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

Average Error: 3.0 → 1.1

Time: 6.8s

Precision: binary64

\[[y, z, t] = \mathsf{sort}([y, z, t]) \\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot 9 \leq -1.6178589077969468 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(27, a \cdot b, \mathsf{fma}\left(2, x, y \cdot \left(-9 \cdot \left(t \cdot z\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(27, a \cdot b, \mathsf{fma}\left(2, x, -9 \cdot \left(z \cdot \left(y \cdot t\right)\right)\right)\right)\\ \end{array} \]

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) -1.6178589077969468e+98)
   (fma 27.0 (* a b) (fma 2.0 x (* y (* -9.0 (* t z)))))
   (fma 27.0 (* a b) (fma 2.0 x (* -9.0 (* z (* y 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) <= -1.6178589077969468e+98) {
		tmp = fma(27.0, (a * b), fma(2.0, x, (y * (-9.0 * (t * z)))));
	} else {
		tmp = fma(27.0, (a * b), fma(2.0, x, (-9.0 * (z * (y * 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	3.0
Target	3.5
Herbie	1.1

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 y 9) < -1.61785890779694683e98

   1. Initial program 7.1

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

   2. Simplified 0.9

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

   3. Taylor expanded in y around 0 0.9

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

   4. Simplified 0.9

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

   5. Applied associate-*l*_binary64 0.9

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

   6. Simplified 0.9

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

   if -1.61785890779694683e98 < (*.f64 y 9)

   1. Initial program 0.7

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

   2. Simplified 4.0

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

   3. Taylor expanded in y around 0 3.9

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

   4. Simplified 3.9

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

   5. Applied associate-*r*_binary64 1.2

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

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot 9 \leq -1.6178589077969468 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(27, a \cdot b, \mathsf{fma}\left(2, x, y \cdot \left(-9 \cdot \left(t \cdot z\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(27, a \cdot b, \mathsf{fma}\left(2, x, -9 \cdot \left(z \cdot \left(y \cdot t\right)\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(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)))