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

Average Error: 6.8 → 0.8

Time: 2.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} = -\infty:\\ \;\;\;\;x + y \cdot \frac{z - x}{t}\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le -9.69488790304777811 \cdot 10^{55}:\\ \;\;\;\;x + \frac{1}{\frac{t}{y \cdot \left(z - x\right)}}\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 8.0502450144782888 \cdot 10^{-243}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 5.6700868646061108 \cdot 10^{271}:\\ \;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r381124 = x;
        double r381125 = y;
        double r381126 = z;
        double r381127 = r381126 - r381124;
        double r381128 = r381125 * r381127;
        double r381129 = t;
        double r381130 = r381128 / r381129;
        double r381131 = r381124 + r381130;
        return r381131;
}

↓

double f(double x, double y, double z, double t) {
        double r381132 = x;
        double r381133 = y;
        double r381134 = z;
        double r381135 = r381134 - r381132;
        double r381136 = r381133 * r381135;
        double r381137 = t;
        double r381138 = r381136 / r381137;
        double r381139 = r381132 + r381138;
        double r381140 = -inf.0;
        bool r381141 = r381139 <= r381140;
        double r381142 = r381135 / r381137;
        double r381143 = r381133 * r381142;
        double r381144 = r381132 + r381143;
        double r381145 = -9.694887903047778e+55;
        bool r381146 = r381139 <= r381145;
        double r381147 = 1.0;
        double r381148 = r381137 / r381136;
        double r381149 = r381147 / r381148;
        double r381150 = r381132 + r381149;
        double r381151 = 8.050245014478289e-243;
        bool r381152 = r381139 <= r381151;
        double r381153 = r381133 / r381137;
        double r381154 = fma(r381153, r381135, r381132);
        double r381155 = 5.670086864606111e+271;
        bool r381156 = r381139 <= r381155;
        double r381157 = r381156 ? r381139 : r381154;
        double r381158 = r381152 ? r381154 : r381157;
        double r381159 = r381146 ? r381150 : r381158;
        double r381160 = r381141 ? r381144 : r381159;
        return r381160;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.8
Target	2.1
Herbie	0.8

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

Derivation

1. Split input into 4 regimes

2. if (+ x (/ (* y (- z x)) t)) < -inf.0

   1. Initial program 64.0

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

   3. Applied *-un-lft-identity 64.0

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

   4. Applied times-frac 0.2

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

   5. Simplified 0.2

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

   if -inf.0 < (+ x (/ (* y (- z x)) t)) < -9.694887903047778e+55

   1. Initial program 0.1

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

   3. Applied clear-num 0.2

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

   if -9.694887903047778e+55 < (+ x (/ (* y (- z x)) t)) < 8.050245014478289e-243 or 5.670086864606111e+271 < (+ x (/ (* y (- z x)) t))

   1. Initial program 11.8

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

   2. Simplified 1.7

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

   if 8.050245014478289e-243 < (+ x (/ (* y (- z x)) t)) < 5.670086864606111e+271

   1. Initial program 0.6

      \[x + \frac{y \cdot \left(z - x\right)}{t}\]
3. Recombined 4 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} = -\infty:\\ \;\;\;\;x + y \cdot \frac{z - x}{t}\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le -9.69488790304777811 \cdot 10^{55}:\\ \;\;\;\;x + \frac{1}{\frac{t}{y \cdot \left(z - x\right)}}\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 8.0502450144782888 \cdot 10^{-243}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 5.6700868646061108 \cdot 10^{271}:\\ \;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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)))