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

Average Error: 6.6 → 1.8

Time: 5.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le 4.1052985606206364 \cdot 10^{-305}:\\ \;\;\;\;\frac{y}{t} \cdot \left(z - x\right) + x\\ \mathbf{elif}\;t \le 3.4271601891193827 \cdot 10^{131}:\\ \;\;\;\;\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt{t}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt{t}} \cdot \left(z - x\right)\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{{z}^{1}}{\frac{t}{y}} - \frac{y}{t} \cdot x\right) + \frac{y}{t} \cdot \mathsf{fma}\left(-x, 1, x\right)\right) + x\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r352290 = x;
        double r352291 = y;
        double r352292 = z;
        double r352293 = r352292 - r352290;
        double r352294 = r352291 * r352293;
        double r352295 = t;
        double r352296 = r352294 / r352295;
        double r352297 = r352290 + r352296;
        return r352297;
}

↓

double f(double x, double y, double z, double t) {
        double r352298 = t;
        double r352299 = 4.1052985606206364e-305;
        bool r352300 = r352298 <= r352299;
        double r352301 = y;
        double r352302 = r352301 / r352298;
        double r352303 = z;
        double r352304 = x;
        double r352305 = r352303 - r352304;
        double r352306 = r352302 * r352305;
        double r352307 = r352306 + r352304;
        double r352308 = 3.4271601891193827e+131;
        bool r352309 = r352298 <= r352308;
        double r352310 = cbrt(r352301);
        double r352311 = r352310 * r352310;
        double r352312 = sqrt(r352298);
        double r352313 = r352311 / r352312;
        double r352314 = r352310 / r352312;
        double r352315 = r352314 * r352305;
        double r352316 = r352313 * r352315;
        double r352317 = r352316 + r352304;
        double r352318 = 1.0;
        double r352319 = pow(r352303, r352318);
        double r352320 = r352298 / r352301;
        double r352321 = r352319 / r352320;
        double r352322 = r352302 * r352304;
        double r352323 = r352321 - r352322;
        double r352324 = -r352304;
        double r352325 = fma(r352324, r352318, r352304);
        double r352326 = r352302 * r352325;
        double r352327 = r352323 + r352326;
        double r352328 = r352327 + r352304;
        double r352329 = r352309 ? r352317 : r352328;
        double r352330 = r352300 ? r352307 : r352329;
        return r352330;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.6
Target	2.2
Herbie	1.8

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

Derivation

1. Split input into 3 regimes

2. if t < 4.1052985606206364e-305

   1. Initial program 6.4

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

   2. Simplified 2.3

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

   4. Applied fma-udef 2.3

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

   if 4.1052985606206364e-305 < t < 3.4271601891193827e+131

   1. Initial program 3.2

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

   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. Using strategy rm

   6. Applied add-sqr-sqrt 2.8

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

   7. Applied add-cube-cbrt 3.3

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

   8. Applied times-frac 3.3

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

   9. Applied associate-*l* 1.4

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

   if 3.4271601891193827e+131 < t

   1. Initial program 12.8

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

   2. Simplified 1.3

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

   4. Applied fma-udef 1.3

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

   6. Applied add-cube-cbrt 1.4

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

   7. Applied add-cube-cbrt 1.6

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

   8. Applied prod-diff 1.6

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

   9. Applied distribute-lft-in 1.6

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

   10. Simplified 1.7

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

   11. Simplified 1.7

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

   13. Applied pow1/3 33.2

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

   14. Applied pow-pow 1.5

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

   15. Simplified 1.5

       \[\leadsto \left(\left(\frac{{z}^{\color{blue}{1}}}{\frac{t}{y}} - \frac{y}{t} \cdot x\right) + \frac{y}{t} \cdot \mathsf{fma}\left(-x, 1, x\right)\right) + x\]
3. Recombined 3 regimes into one program.

4. Final simplification 1.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 4.1052985606206364 \cdot 10^{-305}:\\ \;\;\;\;\frac{y}{t} \cdot \left(z - x\right) + x\\ \mathbf{elif}\;t \le 3.4271601891193827 \cdot 10^{131}:\\ \;\;\;\;\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt{t}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt{t}} \cdot \left(z - x\right)\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{{z}^{1}}{\frac{t}{y}} - \frac{y}{t} \cdot x\right) + \frac{y}{t} \cdot \mathsf{fma}\left(-x, 1, x\right)\right) + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +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)))