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

Average Error: 6.6 → 2.3

Time: 3.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.8300468342836098 \cdot 10^{-67} \lor \neg \left(x \le 7.1410784377771534 \cdot 10^{-250}\right):\\ \;\;\;\;x + \frac{y}{t} \cdot \left(z - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z + \left(-x\right)}{t}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t) {
	double temp;
	if (((x <= -1.83004683428361e-67) || !(x <= 7.141078437777153e-250))) {
		temp = (x + ((y / t) * (z - x)));
	} else {
		temp = (x + (y * ((z + -x) / t)));
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.6
Target	2.4
Herbie	2.3

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

Derivation

1. Split input into 2 regimes

2. if x < -1.83004683428361e-67 or 7.141078437777153e-250 < x

   1. Initial program 7.0

      \[x + \frac{y \cdot \left(z - x\right)}{t}\]
   2. Using strategy rm

   3. Applied sub-neg 7.0

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

   4. Applied distribute-lft-in 7.0

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

   6. Applied clear-num 7.0

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

   7. Simplified 1.3

      \[\leadsto x + \frac{1}{\color{blue}{\frac{\frac{t}{y}}{z - x}}}\]
   8. Using strategy rm

   9. Applied div-inv 1.4

      \[\leadsto x + \frac{1}{\color{blue}{\frac{t}{y} \cdot \frac{1}{z - x}}}\]

   10. Applied add-cube-cbrt 1.4

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

   11. Applied times-frac 1.4

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

   12. Simplified 1.4

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

   13. Simplified 1.3

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

   if -1.83004683428361e-67 < x < 7.141078437777153e-250

   1. Initial program 5.4

      \[x + \frac{y \cdot \left(z - x\right)}{t}\]
   2. Using strategy rm

   3. Applied sub-neg 5.4

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

   4. Applied distribute-lft-in 5.4

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

   6. Applied *-un-lft-identity 5.4

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

   7. Applied distribute-lft-out 5.4

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

   8. Applied times-frac 4.8

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

   9. Simplified 4.8

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

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.8300468342836098 \cdot 10^{-67} \lor \neg \left(x \le 7.1410784377771534 \cdot 10^{-250}\right):\\ \;\;\;\;x + \frac{y}{t} \cdot \left(z - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z + \left(-x\right)}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020058 
(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)) (* (- z) (/ y t))))

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