622

   1. Initial program 13.2

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

   2. Simplified 9.0

      \[\leadsto \color{blue}{\frac{\frac{x \cdot \left(9 \cdot y\right) + b}{z} - 4 \cdot \left(t \cdot a\right)}{c}}\]

   3. Taylor expanded around 0 8.9

      \[\leadsto \frac{\color{blue}{\left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)} - 4 \cdot \left(t \cdot a\right)}{c}\]

   4. Simplified 8.9

      \[\leadsto \frac{\color{blue}{\left(x \cdot \frac{9 \cdot y}{z} + \frac{b}{z}\right)} - 4 \cdot \left(t \cdot a\right)}{c}\]

   if 7.27959378407285776e-268 < t < 3.14191028065253483e-240

   1. Initial program 11.4

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

   2. Simplified 10.0

      \[\leadsto \color{blue}{\frac{\frac{x \cdot \left(9 \cdot y\right) + b}{z} - 4 \cdot \left(t \cdot a\right)}{c}}\]

   3. Taylor expanded around 0 10.1

      \[\leadsto \color{blue}{\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a \cdot t}{c}}\]

   4. Simplified 14.9

      \[\leadsto \color{blue}{\frac{b}{z \cdot c} + \left(x \cdot \frac{9 \cdot y}{z \cdot c} - 4 \cdot \left(t \cdot \frac{a}{c}\right)\right)}\]
   5. Using strategy rm

   6. Applied times-frac 13.8

      \[\leadsto \frac{b}{z \cdot c} + \left(x \cdot \color{blue}{\left(\frac{9}{z} \cdot \frac{y}{c}\right)} - 4 \cdot \left(t \cdot \frac{a}{c}\right)\right)\]

   7. Applied associate-*r* 15.7

      \[\leadsto \frac{b}{z \cdot c} + \left(\color{blue}{\left(x \cdot \frac{9}{z}\right) \cdot \frac{y}{c}} - 4 \cdot \left(t \cdot \frac{a}{c}\right)\right)\]

   if 0.06196880829952622 < t

   1. Initial program 28.4

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

   2. Simplified 18.0

      \[\leadsto \color{blue}{\frac{\frac{x \cdot \left(9 \cdot y\right) + b}{z} - 4 \cdot \left(t \cdot a\right)}{c}}\]

   3. Taylor expanded around 0 15.2

      \[\leadsto \color{blue}{\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a \cdot t}{c}}\]

   4. Simplified 7.7

      \[\leadsto \color{blue}{\frac{b}{z \cdot c} + \left(x \cdot \frac{9 \cdot y}{z \cdot c} - 4 \cdot \left(t \cdot \frac{a}{c}\right)\right)}\]
   5. Using strategy rm

   6. Applied div-inv 7.8

      \[\leadsto \frac{b}{z \cdot c} + \left(x \cdot \color{blue}{\left(\left(9 \cdot y\right) \cdot \frac{1}{z \cdot c}\right)} - 4 \cdot \left(t \cdot \frac{a}{c}\right)\right)\]
3. Recombined 4 regimes into one program.

4. Final simplification 9.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -8.34880938412968186 \cdot 10^{-171}:\\ \;\;\;\;\frac{b}{z \cdot c} + \left(x \cdot \frac{9 \cdot y}{z \cdot c} - 4 \cdot \left(\frac{t}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{a}{\sqrt[3]{c}}\right)\right)\\ \mathbf{elif}\;t \le 7.27959378407285776 \cdot 10^{-268}:\\ \;\;\;\;\frac{\left(x \cdot \frac{9 \cdot y}{z} + \frac{b}{z}\right) - 4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{elif}\;t \le 3.14191028065253483 \cdot 10^{-240}:\\ \;\;\;\;\frac{b}{z \cdot c} + \left(\left(x \cdot \frac{9}{z}\right) \cdot \frac{y}{c} - 4 \cdot \left(t \cdot \frac{a}{c}\right)\right)\\ \mathbf{elif}\;t \le 0.06196880829952622:\\ \;\;\;\;\frac{\left(x \cdot \frac{9 \cdot y}{z} + \frac{b}{z}\right) - 4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{z \cdot c} + \left(x \cdot \left(\left(9 \cdot y\right) \cdot \frac{1}{z \cdot c}\right) - 4 \cdot \left(t \cdot \frac{a}{c}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, J"
  :precision binary64

  :herbie-target
  (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) -1.1001567408041051e-171) (/ (+ (- (* (* x 9.0) y) (* (* z 4.0) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) -0.0) (/ (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) z) c) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 1.1708877911747488e-53) (/ (+ (- (* (* x 9.0) y) (* (* z 4.0) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 2.876823679546137e+130) (- (+ (* (* 9.0 (/ y c)) (/ x z)) (/ b (* c z))) (* 4.0 (/ (* a t) c))) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 1.3838515042456319e+158) (/ (+ (- (* (* x 9.0) y) (* (* z 4.0) (* t a))) b) (* z c)) (- (+ (* 9.0 (* (/ y (* c z)) x)) (/ b (* c z))) (* 4.0 (/ (* a t) c))))))))

  (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)))