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

Average Error: 3.7 → 0.7

Time: 6.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(y \cdot 9\right) \cdot z \leq -3.0471891562604393 \cdot 10^{+100}:\\ \;\;\;\;\left(x \cdot 2 - y \cdot \left(\left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \left(\sqrt[3]{9} \cdot \left(z \cdot t\right)\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \mathbf{elif}\;\left(y \cdot 9\right) \cdot z \leq 3.095409634969567 \cdot 10^{+161}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(27 \cdot b\right) + \left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right)\\ \end{array}\]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b)))

↓

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* (* y 9.0) z) -3.0471891562604393e+100)
   (+
    (- (* x 2.0) (* y (* (* (cbrt 9.0) (cbrt 9.0)) (* (cbrt 9.0) (* z t)))))
    (* (* a 27.0) b))
   (if (<= (* (* y 9.0) z) 3.095409634969567e+161)
     (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* a (* 27.0 b)))
     (+ (* a (* 27.0 b)) (- (* x 2.0) (* (* y 9.0) (* z t)))))))

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 tmp;
	if (((y * 9.0) * z) <= -3.0471891562604393e+100) {
		tmp = ((x * 2.0) - (y * ((cbrt(9.0) * cbrt(9.0)) * (cbrt(9.0) * (z * t))))) + ((a * 27.0) * b);
	} else if (((y * 9.0) * z) <= 3.095409634969567e+161) {
		tmp = ((x * 2.0) - (((y * 9.0) * z) * t)) + (a * (27.0 * b));
	} else {
		tmp = (a * (27.0 * b)) + ((x * 2.0) - ((y * 9.0) * (z * t)));
	}
	return tmp;
}

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	0.7

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y 9) z) < -3.0471891562604393e100

   1. Initial program 15.2

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

   3. Applied associate-*l*_binary64_18774 2.2

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

   5. Applied associate-*l*_binary64_18774 1.9

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

   7. Applied add-cube-cbrt_binary64_18868 1.9

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

   8. Applied associate-*l*_binary64_18774 2.0

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

   9. Simplified 2.0

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

   if -3.0471891562604393e100 < (*.f64 (*.f64 y 9) z) < 3.09540963496956685e161

   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 associate-*l*_binary64_18774 0.4

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

   if 3.09540963496956685e161 < (*.f64 (*.f64 y 9) z)

   1. Initial program 18.3

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

   3. Applied associate-*l*_binary64_18774 1.9

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

   5. Applied associate-*l*_binary64_18774 1.9

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot 9\right) \cdot z \leq -3.0471891562604393 \cdot 10^{+100}:\\ \;\;\;\;\left(x \cdot 2 - y \cdot \left(\left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \left(\sqrt[3]{9} \cdot \left(z \cdot t\right)\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \mathbf{elif}\;\left(y \cdot 9\right) \cdot z \leq 3.095409634969567 \cdot 10^{+161}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(27 \cdot b\right) + \left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020355 
(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)))