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

Average Error: 3.5 → 0.7

Time: 3.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(y \cdot 9\right) \cdot z\right) \cdot t = -\infty:\\ \;\;\;\;x \cdot 2 + \left(27 \cdot \left(a \cdot b\right) - \left(9 \cdot \left(t \cdot y\right)\right) \cdot z\right)\\ \mathbf{elif}\;\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \le 3.271497844840803 \cdot 10^{271}:\\ \;\;\;\;x \cdot 2 + \left(27 \cdot \left(a \cdot b\right) - \left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \left(\left(\sqrt[3]{9} \cdot t\right) \cdot \left(z \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2 + \left(27 \cdot \left(a \cdot b\right) - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right)\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a, double b) {
	double VAR;
	if (((((y * 9.0) * z) * t) <= -inf.0)) {
		VAR = ((x * 2.0) + ((27.0 * (a * b)) - ((9.0 * (t * y)) * z)));
	} else {
		double VAR_1;
		if (((((y * 9.0) * z) * t) <= 3.2714978448408032e+271)) {
			VAR_1 = ((x * 2.0) + ((27.0 * (a * b)) - ((cbrt(9.0) * cbrt(9.0)) * ((cbrt(9.0) * t) * (z * y)))));
		} else {
			VAR_1 = ((x * 2.0) + ((27.0 * (a * b)) - ((y * 9.0) * (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	3.5
Target	2.8
Herbie	0.7

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

Derivation

1. Split input into 3 regimes

2. if (* (* (* y 9.0) z) t) < -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. Using strategy rm

   3. Applied sub-neg 64.0

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

   4. Applied associate-+l+ 64.0

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

   5. Simplified 64.0

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

   6. Taylor expanded around inf 63.0

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

   8. Applied add-cube-cbrt 63.0

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

   9. Applied associate-*l* 63.0

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

   10. Taylor expanded around inf 63.0

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

   11. Simplified 0.6

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

   if -inf.0 < (* (* (* y 9.0) z) t) < 3.2714978448408032e+271

   1. Initial program 0.4

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

   3. Applied sub-neg 0.4

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

   4. Applied associate-+l+ 0.4

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

   5. Simplified 0.4

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

   6. Taylor expanded around inf 0.4

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

   8. Applied add-cube-cbrt 0.4

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

   9. Applied associate-*l* 0.4

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

   11. Applied associate-*r* 0.4

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

   if 3.2714978448408032e+271 < (* (* (* y 9.0) z) t)

   1. Initial program 41.3

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

   3. Applied sub-neg 41.3

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

   4. Applied associate-+l+ 41.3

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

   5. Simplified 41.1

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

   7. Applied associate-*l* 7.7

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y \cdot 9\right) \cdot z\right) \cdot t = -\infty:\\ \;\;\;\;x \cdot 2 + \left(27 \cdot \left(a \cdot b\right) - \left(9 \cdot \left(t \cdot y\right)\right) \cdot z\right)\\ \mathbf{elif}\;\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \le 3.271497844840803 \cdot 10^{271}:\\ \;\;\;\;x \cdot 2 + \left(27 \cdot \left(a \cdot b\right) - \left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \left(\left(\sqrt[3]{9} \cdot t\right) \cdot \left(z \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2 + \left(27 \cdot \left(a \cdot b\right) - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020105 
(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) (* (* (* y 9) z) t)) (* a (* 27 b))) (+ (- (* x 2) (* 9 (* y (* t z)))) (* (* a 27) b)))

  (+ (- (* x 2) (* (* (* y 9) z) t)) (* (* a 27) b)))