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

Average Error: 6.3 → 6.1

Time: 11.9s

Precision: 64

Math

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

↓

\[x + \frac{y}{\frac{t}{z - x}}\]

C

double f(double x, double y, double z, double t) {
        double r217252 = x;
        double r217253 = y;
        double r217254 = z;
        double r217255 = r217254 - r217252;
        double r217256 = r217253 * r217255;
        double r217257 = t;
        double r217258 = r217256 / r217257;
        double r217259 = r217252 + r217258;
        return r217259;
}

↓

double f(double x, double y, double z, double t) {
        double r217260 = x;
        double r217261 = y;
        double r217262 = t;
        double r217263 = z;
        double r217264 = r217263 - r217260;
        double r217265 = r217262 / r217264;
        double r217266 = r217261 / r217265;
        double r217267 = r217260 + r217266;
        return r217267;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.3
Target	2.2
Herbie	6.1

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

Derivation

1. Split input into 2 regimes

2. if x < -3.908231963040019e-294 or 4.1676572019103e-54 < x

   1. Initial program 6.8

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

   3. Applied clear-num 6.8

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

   5. Applied div-inv 6.9

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

   6. Applied associate-/r* 7.0

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

   8. Applied *-un-lft-identity 7.0

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

   9. Applied *-un-lft-identity 7.0

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

   10. Applied add-cube-cbrt 7.0

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

   11. Applied times-frac 7.0

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

   12. Applied times-frac 7.0

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

   13. Simplified 7.0

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

   14. Simplified 1.4

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

   16. Applied associate-*l/ 1.3

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

   if -3.908231963040019e-294 < x < 4.1676572019103e-54

   1. Initial program 4.8

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

   3. Applied associate-/l* 4.1

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

4. Final simplification 6.1

   \[\leadsto x + \frac{y}{\frac{t}{z - x}}\]

Reproduce

herbie shell --seed 2019298 
(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)))