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

Average Error: 20.1 → 4.6

Time: 6.8s

Precision: 64

Math

\[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

↓

\[\begin{array}{l} \mathbf{if}\;c \le -8.92080495788657361 \cdot 10^{34} \lor \neg \left(c \le 1.376600657529797 \cdot 10^{29}\right):\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t \cdot 1}{\frac{c}{a}}, \frac{\mathsf{fma}\left(9, y \cdot \frac{x}{c}, \frac{b}{c}\right)}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t \cdot a}{c}, \frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z}}{c}\right)\\ \end{array}\]

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return ((double) (((double) (((double) (((double) (((double) (x * 9.0)) * y)) - ((double) (((double) (((double) (z * 4.0)) * t)) * a)))) + b)) / ((double) (z * c))));
}

↓

double code(double x, double y, double z, double t, double a, double b, double c) {
	double VAR;
	if (((c <= -8.920804957886574e+34) || !(c <= 1.376600657529797e+29))) {
		VAR = ((double) fma(((double) -(4.0)), ((double) (((double) (t * 1.0)) / ((double) (c / a)))), ((double) (((double) fma(9.0, ((double) (y * ((double) (x / c)))), ((double) (b / c)))) / z))));
	} else {
		VAR = ((double) fma(((double) -(4.0)), ((double) (((double) (t * a)) / c)), ((double) (((double) (((double) fma(x, ((double) (9.0 * y)), b)) / z)) / c))));
	}
	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

Bits error versus c

Target

Original	20.1
Target	14.5
Herbie	4.6

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.10015674080410512 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.17088779117474882 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.8768236795461372 \cdot 10^{130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.3838515042456319 \cdot 10^{158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if c < -8.920804957886574e+34 or 1.376600657529797e+29 < c

   1. Initial program 23.3

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

   2. Simplified 15.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4, \frac{t \cdot a}{c}, \frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z \cdot c}\right)}\]
   3. Using strategy rm

   4. Applied *-commutative 15.2

      \[\leadsto \mathsf{fma}\left(-4, \frac{t \cdot a}{c}, \frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{\color{blue}{c \cdot z}}\right)\]

   5. Applied associate-/r* 12.2

      \[\leadsto \mathsf{fma}\left(-4, \frac{t \cdot a}{c}, \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{c}}{z}}\right)\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 12.2

      \[\leadsto \mathsf{fma}\left(-4, \frac{t \cdot \color{blue}{\left(1 \cdot a\right)}}{c}, \frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{c}}{z}\right)\]

   8. Applied associate-*r* 12.2

      \[\leadsto \mathsf{fma}\left(-4, \frac{\color{blue}{\left(t \cdot 1\right) \cdot a}}{c}, \frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{c}}{z}\right)\]

   9. Applied associate-/l* 8.6

      \[\leadsto \mathsf{fma}\left(-4, \color{blue}{\frac{t \cdot 1}{\frac{c}{a}}}, \frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{c}}{z}\right)\]

   10. Taylor expanded around 0 8.5

       \[\leadsto \mathsf{fma}\left(-4, \frac{t \cdot 1}{\frac{c}{a}}, \frac{\color{blue}{9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}}}{z}\right)\]

   11. Simplified 8.5

       \[\leadsto \mathsf{fma}\left(-4, \frac{t \cdot 1}{\frac{c}{a}}, \frac{\color{blue}{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{b}{c}\right)}}{z}\right)\]
   12. Using strategy rm

   13. Applied *-un-lft-identity 8.5

       \[\leadsto \mathsf{fma}\left(-4, \frac{t \cdot 1}{\frac{c}{a}}, \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{\color{blue}{1 \cdot c}}, \frac{b}{c}\right)}{z}\right)\]

   14. Applied *-commutative 8.5

       \[\leadsto \mathsf{fma}\left(-4, \frac{t \cdot 1}{\frac{c}{a}}, \frac{\mathsf{fma}\left(9, \frac{\color{blue}{y \cdot x}}{1 \cdot c}, \frac{b}{c}\right)}{z}\right)\]

   15. Applied times-frac 5.4

       \[\leadsto \mathsf{fma}\left(-4, \frac{t \cdot 1}{\frac{c}{a}}, \frac{\mathsf{fma}\left(9, \color{blue}{\frac{y}{1} \cdot \frac{x}{c}}, \frac{b}{c}\right)}{z}\right)\]

   16. Simplified 5.4

       \[\leadsto \mathsf{fma}\left(-4, \frac{t \cdot 1}{\frac{c}{a}}, \frac{\mathsf{fma}\left(9, \color{blue}{y} \cdot \frac{x}{c}, \frac{b}{c}\right)}{z}\right)\]

   if -8.920804957886574e+34 < c < 1.376600657529797e+29

   1. Initial program 14.8

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

   2. Simplified 5.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4, \frac{t \cdot a}{c}, \frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z \cdot c}\right)}\]
   3. Using strategy rm

   4. Applied associate-/r* 3.2

      \[\leadsto \mathsf{fma}\left(-4, \frac{t \cdot a}{c}, \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z}}{c}}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 4.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -8.92080495788657361 \cdot 10^{34} \lor \neg \left(c \le 1.376600657529797 \cdot 10^{29}\right):\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t \cdot 1}{\frac{c}{a}}, \frac{\mathsf{fma}\left(9, y \cdot \frac{x}{c}, \frac{b}{c}\right)}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t \cdot a}{c}, \frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z}}{c}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, J"
  :precision binary64

  :herbie-target
  (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) -1.1001567408041051e-171) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) -0.0) (/ (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) z) c) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 1.1708877911747488e-53) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 2.876823679546137e+130) (- (+ (* (* 9 (/ y c)) (/ x z)) (/ b (* c z))) (* 4 (/ (* a t) c))) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 1.3838515042456319e+158) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (- (+ (* 9 (* (/ y (* c z)) x)) (/ b (* c z))) (* 4 (/ (* a t) c))))))))

  (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)))