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

Average Error: 6.5 → 1.6

Time: 26.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.587122080323390289480100433348702966458 \cdot 10^{58}:\\ \;\;\;\;y \cdot \frac{z - x}{t} + x\\ \mathbf{elif}\;t \le 6.204649366460223122009107028759053602807 \cdot 10^{83}:\\ \;\;\;\;x + \frac{\left(z - x\right) \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{z - x} \cdot \sqrt[3]{z - x}\right) \cdot \left(\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{z - x}\right) \cdot \frac{\sqrt[3]{y}}{t}\right) + x\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r206076 = x;
        double r206077 = y;
        double r206078 = z;
        double r206079 = r206078 - r206076;
        double r206080 = r206077 * r206079;
        double r206081 = t;
        double r206082 = r206080 / r206081;
        double r206083 = r206076 + r206082;
        return r206083;
}

↓

double f(double x, double y, double z, double t) {
        double r206084 = t;
        double r206085 = -1.5871220803233903e+58;
        bool r206086 = r206084 <= r206085;
        double r206087 = y;
        double r206088 = z;
        double r206089 = x;
        double r206090 = r206088 - r206089;
        double r206091 = r206090 / r206084;
        double r206092 = r206087 * r206091;
        double r206093 = r206092 + r206089;
        double r206094 = 6.204649366460223e+83;
        bool r206095 = r206084 <= r206094;
        double r206096 = r206090 * r206087;
        double r206097 = r206096 / r206084;
        double r206098 = r206089 + r206097;
        double r206099 = cbrt(r206090);
        double r206100 = r206099 * r206099;
        double r206101 = cbrt(r206087);
        double r206102 = r206101 * r206101;
        double r206103 = r206102 * r206099;
        double r206104 = r206101 / r206084;
        double r206105 = r206103 * r206104;
        double r206106 = r206100 * r206105;
        double r206107 = r206106 + r206089;
        double r206108 = r206095 ? r206098 : r206107;
        double r206109 = r206086 ? r206093 : r206108;
        return r206109;
}

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	1.6

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

Derivation

1. Split input into 3 regimes

2. if t < -1.5871220803233903e+58

   1. Initial program 11.1

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

   2. Simplified 1.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)}\]
   3. Using strategy rm

   4. Applied fma-udef 1.2

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

   5. Simplified 1.2

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

   7. Applied add-cube-cbrt 1.5

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

   8. Applied associate-*l* 1.5

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

   9. Taylor expanded around 0 11.1

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

   10. Simplified 1.2

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

   if -1.5871220803233903e+58 < t < 6.204649366460223e+83

   1. Initial program 2.0

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

   2. Simplified 2.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)}\]
   3. Using strategy rm

   4. Applied fma-udef 2.6

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

   5. Simplified 2.6

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

   7. Applied pow1 2.6

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

   8. Applied pow1 2.6

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

   9. Applied pow-prod-down 2.6

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

   10. Simplified 2.0

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

   if 6.204649366460223e+83 < t

   1. Initial program 11.1

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

   2. Simplified 1.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)}\]
   3. Using strategy rm

   4. Applied fma-udef 1.4

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

   5. Simplified 1.4

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

   7. Applied add-cube-cbrt 1.7

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

   8. Applied associate-*l* 1.7

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

   10. Applied *-un-lft-identity 1.7

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

   11. Applied add-cube-cbrt 1.8

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

   12. Applied times-frac 1.8

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

   13. Applied associate-*r* 1.2

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

   14. Simplified 1.2

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

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.587122080323390289480100433348702966458 \cdot 10^{58}:\\ \;\;\;\;y \cdot \frac{z - x}{t} + x\\ \mathbf{elif}\;t \le 6.204649366460223122009107028759053602807 \cdot 10^{83}:\\ \;\;\;\;x + \frac{\left(z - x\right) \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{z - x} \cdot \sqrt[3]{z - x}\right) \cdot \left(\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{z - x}\right) \cdot \frac{\sqrt[3]{y}}{t}\right) + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(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)))