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

Average Error: 5.9 → 0.8

Time: 2.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -30843645.5034677498:\\ \;\;\;\;y \cdot \frac{z - t}{a} + x\\ \mathbf{elif}\;a \le 5.6925026423385466 \cdot 10^{-40}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\sqrt{a}} \cdot \frac{z - t}{\sqrt{a}} + x\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r360816 = x;
        double r360817 = y;
        double r360818 = z;
        double r360819 = t;
        double r360820 = r360818 - r360819;
        double r360821 = r360817 * r360820;
        double r360822 = a;
        double r360823 = r360821 / r360822;
        double r360824 = r360816 + r360823;
        return r360824;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r360825 = a;
        double r360826 = -30843645.50346775;
        bool r360827 = r360825 <= r360826;
        double r360828 = y;
        double r360829 = z;
        double r360830 = t;
        double r360831 = r360829 - r360830;
        double r360832 = r360831 / r360825;
        double r360833 = r360828 * r360832;
        double r360834 = x;
        double r360835 = r360833 + r360834;
        double r360836 = 5.692502642338547e-40;
        bool r360837 = r360825 <= r360836;
        double r360838 = r360831 * r360828;
        double r360839 = r360838 / r360825;
        double r360840 = r360839 + r360834;
        double r360841 = sqrt(r360825);
        double r360842 = r360828 / r360841;
        double r360843 = r360831 / r360841;
        double r360844 = r360842 * r360843;
        double r360845 = r360844 + r360834;
        double r360846 = r360837 ? r360840 : r360845;
        double r360847 = r360827 ? r360835 : r360846;
        return r360847;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	5.9
Target	0.8
Herbie	0.8

\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if a < -30843645.50346775

   1. Initial program 9.4

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

   2. Simplified 1.8

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

   4. Applied fma-udef 1.8

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

   6. Applied div-inv 1.8

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

   7. Applied associate-*l* 0.8

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

   8. Simplified 0.8

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

   if -30843645.50346775 < a < 5.692502642338547e-40

   1. Initial program 0.9

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

   2. Simplified 4.4

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

   4. Applied fma-udef 4.4

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

   5. Taylor expanded around 0 0.9

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

   6. Simplified 0.9

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

   if 5.692502642338547e-40 < a

   1. Initial program 8.0

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

   2. Simplified 1.9

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

   4. Applied fma-udef 1.9

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

   6. Applied div-inv 1.9

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

   7. Applied associate-*l* 0.7

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

   8. Simplified 0.7

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

   10. Applied add-sqr-sqrt 0.8

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

   11. Applied *-un-lft-identity 0.8

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

   12. Applied times-frac 0.8

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

   13. Applied associate-*r* 0.9

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

   14. Simplified 0.8

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -30843645.5034677498:\\ \;\;\;\;y \cdot \frac{z - t}{a} + x\\ \mathbf{elif}\;a \le 5.6925026423385466 \cdot 10^{-40}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\sqrt{a}} \cdot \frac{z - t}{\sqrt{a}} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :precision binary64

  :herbie-target
  (if (< y -1.0761266216389975e-10) (+ x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

  (+ x (/ (* y (- z t)) a)))