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

Average Error: 3.7 → 0.5

Time: 6.8s

Precision: binary64

Math

\[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]

↓

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \leq -\infty:\\ \;\;\;\;x \cdot 2 + \left(a \cdot \left(27 \cdot b\right) - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right)\\ \mathbf{elif}\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \leq 3.540025357460908 \cdot 10^{+303}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2 + \left(\sqrt[3]{a \cdot \left(27 \cdot b\right)} \cdot \left(\sqrt[3]{a \cdot \left(27 \cdot b\right)} \cdot \sqrt[3]{a \cdot \left(27 \cdot b\right)}\right) - y \cdot \left(9 \cdot \left(z \cdot t\right)\right)\right)\\ \end{array}\]

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((double) (((double) (((double) (x * 2.0)) - ((double) (((double) (((double) (y * 9.0)) * z)) * t)))) + ((double) (((double) (a * 27.0)) * b))));
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	double VAR;
	if ((((double) (((double) (((double) (x * 2.0)) - ((double) (((double) (((double) (y * 9.0)) * z)) * t)))) + ((double) (((double) (a * 27.0)) * b)))) <= ((double) -(((double) INFINITY))))) {
		VAR = ((double) (((double) (x * 2.0)) + ((double) (((double) (a * ((double) (27.0 * b)))) - ((double) (((double) (y * 9.0)) * ((double) (z * t))))))));
	} else {
		double VAR_1;
		if ((((double) (((double) (((double) (x * 2.0)) - ((double) (((double) (((double) (y * 9.0)) * z)) * t)))) + ((double) (((double) (a * 27.0)) * b)))) <= 3.540025357460908e+303)) {
			VAR_1 = ((double) (((double) (((double) (x * 2.0)) - ((double) (((double) (((double) (y * 9.0)) * z)) * t)))) + ((double) (((double) (a * 27.0)) * b))));
		} else {
			VAR_1 = ((double) (((double) (x * 2.0)) + ((double) (((double) (((double) cbrt(((double) (a * ((double) (27.0 * b)))))) * ((double) (((double) cbrt(((double) (a * ((double) (27.0 * b)))))) * ((double) cbrt(((double) (a * ((double) (27.0 * b)))))))))) - ((double) (y * ((double) (9.0 * ((double) (z * t))))))))));
		}
		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	3.7
Target	2.7
Herbie	0.5

\[\begin{array}{l} \mathbf{if}\;y < 7.590524218811189 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b)) < -inf.0

   1. Initial program 64.0

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]

   2. Simplified 0.3

      \[\leadsto \color{blue}{x \cdot 2 + \left(a \cdot \left(27 \cdot b\right) - y \cdot \left(9 \cdot \left(z \cdot t\right)\right)\right)}\]
   3. Using strategy rm

   4. Applied associate-*r* 1.0

      \[\leadsto x \cdot 2 + \left(a \cdot \left(27 \cdot b\right) - \color{blue}{\left(y \cdot 9\right) \cdot \left(z \cdot t\right)}\right)\]

   if -inf.0 < (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b)) < 3.54002535746090829e303

   1. Initial program 0.4

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]

   if 3.54002535746090829e303 < (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b))

   1. Initial program 50.8

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]

   2. Simplified 1.4

      \[\leadsto \color{blue}{x \cdot 2 + \left(a \cdot \left(27 \cdot b\right) - y \cdot \left(9 \cdot \left(z \cdot t\right)\right)\right)}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 1.5

      \[\leadsto x \cdot 2 + \left(\color{blue}{\left(\sqrt[3]{a \cdot \left(27 \cdot b\right)} \cdot \sqrt[3]{a \cdot \left(27 \cdot b\right)}\right) \cdot \sqrt[3]{a \cdot \left(27 \cdot b\right)}} - y \cdot \left(9 \cdot \left(z \cdot t\right)\right)\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \leq -\infty:\\ \;\;\;\;x \cdot 2 + \left(a \cdot \left(27 \cdot b\right) - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right)\\ \mathbf{elif}\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \leq 3.540025357460908 \cdot 10^{+303}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2 + \left(\sqrt[3]{a \cdot \left(27 \cdot b\right)} \cdot \left(\sqrt[3]{a \cdot \left(27 \cdot b\right)} \cdot \sqrt[3]{a \cdot \left(27 \cdot b\right)}\right) - y \cdot \left(9 \cdot \left(z \cdot t\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020199 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< y 7.590524218811189e-161) (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* a (* 27.0 b))) (+ (- (* x 2.0) (* 9.0 (* y (* t z)))) (* (* a 27.0) b)))

  (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b)))