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

Average Error: 20.5 → 9.7

Time: 1.2min

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -4.12533014972370793 \cdot 10^{-111}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{elif}\;t \le -3.5511642756597985 \cdot 10^{-215}:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;t \le 6.1338338921947738 \cdot 10^{-218}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(x \cdot \frac{1}{\frac{z \cdot c}{y}}\right)\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;t \le 1.4271930213022396 \cdot 10^{-183}:\\ \;\;\;\;\left(\frac{\frac{b}{z}}{c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;t \le 4.38316265159631932 \cdot 10^{-156}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + \left(9 \cdot \frac{x}{z}\right) \cdot \frac{y}{c}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;t \le 2.35065710944335856 \cdot 10^{39}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(x \cdot \frac{1}{\frac{z \cdot c}{y}}\right)\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;t \le 9.7887978917909521 \cdot 10^{288}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + \left(\sqrt[3]{9 \cdot \frac{x \cdot y}{z \cdot c}} \cdot \sqrt[3]{9 \cdot \frac{x \cdot y}{z \cdot c}}\right) \cdot \sqrt[3]{9 \cdot \frac{x \cdot y}{z \cdot c}}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

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 <= -4.125330149723708e-111)) {
		VAR = ((double) (((double) (((double) (b / ((double) (z * c)))) + ((double) (9.0 * ((double) (x / ((double) (((double) (z * c)) / y)))))))) - ((double) (4.0 * ((double) (a * ((double) (t / c))))))));
	} else {
		double VAR_1;
		if ((t <= -3.5511642756597985e-215)) {
			VAR_1 = ((double) (((double) (((double) (((double) (((double) (((double) (x * 9.0)) * y)) - ((double) (((double) (((double) (z * 4.0)) * t)) * a)))) + b)) / z)) / c));
		} else {
			double VAR_2;
			if ((t <= 6.133833892194774e-218)) {
				VAR_2 = ((double) (((double) (((double) (b / ((double) (z * c)))) + ((double) (9.0 * ((double) (x * ((double) (1.0 / ((double) (((double) (z * c)) / y)))))))))) - ((double) (4.0 * ((double) (((double) (a * t)) / c))))));
			} else {
				double VAR_3;
				if ((t <= 1.4271930213022396e-183)) {
					VAR_3 = ((double) (((double) (((double) (((double) (b / z)) / c)) + ((double) (9.0 * ((double) (((double) (x * y)) / ((double) (z * c)))))))) - ((double) (4.0 * ((double) (((double) (a * t)) / c))))));
				} else {
					double VAR_4;
					if ((t <= 4.3831626515963193e-156)) {
						VAR_4 = ((double) (((double) (((double) (b / ((double) (z * c)))) + ((double) (((double) (9.0 * ((double) (x / z)))) * ((double) (y / c)))))) - ((double) (4.0 * ((double) (((double) (a * t)) / c))))));
					} else {
						double VAR_5;
						if ((t <= 2.3506571094433586e+39)) {
							VAR_5 = ((double) (((double) (((double) (b / ((double) (z * c)))) + ((double) (9.0 * ((double) (x * ((double) (1.0 / ((double) (((double) (z * c)) / y)))))))))) - ((double) (4.0 * ((double) (((double) (a * t)) / c))))));
						} else {
							double VAR_6;
							if ((t <= 9.788797891790952e+288)) {
								VAR_6 = ((double) (((double) (((double) (b / ((double) (z * c)))) + ((double) (9.0 * ((double) (x / ((double) (((double) (z * c)) / y)))))))) - ((double) (4.0 * ((double) (a * ((double) (t / c))))))));
							} else {
								VAR_6 = ((double) (((double) (((double) (b / ((double) (z * c)))) + ((double) (((double) (((double) cbrt(((double) (9.0 * ((double) (((double) (x * y)) / ((double) (z * c)))))))) * ((double) cbrt(((double) (9.0 * ((double) (((double) (x * y)) / ((double) (z * c)))))))))) * ((double) cbrt(((double) (9.0 * ((double) (((double) (x * y)) / ((double) (z * c)))))))))))) - ((double) (4.0 * ((double) (((double) (a * t)) / c))))));
							}
							VAR_5 = VAR_6;
						}
						VAR_4 = VAR_5;
					}
					VAR_3 = VAR_4;
				}
				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	20.5
Target	14.3
Herbie	9.7

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

Derivation

1. Split input into 6 regimes

2. if t < -4.12533014972370793e-111 or 2.35065710944335856e39 < t < 9.7887978917909521e288

   1. Initial program 27.5

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

   2. Taylor expanded around 0 14.1

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

   4. Applied associate-/l* 12.7

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

   6. Applied *-un-lft-identity 12.7

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

   7. Applied times-frac 9.6

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

   8. Simplified 9.6

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

   if -4.12533014972370793e-111 < t < -3.5511642756597985e-215

   1. Initial program 14.3

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

   3. Applied associate-/r* 11.2

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

   if -3.5511642756597985e-215 < t < 6.1338338921947738e-218 or 4.38316265159631932e-156 < t < 2.35065710944335856e39

   1. Initial program 13.1

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

   2. Taylor expanded around 0 8.9

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

   4. Applied associate-/l* 8.8

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

   6. Applied div-inv 9.0

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

   if 6.1338338921947738e-218 < t < 1.4271930213022396e-183

   1. Initial program 10.7

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

   2. Taylor expanded around 0 6.5

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

   4. Applied associate-/r* 8.8

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

   if 1.4271930213022396e-183 < t < 4.38316265159631932e-156

   1. Initial program 14.8

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

   2. Taylor expanded around 0 12.2

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

   4. Applied times-frac 11.9

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

   5. Applied associate-*r* 11.9

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

   if 9.7887978917909521e288 < t

   1. Initial program 41.4

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

   2. Taylor expanded around 0 19.2

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

   4. Applied add-cube-cbrt 19.4

      \[\leadsto \left(\frac{b}{z \cdot c} + \color{blue}{\left(\sqrt[3]{9 \cdot \frac{x \cdot y}{z \cdot c}} \cdot \sqrt[3]{9 \cdot \frac{x \cdot y}{z \cdot c}}\right) \cdot \sqrt[3]{9 \cdot \frac{x \cdot y}{z \cdot c}}}\right) - 4 \cdot \frac{a \cdot t}{c}\]
3. Recombined 6 regimes into one program.

4. Final simplification 9.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.12533014972370793 \cdot 10^{-111}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{elif}\;t \le -3.5511642756597985 \cdot 10^{-215}:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;t \le 6.1338338921947738 \cdot 10^{-218}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(x \cdot \frac{1}{\frac{z \cdot c}{y}}\right)\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;t \le 1.4271930213022396 \cdot 10^{-183}:\\ \;\;\;\;\left(\frac{\frac{b}{z}}{c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;t \le 4.38316265159631932 \cdot 10^{-156}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + \left(9 \cdot \frac{x}{z}\right) \cdot \frac{y}{c}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;t \le 2.35065710944335856 \cdot 10^{39}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(x \cdot \frac{1}{\frac{z \cdot c}{y}}\right)\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;t \le 9.7887978917909521 \cdot 10^{288}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + \left(\sqrt[3]{9 \cdot \frac{x \cdot y}{z \cdot c}} \cdot \sqrt[3]{9 \cdot \frac{x \cdot y}{z \cdot c}}\right) \cdot \sqrt[3]{9 \cdot \frac{x \cdot y}{z \cdot c}}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Reproduce

herbie shell --seed 2020163 
(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)))