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

Average Error: 6.6 → 1.8

Time: 5.2s

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 r330902 = x;
        double r330903 = y;
        double r330904 = z;
        double r330905 = r330904 - r330902;
        double r330906 = r330903 * r330905;
        double r330907 = t;
        double r330908 = r330906 / r330907;
        double r330909 = r330902 + r330908;
        return r330909;
}

↓

double f(double x, double y, double z, double t) {
        double r330910 = t;
        double r330911 = 4.1052985606206364e-305;
        bool r330912 = r330910 <= r330911;
        double r330913 = y;
        double r330914 = r330913 / r330910;
        double r330915 = z;
        double r330916 = x;
        double r330917 = r330915 - r330916;
        double r330918 = r330914 * r330917;
        double r330919 = r330918 + r330916;
        double r330920 = 3.4271601891193827e+131;
        bool r330921 = r330910 <= r330920;
        double r330922 = cbrt(r330913);
        double r330923 = r330922 * r330922;
        double r330924 = sqrt(r330910);
        double r330925 = r330923 / r330924;
        double r330926 = r330922 / r330924;
        double r330927 = r330926 * r330917;
        double r330928 = r330925 * r330927;
        double r330929 = r330928 + r330916;
        double r330930 = 1.0;
        double r330931 = pow(r330915, r330930);
        double r330932 = r330910 / r330913;
        double r330933 = r330931 / r330932;
        double r330934 = r330914 * r330916;
        double r330935 = r330933 - r330934;
        double r330936 = -r330916;
        double r330937 = fma(r330936, r330930, r330916);
        double r330938 = r330914 * r330937;
        double r330939 = r330935 + r330938;
        double r330940 = r330939 + r330916;
        double r330941 = r330921 ? r330929 : r330940;
        double r330942 = r330912 ? r330919 : r330941;
        return r330942;
}

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)))