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

Average Error: 20.8 → 4.6

Time: 7.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -1.77373236189849584 \cdot 10^{-276}:\\ \;\;\;\;\frac{b}{z \cdot c} + \left(9 \cdot \left(y \cdot \frac{x}{z \cdot c}\right) - 4 \cdot \left(\left(t \cdot \left(\sqrt[3]{a} \cdot \frac{\sqrt[3]{a}}{\sqrt[3]{c} \cdot \sqrt[3]{c}}\right)\right) \cdot \frac{\sqrt[3]{a}}{\sqrt[3]{c}}\right)\right)\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 3.6756499733293384 \cdot 10^{25}:\\ \;\;\;\;\left(\frac{b + x \cdot \left(9 \cdot y\right)}{z} - 4 \cdot \left(t \cdot a\right)\right) \cdot \frac{1}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 3.21424165622918218 \cdot 10^{303}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{z \cdot c} + \left(9 \cdot \left(y \cdot \frac{x}{z \cdot c}\right) - 4 \cdot \left(\left(t \cdot \left(\sqrt[3]{a} \cdot \frac{\sqrt[3]{a}}{\sqrt[3]{c} \cdot \sqrt[3]{c}}\right)\right) \cdot \frac{\sqrt[3]{a}}{\sqrt[3]{c}}\right)\right)\\ \end{array}\]

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((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) (x * 9.0)) * y)) - ((double) (((double) (((double) (z * 4.0)) * t)) * a)))) + b)) / ((double) (z * c))) <= -1.7737323618984958e-276)) {
		VAR = ((double) ((b / ((double) (z * c))) + ((double) (((double) (9.0 * ((double) (y * (x / ((double) (z * c))))))) - ((double) (4.0 * ((double) (((double) (t * ((double) (((double) cbrt(a)) * (((double) cbrt(a)) / ((double) (((double) cbrt(c)) * ((double) cbrt(c))))))))) * (((double) cbrt(a)) / ((double) cbrt(c)))))))))));
	} else {
		double VAR_1;
		if (((((double) (((double) (((double) (((double) (x * 9.0)) * y)) - ((double) (((double) (((double) (z * 4.0)) * t)) * a)))) + b)) / ((double) (z * c))) <= 3.6756499733293384e+25)) {
			VAR_1 = ((double) (((double) ((((double) (b + ((double) (x * ((double) (9.0 * y)))))) / z) - ((double) (4.0 * ((double) (t * a)))))) * (1.0 / c)));
		} else {
			double VAR_2;
			if (((((double) (((double) (((double) (((double) (x * 9.0)) * y)) - ((double) (((double) (((double) (z * 4.0)) * t)) * a)))) + b)) / ((double) (z * c))) <= 3.214241656229182e+303)) {
				VAR_2 = (((double) (((double) (((double) (((double) (x * 9.0)) * y)) - ((double) (((double) (((double) (z * 4.0)) * t)) * a)))) + b)) / ((double) (z * c)));
			} else {
				VAR_2 = ((double) ((b / ((double) (z * c))) + ((double) (((double) (9.0 * ((double) (y * (x / ((double) (z * c))))))) - ((double) (4.0 * ((double) (((double) (t * ((double) (((double) cbrt(a)) * (((double) cbrt(a)) / ((double) (((double) cbrt(c)) * ((double) cbrt(c))))))))) * (((double) cbrt(a)) / ((double) cbrt(c)))))))))));
			}
			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	20.8
Target	14.7
Herbie	4.6

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

Derivation

1. Split input into 3 regimes

2. if (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -1.77373236189849584e-276 or 3.21424165622918218e303 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))

   1. Initial program 28.5

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

   2. Simplified 17.3

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

   3. Taylor expanded around 0 15.0

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

   4. Simplified 9.8

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

   6. Applied add-cube-cbrt 10.2

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

   7. Applied add-cube-cbrt 10.3

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

   8. Applied times-frac 10.3

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

   9. Applied associate-*r* 7.3

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

   10. Simplified 7.3

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

   if -1.77373236189849584e-276 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 3.6756499733293384e25

   1. Initial program 18.8

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

   2. Simplified 1.0

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

   4. Applied div-inv 1.1

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

   if 3.6756499733293384e25 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 3.21424165622918218e303

   1. Initial program 0.7

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

4. Final simplification 4.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -1.77373236189849584 \cdot 10^{-276}:\\ \;\;\;\;\frac{b}{z \cdot c} + \left(9 \cdot \left(y \cdot \frac{x}{z \cdot c}\right) - 4 \cdot \left(\left(t \cdot \left(\sqrt[3]{a} \cdot \frac{\sqrt[3]{a}}{\sqrt[3]{c} \cdot \sqrt[3]{c}}\right)\right) \cdot \frac{\sqrt[3]{a}}{\sqrt[3]{c}}\right)\right)\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 3.6756499733293384 \cdot 10^{25}:\\ \;\;\;\;\left(\frac{b + x \cdot \left(9 \cdot y\right)}{z} - 4 \cdot \left(t \cdot a\right)\right) \cdot \frac{1}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 3.21424165622918218 \cdot 10^{303}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{z \cdot c} + \left(9 \cdot \left(y \cdot \frac{x}{z \cdot c}\right) - 4 \cdot \left(\left(t \cdot \left(\sqrt[3]{a} \cdot \frac{\sqrt[3]{a}}{\sqrt[3]{c} \cdot \sqrt[3]{c}}\right)\right) \cdot \frac{\sqrt[3]{a}}{\sqrt[3]{c}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(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)))