Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3

Average Error: 11.5 → 2.3

Time: 9.2s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -2.7061335562483838 \cdot 10^{-235}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y - z}}\\ \mathbf{elif}\;z \le 1.03830327655720515 \cdot 10^{-10}:\\ \;\;\;\;\left(x \cdot \left(y - z\right)\right) \cdot \frac{1}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - z}{t - z}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r704435 = x;
        double r704436 = y;
        double r704437 = z;
        double r704438 = r704436 - r704437;
        double r704439 = r704435 * r704438;
        double r704440 = t;
        double r704441 = r704440 - r704437;
        double r704442 = r704439 / r704441;
        return r704442;
}

↓

double f(double x, double y, double z, double t) {
        double r704443 = z;
        double r704444 = -2.7061335562483838e-235;
        bool r704445 = r704443 <= r704444;
        double r704446 = x;
        double r704447 = t;
        double r704448 = r704447 - r704443;
        double r704449 = y;
        double r704450 = r704449 - r704443;
        double r704451 = r704448 / r704450;
        double r704452 = r704446 / r704451;
        double r704453 = 1.0383032765572051e-10;
        bool r704454 = r704443 <= r704453;
        double r704455 = r704446 * r704450;
        double r704456 = 1.0;
        double r704457 = r704456 / r704448;
        double r704458 = r704455 * r704457;
        double r704459 = r704450 / r704448;
        double r704460 = r704446 * r704459;
        double r704461 = r704454 ? r704458 : r704460;
        double r704462 = r704445 ? r704452 : r704461;
        return r704462;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	11.5
Target	2.4
Herbie	2.3

\[\frac{x}{\frac{t - z}{y - z}}\]

Derivation

1. Split input into 3 regimes

2. if z < -2.7061335562483838e-235

   1. Initial program 12.2

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

   3. Applied associate-/l* 1.8

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

   if -2.7061335562483838e-235 < z < 1.0383032765572051e-10

   1. Initial program 5.1

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

   3. Applied div-inv 5.2

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

   if 1.0383032765572051e-10 < z

   1. Initial program 16.8

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

   3. Applied *-un-lft-identity 16.8

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

   4. Applied times-frac 0.1

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

   5. Simplified 0.1

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

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.7061335562483838 \cdot 10^{-235}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y - z}}\\ \mathbf{elif}\;z \le 1.03830327655720515 \cdot 10^{-10}:\\ \;\;\;\;\left(x \cdot \left(y - z\right)\right) \cdot \frac{1}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - z}{t - z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3"
  :precision binary64

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

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