Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D

Average Error: 6.5 → 1.5

Time: 4.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -72395.033720331005:\\ \;\;\;\;x + \left(\frac{y}{\frac{t}{z}} - \frac{y}{\frac{t}{x}}\right)\\ \mathbf{elif}\;y \le -1.78307652196547843 \cdot 10^{-183}:\\ \;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{y}{{\left(\sqrt[3]{t}\right)}^{2}} \cdot \left(\sqrt[3]{z - x} \cdot \frac{\sqrt[3]{z - x}}{\sqrt[3]{{\left(\sqrt[3]{t}\right)}^{2}}}\right)\right) \cdot \frac{\sqrt[3]{z - x}}{\sqrt[3]{\sqrt[3]{t}}}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t) {
	double VAR;
	if ((y <= -72395.033720331)) {
		VAR = ((double) (x + ((double) (((double) (y / ((double) (t / z)))) - ((double) (y / ((double) (t / x))))))));
	} else {
		double VAR_1;
		if ((y <= -1.7830765219654784e-183)) {
			VAR_1 = ((double) (x + ((double) (((double) (y * ((double) (z - x)))) / t))));
		} else {
			VAR_1 = ((double) (x + ((double) (((double) (((double) (y / ((double) pow(((double) cbrt(t)), 2.0)))) * ((double) (((double) cbrt(((double) (z - x)))) * ((double) (((double) cbrt(((double) (z - x)))) / ((double) cbrt(((double) pow(((double) cbrt(t)), 2.0)))))))))) * ((double) (((double) cbrt(((double) (z - x)))) / ((double) cbrt(((double) cbrt(t))))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.5
Target	2.0
Herbie	1.5

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

Derivation

1. Split input into 3 regimes

2. if y < -72395.033720331005

   1. Initial program 15.9

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

   2. Simplified 1.3

      \[\leadsto \color{blue}{x + y \cdot \frac{z - x}{t}}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 2.1

      \[\leadsto x + y \cdot \frac{z - x}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}\]

   5. Applied *-un-lft-identity 2.1

      \[\leadsto x + y \cdot \frac{\color{blue}{1 \cdot \left(z - x\right)}}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}\]

   6. Applied times-frac 2.1

      \[\leadsto x + y \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z - x}{\sqrt[3]{t}}\right)}\]

   7. Applied associate-*r* 2.4

      \[\leadsto x + \color{blue}{\left(y \cdot \frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right) \cdot \frac{z - x}{\sqrt[3]{t}}}\]

   8. Simplified 2.4

      \[\leadsto x + \color{blue}{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \frac{z - x}{\sqrt[3]{t}}\]
   9. Using strategy rm

   10. Applied add-cube-cbrt 2.5

       \[\leadsto x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z - x}{\sqrt[3]{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}}\]

   11. Applied cbrt-prod 2.5

       \[\leadsto x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z - x}{\color{blue}{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}}}\]

   12. Applied add-cube-cbrt 2.7

       \[\leadsto x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\color{blue}{\left(\sqrt[3]{z - x} \cdot \sqrt[3]{z - x}\right) \cdot \sqrt[3]{z - x}}}{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}}\]

   13. Applied times-frac 2.7

       \[\leadsto x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \color{blue}{\left(\frac{\sqrt[3]{z - x} \cdot \sqrt[3]{z - x}}{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \frac{\sqrt[3]{z - x}}{\sqrt[3]{\sqrt[3]{t}}}\right)}\]

   14. Applied associate-*r* 2.6

       \[\leadsto x + \color{blue}{\left(\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt[3]{z - x} \cdot \sqrt[3]{z - x}}{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}}\right) \cdot \frac{\sqrt[3]{z - x}}{\sqrt[3]{\sqrt[3]{t}}}}\]

   15. Simplified 2.6

       \[\leadsto x + \color{blue}{\left(\frac{y}{{\left(\sqrt[3]{t}\right)}^{2}} \cdot \left(\sqrt[3]{z - x} \cdot \frac{\sqrt[3]{z - x}}{\sqrt[3]{{\left(\sqrt[3]{t}\right)}^{2}}}\right)\right)} \cdot \frac{\sqrt[3]{z - x}}{\sqrt[3]{\sqrt[3]{t}}}\]

   16. Taylor expanded around 0 15.9

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

   17. Simplified 1.6

       \[\leadsto x + \color{blue}{\left(\frac{y}{\frac{t}{z}} - \frac{y}{\frac{t}{x}}\right)}\]

   if -72395.033720331005 < y < -1.78307652196547843e-183

   1. Initial program 0.5

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

   if -1.78307652196547843e-183 < y

   1. Initial program 5.7

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

   2. Simplified 7.4

      \[\leadsto \color{blue}{x + y \cdot \frac{z - x}{t}}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 7.9

      \[\leadsto x + y \cdot \frac{z - x}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}\]

   5. Applied *-un-lft-identity 7.9

      \[\leadsto x + y \cdot \frac{\color{blue}{1 \cdot \left(z - x\right)}}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}\]

   6. Applied times-frac 7.9

      \[\leadsto x + y \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z - x}{\sqrt[3]{t}}\right)}\]

   7. Applied associate-*r* 3.5

      \[\leadsto x + \color{blue}{\left(y \cdot \frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right) \cdot \frac{z - x}{\sqrt[3]{t}}}\]

   8. Simplified 3.5

      \[\leadsto x + \color{blue}{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \frac{z - x}{\sqrt[3]{t}}\]
   9. Using strategy rm

   10. Applied add-cube-cbrt 3.6

       \[\leadsto x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z - x}{\sqrt[3]{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}}\]

   11. Applied cbrt-prod 3.6

       \[\leadsto x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z - x}{\color{blue}{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}}}\]

   12. Applied add-cube-cbrt 3.7

       \[\leadsto x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\color{blue}{\left(\sqrt[3]{z - x} \cdot \sqrt[3]{z - x}\right) \cdot \sqrt[3]{z - x}}}{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}}\]

   13. Applied times-frac 3.7

       \[\leadsto x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \color{blue}{\left(\frac{\sqrt[3]{z - x} \cdot \sqrt[3]{z - x}}{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \frac{\sqrt[3]{z - x}}{\sqrt[3]{\sqrt[3]{t}}}\right)}\]

   14. Applied associate-*r* 1.8

       \[\leadsto x + \color{blue}{\left(\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt[3]{z - x} \cdot \sqrt[3]{z - x}}{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}}\right) \cdot \frac{\sqrt[3]{z - x}}{\sqrt[3]{\sqrt[3]{t}}}}\]

   15. Simplified 1.8

       \[\leadsto x + \color{blue}{\left(\frac{y}{{\left(\sqrt[3]{t}\right)}^{2}} \cdot \left(\sqrt[3]{z - x} \cdot \frac{\sqrt[3]{z - x}}{\sqrt[3]{{\left(\sqrt[3]{t}\right)}^{2}}}\right)\right)} \cdot \frac{\sqrt[3]{z - x}}{\sqrt[3]{\sqrt[3]{t}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -72395.033720331005:\\ \;\;\;\;x + \left(\frac{y}{\frac{t}{z}} - \frac{y}{\frac{t}{x}}\right)\\ \mathbf{elif}\;y \le -1.78307652196547843 \cdot 10^{-183}:\\ \;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{y}{{\left(\sqrt[3]{t}\right)}^{2}} \cdot \left(\sqrt[3]{z - x} \cdot \frac{\sqrt[3]{z - x}}{\sqrt[3]{{\left(\sqrt[3]{t}\right)}^{2}}}\right)\right) \cdot \frac{\sqrt[3]{z - x}}{\sqrt[3]{\sqrt[3]{t}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"
  :precision binary64

  :herbie-target
  (- x (+ (* x (/ y t)) (* (neg z) (/ y t))))

  (+ x (/ (* y (- z x)) t)))