Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3

Average Error: 10.0 → 0.3

Time: 16.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.961876288868113535790785433136922837478 \cdot 10^{-84} \lor \neg \left(z \le 6.777650331950523678129413213381219012915 \cdot 10^{-29}\right):\\ \;\;\;\;\left(y - x \cdot \frac{y}{z}\right) + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{x + y \cdot \left(z - x\right)}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r502245 = x;
        double r502246 = y;
        double r502247 = z;
        double r502248 = r502247 - r502245;
        double r502249 = r502246 * r502248;
        double r502250 = r502245 + r502249;
        double r502251 = r502250 / r502247;
        return r502251;
}

↓

double f(double x, double y, double z) {
        double r502252 = z;
        double r502253 = -1.9618762888681135e-84;
        bool r502254 = r502252 <= r502253;
        double r502255 = 6.777650331950524e-29;
        bool r502256 = r502252 <= r502255;
        double r502257 = !r502256;
        bool r502258 = r502254 || r502257;
        double r502259 = y;
        double r502260 = x;
        double r502261 = r502259 / r502252;
        double r502262 = r502260 * r502261;
        double r502263 = r502259 - r502262;
        double r502264 = r502260 / r502252;
        double r502265 = r502263 + r502264;
        double r502266 = 1.0;
        double r502267 = r502252 - r502260;
        double r502268 = r502259 * r502267;
        double r502269 = r502260 + r502268;
        double r502270 = r502252 / r502269;
        double r502271 = r502266 / r502270;
        double r502272 = r502258 ? r502265 : r502271;
        return r502272;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.0
Target	0.0
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if z < -1.9618762888681135e-84 or 6.777650331950524e-29 < z

   1. Initial program 14.2

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

   2. Taylor expanded around 0 4.9

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

   3. Simplified 4.9

      \[\leadsto \color{blue}{y - \frac{x \cdot y - x}{z}}\]
   4. Using strategy rm

   5. Applied div-sub 4.9

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

   6. Applied associate--r- 4.9

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

   8. Applied *-un-lft-identity 4.9

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

   9. Applied times-frac 0.3

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

   10. Simplified 0.3

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

   if -1.9618762888681135e-84 < z < 6.777650331950524e-29

   1. Initial program 0.1

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

   3. Applied clear-num 0.3

      \[\leadsto \color{blue}{\frac{1}{\frac{z}{x + y \cdot \left(z - x\right)}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.961876288868113535790785433136922837478 \cdot 10^{-84} \lor \neg \left(z \le 6.777650331950523678129413213381219012915 \cdot 10^{-29}\right):\\ \;\;\;\;\left(y - x \cdot \frac{y}{z}\right) + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{x + y \cdot \left(z - x\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
  :precision binary64

  :herbie-target
  (- (+ y (/ x z)) (/ y (/ z x)))

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