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

Average Error: 11.6 → 2.4

Time: 10.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -9.209930139418709416077035712078213691711 \lor \neg \left(z \le -5.98376643875231480812769882703216664937 \cdot 10^{-268}\right):\\ \;\;\;\;x \cdot \frac{y - z}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{\frac{t - z}{x}}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r374684 = x;
        double r374685 = y;
        double r374686 = z;
        double r374687 = r374685 - r374686;
        double r374688 = r374684 * r374687;
        double r374689 = t;
        double r374690 = r374689 - r374686;
        double r374691 = r374688 / r374690;
        return r374691;
}

↓

double f(double x, double y, double z, double t) {
        double r374692 = z;
        double r374693 = -9.20993013941871;
        bool r374694 = r374692 <= r374693;
        double r374695 = -5.983766438752315e-268;
        bool r374696 = r374692 <= r374695;
        double r374697 = !r374696;
        bool r374698 = r374694 || r374697;
        double r374699 = x;
        double r374700 = y;
        double r374701 = r374700 - r374692;
        double r374702 = t;
        double r374703 = r374702 - r374692;
        double r374704 = r374701 / r374703;
        double r374705 = r374699 * r374704;
        double r374706 = r374703 / r374699;
        double r374707 = r374701 / r374706;
        double r374708 = r374698 ? r374705 : r374707;
        return r374708;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	11.6
Target	2.2
Herbie	2.4

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

Derivation

1. Split input into 2 regimes

2. if z < -9.20993013941871 or -5.983766438752315e-268 < z

   1. Initial program 13.3

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

   3. Applied *-un-lft-identity 13.3

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

   4. Applied times-frac 1.7

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

   5. Simplified 1.7

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

   if -9.20993013941871 < z < -5.983766438752315e-268

   1. Initial program 5.1

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

   3. Applied clear-num 5.5

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

   5. Applied *-un-lft-identity 5.5

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

   6. Applied add-cube-cbrt 5.5

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot \frac{t - z}{x \cdot \left(y - z\right)}}\]

   7. Applied times-frac 5.5

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{\frac{t - z}{x \cdot \left(y - z\right)}}}\]

   8. Simplified 5.5

      \[\leadsto \color{blue}{1} \cdot \frac{\sqrt[3]{1}}{\frac{t - z}{x \cdot \left(y - z\right)}}\]

   9. Simplified 5.1

      \[\leadsto 1 \cdot \color{blue}{\frac{y - z}{\frac{t - z}{x}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -9.209930139418709416077035712078213691711 \lor \neg \left(z \le -5.98376643875231480812769882703216664937 \cdot 10^{-268}\right):\\ \;\;\;\;x \cdot \frac{y - z}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{\frac{t - z}{x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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)))