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

Average Error: 19.8 → 3.8

Time: 9.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} = -inf.0:\\ \;\;\;\;\frac{b}{z \cdot c} + \left(9 \cdot \left(y \cdot \frac{x}{z \cdot c}\right) - 4 \cdot \left(\left(\sqrt[3]{t \cdot \frac{a}{c}} \cdot \sqrt[3]{t \cdot \frac{a}{c}}\right) \cdot \sqrt[3]{\left(t \cdot \left(\sqrt[3]{a} \cdot \frac{\sqrt[3]{a}}{\sqrt[3]{c} \cdot \sqrt[3]{c}}\right)\right) \cdot \frac{\sqrt[3]{a}}{\sqrt[3]{c}}}\right)\right)\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -1.4822 \cdot 10^{-323}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 8.0594233453660248 \cdot 10^{54}:\\ \;\;\;\;\frac{\frac{1}{\frac{z}{b + x \cdot \left(9 \cdot y\right)}} - 4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 1.215755384673848 \cdot 10^{294}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{z \cdot c} + \left(9 \cdot \left(y \cdot \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{c \cdot \frac{z}{\sqrt[3]{x}}}\right) - 4 \cdot \left(t \cdot \frac{a}{c}\right)\right)\\ \end{array}\]

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 ((((double) (((double) (((double) (((double) (((double) (x * 9.0)) * y)) - ((double) (((double) (((double) (z * 4.0)) * t)) * a)))) + b)) / ((double) (z * c)))) <= -inf.0)) {
		VAR = ((double) (((double) (b / ((double) (z * c)))) + ((double) (((double) (9.0 * ((double) (y * ((double) (x / ((double) (z * c)))))))) - ((double) (4.0 * ((double) (((double) (((double) cbrt(((double) (t * ((double) (a / c)))))) * ((double) cbrt(((double) (t * ((double) (a / c)))))))) * ((double) cbrt(((double) (((double) (t * ((double) (((double) cbrt(a)) * ((double) (((double) cbrt(a)) / ((double) (((double) cbrt(c)) * ((double) cbrt(c)))))))))) * ((double) (((double) cbrt(a)) / ((double) cbrt(c))))))))))))))));
	} else {
		double VAR_1;
		if ((((double) (((double) (((double) (((double) (((double) (x * 9.0)) * y)) - ((double) (((double) (((double) (z * 4.0)) * t)) * a)))) + b)) / ((double) (z * c)))) <= -1.4821969375237e-323)) {
			VAR_1 = ((double) (((double) (((double) (((double) (((double) (x * 9.0)) * y)) - ((double) (((double) (((double) (z * 4.0)) * t)) * a)))) + b)) / ((double) (z * c))));
		} else {
			double VAR_2;
			if ((((double) (((double) (((double) (((double) (((double) (x * 9.0)) * y)) - ((double) (((double) (((double) (z * 4.0)) * t)) * a)))) + b)) / ((double) (z * c)))) <= 8.059423345366025e+54)) {
				VAR_2 = ((double) (((double) (((double) (1.0 / ((double) (z / ((double) (b + ((double) (x * ((double) (9.0 * y)))))))))) - ((double) (4.0 * ((double) (t * a)))))) / c));
			} else {
				double VAR_3;
				if ((((double) (((double) (((double) (((double) (((double) (x * 9.0)) * y)) - ((double) (((double) (((double) (z * 4.0)) * t)) * a)))) + b)) / ((double) (z * c)))) <= 1.215755384673848e+294)) {
					VAR_3 = ((double) (((double) (((double) (((double) (((double) (x * 9.0)) * y)) - ((double) (((double) (((double) (z * 4.0)) * t)) * a)))) + b)) / ((double) (z * c))));
				} else {
					VAR_3 = ((double) (((double) (b / ((double) (z * c)))) + ((double) (((double) (9.0 * ((double) (y * ((double) (((double) (((double) cbrt(x)) * ((double) cbrt(x)))) / ((double) (c * ((double) (z / ((double) cbrt(x)))))))))))) - ((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	19.8
Target	14.0
Herbie	3.8

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

Derivation

1. Split input into 4 regimes

2. if (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 24.9

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

   3. Taylor expanded around 0 31.9

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

   4. Simplified 9.7

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

   6. Applied add-cube-cbrt 10.3

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

   8. Applied add-cube-cbrt 10.2

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

   9. Applied add-cube-cbrt 10.3

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

   10. Applied times-frac 10.3

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

   11. Applied associate-*r* 10.3

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

   12. Simplified 10.3

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

   if -inf.0 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -1.4822e-323 or 8.0594233453660248e54 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 1.215755384673848e294

   1. Initial program 0.6

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

   if -1.4822e-323 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 8.0594233453660248e54

   1. Initial program 17.3

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

   2. Simplified 1.5

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

   4. Applied clear-num 1.6

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

   if 1.215755384673848e294 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))

   1. Initial program 61.3

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

   2. Simplified 26.3

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

   3. Taylor expanded around 0 29.3

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

   4. Simplified 15.8

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

   6. Applied add-cube-cbrt 16.0

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

   7. Applied associate-/l* 16.0

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

   8. Simplified 13.4

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

4. Final simplification 3.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} = -inf.0:\\ \;\;\;\;\frac{b}{z \cdot c} + \left(9 \cdot \left(y \cdot \frac{x}{z \cdot c}\right) - 4 \cdot \left(\left(\sqrt[3]{t \cdot \frac{a}{c}} \cdot \sqrt[3]{t \cdot \frac{a}{c}}\right) \cdot \sqrt[3]{\left(t \cdot \left(\sqrt[3]{a} \cdot \frac{\sqrt[3]{a}}{\sqrt[3]{c} \cdot \sqrt[3]{c}}\right)\right) \cdot \frac{\sqrt[3]{a}}{\sqrt[3]{c}}}\right)\right)\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -1.4822 \cdot 10^{-323}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 8.0594233453660248 \cdot 10^{54}:\\ \;\;\;\;\frac{\frac{1}{\frac{z}{b + x \cdot \left(9 \cdot y\right)}} - 4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 1.215755384673848 \cdot 10^{294}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{z \cdot c} + \left(9 \cdot \left(y \cdot \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{c \cdot \frac{z}{\sqrt[3]{x}}}\right) - 4 \cdot \left(t \cdot \frac{a}{c}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(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)))