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

Average Error: 7.4 → 7.2

Time: 5.5s

Precision: binary64

Math

\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -8.946880921470724 \cdot 10^{+237}:\\ \;\;\;\;\frac{y}{a \cdot \sqrt{2}} \cdot \frac{x}{\sqrt{2}} - \left(9 \cdot t\right) \cdot \frac{z}{a \cdot 2}\\ \mathbf{elif}\;z \leq -8.813908218785836 \cdot 10^{+56}:\\ \;\;\;\;\frac{1}{2 \cdot \frac{a}{y \cdot x - z \cdot \left(9 \cdot t\right)}}\\ \mathbf{elif}\;z \leq -3.898581623228132 \cdot 10^{-126}:\\ \;\;\;\;y \cdot \frac{x}{a \cdot 2} - \left(9 \cdot \frac{t}{a}\right) \cdot \frac{z}{2}\\ \mathbf{elif}\;z \leq -1.3426129749595774 \cdot 10^{-273}:\\ \;\;\;\;\frac{y}{a \cdot \sqrt{2}} \cdot \frac{x}{\sqrt{2}} - \left(9 \cdot t\right) \cdot \frac{z}{a \cdot 2}\\ \mathbf{elif}\;z \leq 6.1237664107480045 \cdot 10^{+54}:\\ \;\;\;\;\frac{1}{2 \cdot \frac{a}{y \cdot x - z \cdot \left(9 \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \end{array}\]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))

↓

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -8.946880921470724e+237)
   (-
    (* (/ y (* a (sqrt 2.0))) (/ x (sqrt 2.0)))
    (* (* 9.0 t) (/ z (* a 2.0))))
   (if (<= z -8.813908218785836e+56)
     (/ 1.0 (* 2.0 (/ a (- (* y x) (* z (* 9.0 t))))))
     (if (<= z -3.898581623228132e-126)
       (- (* y (/ x (* a 2.0))) (* (* 9.0 (/ t a)) (/ z 2.0)))
       (if (<= z -1.3426129749595774e-273)
         (-
          (* (/ y (* a (sqrt 2.0))) (/ x (sqrt 2.0)))
          (* (* 9.0 t) (/ z (* a 2.0))))
         (if (<= z 6.1237664107480045e+54)
           (/ 1.0 (* 2.0 (/ a (- (* y x) (* z (* 9.0 t))))))
           (- (* 0.5 (* x (/ y a))) (* 4.5 (* z (/ t a))))))))))

C

double code(double x, double y, double z, double t, double a) {
	return (((double) (((double) (x * y)) - ((double) (((double) (z * 9.0)) * t)))) / ((double) (a * 2.0)));
}

↓

double code(double x, double y, double z, double t, double a) {
	double VAR;
	if ((z <= -8.946880921470724e+237)) {
		VAR = ((double) (((double) ((y / ((double) (a * ((double) sqrt(2.0))))) * (x / ((double) sqrt(2.0))))) - ((double) (((double) (9.0 * t)) * (z / ((double) (a * 2.0)))))));
	} else {
		double VAR_1;
		if ((z <= -8.813908218785836e+56)) {
			VAR_1 = (1.0 / ((double) (2.0 * (a / ((double) (((double) (y * x)) - ((double) (z * ((double) (9.0 * t))))))))));
		} else {
			double VAR_2;
			if ((z <= -3.898581623228132e-126)) {
				VAR_2 = ((double) (((double) (y * (x / ((double) (a * 2.0))))) - ((double) (((double) (9.0 * (t / a))) * (z / 2.0)))));
			} else {
				double VAR_3;
				if ((z <= -1.3426129749595774e-273)) {
					VAR_3 = ((double) (((double) ((y / ((double) (a * ((double) sqrt(2.0))))) * (x / ((double) sqrt(2.0))))) - ((double) (((double) (9.0 * t)) * (z / ((double) (a * 2.0)))))));
				} else {
					double VAR_4;
					if ((z <= 6.1237664107480045e+54)) {
						VAR_4 = (1.0 / ((double) (2.0 * (a / ((double) (((double) (y * x)) - ((double) (z * ((double) (9.0 * t))))))))));
					} else {
						VAR_4 = ((double) (((double) (0.5 * ((double) (x * (y / a))))) - ((double) (4.5 * ((double) (z * (t / a)))))));
					}
					VAR_3 = VAR_4;
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	7.4
Target	5.6
Herbie	7.2

\[\begin{array}{l} \mathbf{if}\;a < -2.090464557976709 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a < 2.144030707833976 \cdot 10^{+99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if z < -8.9468809214707244e237 or -3.898581623228132e-126 < z < -1.34261297495957743e-273

   1. Initial program 8.8

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

   2. Simplified 8.6

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

   4. Applied div-sub 8.6

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

   5. Simplified 10.2

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

   6. Simplified 9.8

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

   8. Applied *-un-lft-identity 9.8

      \[\leadsto y \cdot \frac{\color{blue}{1 \cdot x}}{a \cdot 2} - \left(9 \cdot t\right) \cdot \frac{z}{a \cdot 2}\]

   9. Applied times-frac 9.8

      \[\leadsto y \cdot \color{blue}{\left(\frac{1}{a} \cdot \frac{x}{2}\right)} - \left(9 \cdot t\right) \cdot \frac{z}{a \cdot 2}\]

   10. Applied associate-*r* 9.4

       \[\leadsto \color{blue}{\left(y \cdot \frac{1}{a}\right) \cdot \frac{x}{2}} - \left(9 \cdot t\right) \cdot \frac{z}{a \cdot 2}\]

   11. Simplified 9.4

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

   13. Applied add-sqr-sqrt 9.8

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

   14. Applied *-un-lft-identity 9.8

       \[\leadsto \frac{y}{a} \cdot \frac{\color{blue}{1 \cdot x}}{\sqrt{2} \cdot \sqrt{2}} - \left(9 \cdot t\right) \cdot \frac{z}{a \cdot 2}\]

   15. Applied times-frac 9.7

       \[\leadsto \frac{y}{a} \cdot \color{blue}{\left(\frac{1}{\sqrt{2}} \cdot \frac{x}{\sqrt{2}}\right)} - \left(9 \cdot t\right) \cdot \frac{z}{a \cdot 2}\]

   16. Applied associate-*r* 9.7

       \[\leadsto \color{blue}{\left(\frac{y}{a} \cdot \frac{1}{\sqrt{2}}\right) \cdot \frac{x}{\sqrt{2}}} - \left(9 \cdot t\right) \cdot \frac{z}{a \cdot 2}\]

   17. Simplified 9.6

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

   if -8.9468809214707244e237 < z < -8.81390821878583551e56 or -1.34261297495957743e-273 < z < 6.123766410748005e54

   1. Initial program 6.3

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

   2. Simplified 6.4

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

   4. Applied clear-num 6.8

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{x \cdot y - z \cdot \left(9 \cdot t\right)}}}\]

   5. Simplified 6.7

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{a}{x \cdot y - z \cdot \left(9 \cdot t\right)}}}\]

   if -8.81390821878583551e56 < z < -3.898581623228132e-126

   1. Initial program 4.0

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

   2. Simplified 4.1

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

   4. Applied div-sub 4.1

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

   5. Simplified 5.7

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

   6. Simplified 5.7

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

   8. Applied *-un-lft-identity 5.7

      \[\leadsto y \cdot \frac{x}{a \cdot 2} - \left(9 \cdot t\right) \cdot \frac{\color{blue}{1 \cdot z}}{a \cdot 2}\]

   9. Applied times-frac 5.8

      \[\leadsto y \cdot \frac{x}{a \cdot 2} - \left(9 \cdot t\right) \cdot \color{blue}{\left(\frac{1}{a} \cdot \frac{z}{2}\right)}\]

   10. Applied associate-*r* 6.4

       \[\leadsto y \cdot \frac{x}{a \cdot 2} - \color{blue}{\left(\left(9 \cdot t\right) \cdot \frac{1}{a}\right) \cdot \frac{z}{2}}\]

   11. Simplified 6.3

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

   if 6.123766410748005e54 < z

   1. Initial program 12.6

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

   2. Simplified 12.4

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

   4. Applied div-sub 12.4

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

   5. Simplified 12.3

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

   6. Simplified 9.5

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

   7. Taylor expanded around 0 12.3

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}}\]

   8. Simplified 7.0

      \[\leadsto \color{blue}{0.5 \cdot \left(\frac{y}{a} \cdot x\right) - 4.5 \cdot \left(\frac{t}{a} \cdot z\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 7.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.946880921470724 \cdot 10^{+237}:\\ \;\;\;\;\frac{y}{a \cdot \sqrt{2}} \cdot \frac{x}{\sqrt{2}} - \left(9 \cdot t\right) \cdot \frac{z}{a \cdot 2}\\ \mathbf{elif}\;z \leq -8.813908218785836 \cdot 10^{+56}:\\ \;\;\;\;\frac{1}{2 \cdot \frac{a}{y \cdot x - z \cdot \left(9 \cdot t\right)}}\\ \mathbf{elif}\;z \leq -3.898581623228132 \cdot 10^{-126}:\\ \;\;\;\;y \cdot \frac{x}{a \cdot 2} - \left(9 \cdot \frac{t}{a}\right) \cdot \frac{z}{2}\\ \mathbf{elif}\;z \leq -1.3426129749595774 \cdot 10^{-273}:\\ \;\;\;\;\frac{y}{a \cdot \sqrt{2}} \cdot \frac{x}{\sqrt{2}} - \left(9 \cdot t\right) \cdot \frac{z}{a \cdot 2}\\ \mathbf{elif}\;z \leq 6.1237664107480045 \cdot 10^{+54}:\\ \;\;\;\;\frac{1}{2 \cdot \frac{a}{y \cdot x - z \cdot \left(9 \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020198 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :herbie-target
  (if (< a -2.090464557976709e+86) (- (* 0.5 (/ (* y x) a)) (* 4.5 (/ t (/ a z)))) (if (< a 2.144030707833976e+99) (/ (- (* x y) (* z (* 9.0 t))) (* a 2.0)) (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5)))))

  (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))