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

Average Error: 3.1 → 0.5

Time: 6.1s

Precision: binary64

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

\[[a, b]=\mathsf{sort}([a, b])\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq 1.4583759884213824 \cdot 10^{-80}:\\ \;\;\;\;\mathsf{fma}\left(27, a \cdot b, \mathsf{fma}\left(2, x, \left(y \cdot \left(z \cdot t\right)\right) \cdot -9\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot x - t \cdot \left(z \cdot \left(y \cdot 9\right)\right)\right) + b \cdot \left(27 \cdot a\right)\\ \end{array} \]

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 (<= z 1.4583759884213824e-80)
   (fma 27.0 (* a b) (fma 2.0 x (* (* y (* z t)) -9.0)))
   (+ (- (* 2.0 x) (* t (* z (* y 9.0)))) (* b (* 27.0 a)))))

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 (z <= 1.4583759884213824e-80) {
		tmp = fma(27.0, (a * b), fma(2.0, x, ((y * (z * t)) * -9.0)));
	} else {
		tmp = ((2.0 * x) - (t * (z * (y * 9.0)))) + (b * (27.0 * a));
	}
	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	3.1
Target	3.5
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 2 regimes

2. if z < 1.4583759884213824e-80

   1. Initial program 3.7

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

   2. Simplified 0.7

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

   3. Taylor expanded in y around 0 0.6

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

   4. Simplified 0.6

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

   if 1.4583759884213824e-80 < z

   1. Initial program 0.4

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.4583759884213824 \cdot 10^{-80}:\\ \;\;\;\;\mathsf{fma}\left(27, a \cdot b, \mathsf{fma}\left(2, x, \left(y \cdot \left(z \cdot t\right)\right) \cdot -9\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot x - t \cdot \left(z \cdot \left(y \cdot 9\right)\right)\right) + b \cdot \left(27 \cdot a\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021313 
(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)))