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

Average Error: 6.5 → 6.5

Time: 13.1s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r236669 = x;
        double r236670 = y;
        double r236671 = z;
        double r236672 = r236671 - r236669;
        double r236673 = r236670 * r236672;
        double r236674 = t;
        double r236675 = r236673 / r236674;
        double r236676 = r236669 + r236675;
        return r236676;
}

↓

double f(double x, double y, double z, double t) {
        double r236677 = y;
        double r236678 = z;
        double r236679 = t;
        double r236680 = r236678 / r236679;
        double r236681 = x;
        double r236682 = r236681 / r236679;
        double r236683 = r236680 - r236682;
        double r236684 = r236677 * r236683;
        double r236685 = r236684 + r236681;
        return r236685;
}

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	6.5

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

Derivation

1. Split input into 2 regimes

2. if y < -1.258518507926354e-57 or 3.732545607262176e-61 < y

   1. Initial program 12.1

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

   3. Applied add-cube-cbrt 12.7

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

   4. Applied times-frac 2.2

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

   6. Applied add-cube-cbrt 2.4

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

   7. Applied *-un-lft-identity 2.4

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

   8. Applied times-frac 2.4

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

   9. Applied associate-*r* 3.4

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

   10. Simplified 3.4

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

   11. Taylor expanded around 0 12.1

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

   13. Applied *-un-lft-identity 12.1

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

   14. Applied *-un-lft-identity 12.1

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

   15. Applied distribute-lft-out-- 12.1

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

   16. Simplified 1.9

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

   if -1.258518507926354e-57 < y < 3.732545607262176e-61

   1. Initial program 1.2

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

   3. Applied add-cube-cbrt 1.6

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

   4. Applied times-frac 4.0

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

   6. Applied add-cube-cbrt 4.1

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

   7. Applied *-un-lft-identity 4.1

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

   8. Applied times-frac 4.2

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

   9. Applied associate-*r* 2.3

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

   10. Simplified 2.3

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

   11. Taylor expanded around 0 1.2

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

   13. Applied associate-/l* 0.8

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

4. Final simplification 6.5

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

Reproduce

herbie shell --seed 2019297 
(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)))