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

Average Error: 3.5 → 2.1

Time: 4.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(y \cdot 9\right) \cdot z \le 2.5565242382119028 \cdot 10^{108}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \mathsf{fma}\left(27, a \cdot b, -\left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \left(\sqrt[3]{9} \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \mathsf{fma}\left(27, a \cdot b, -9 \cdot \left(\left(t \cdot z\right) \cdot y\right)\right)\right)\\ \end{array}\]

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 temp;
	if ((((y * 9.0) * z) <= 2.556524238211903e+108)) {
		temp = fma(2.0, x, fma(27.0, (a * b), -((cbrt(9.0) * cbrt(9.0)) * (cbrt(9.0) * (t * (z * y))))));
	} else {
		temp = fma(2.0, x, fma(27.0, (a * b), -(9.0 * ((t * z) * y))));
	}
	return temp;
}

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	2.6
Herbie	2.1

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

Derivation

1. Split input into 2 regimes

2. if (* (* y 9.0) z) < 2.556524238211903e+108

   1. Initial program 2.1

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

   2. Simplified 2.1

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

   3. Taylor expanded around inf 2.0

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

   4. Simplified 2.0

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

   6. Applied fma-neg 2.0

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

   8. Applied add-cube-cbrt 2.0

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

   9. Applied associate-*l* 2.0

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

   if 2.556524238211903e+108 < (* (* y 9.0) z)

   1. Initial program 14.4

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

   2. Simplified 14.5

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

   3. Taylor expanded around inf 14.3

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

   4. Simplified 14.3

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

   6. Applied fma-neg 14.3

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

   8. Applied associate-*r* 2.7

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

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot 9\right) \cdot z \le 2.5565242382119028 \cdot 10^{108}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \mathsf{fma}\left(27, a \cdot b, -\left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \left(\sqrt[3]{9} \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \mathsf{fma}\left(27, a \cdot b, -9 \cdot \left(\left(t \cdot z\right) \cdot y\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020058 +o rules:numerics
(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)))