Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3

Average Error: 10.4 → 0.1

Time: 10.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -884510415733809645269240406496772096 \lor \neg \left(z \le 1.138503149755666810409392543726458946196 \cdot 10^{-7}\right):\\ \;\;\;\;x \cdot \frac{\left(y - z\right) + 1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r446976 = x;
        double r446977 = y;
        double r446978 = z;
        double r446979 = r446977 - r446978;
        double r446980 = 1.0;
        double r446981 = r446979 + r446980;
        double r446982 = r446976 * r446981;
        double r446983 = r446982 / r446978;
        return r446983;
}

↓

double f(double x, double y, double z) {
        double r446984 = z;
        double r446985 = -8.845104157338096e+35;
        bool r446986 = r446984 <= r446985;
        double r446987 = 1.1385031497556668e-07;
        bool r446988 = r446984 <= r446987;
        double r446989 = !r446988;
        bool r446990 = r446986 || r446989;
        double r446991 = x;
        double r446992 = y;
        double r446993 = r446992 - r446984;
        double r446994 = 1.0;
        double r446995 = r446993 + r446994;
        double r446996 = r446995 / r446984;
        double r446997 = r446991 * r446996;
        double r446998 = r446991 / r446984;
        double r446999 = r446998 * r446995;
        double r447000 = r446990 ? r446997 : r446999;
        return r447000;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.4
Target	0.5
Herbie	0.1

\[\begin{array}{l} \mathbf{if}\;x \lt -2.714831067134359919650240696134672137284 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.874108816439546156869494499878029491333 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -8.845104157338096e+35 or 1.1385031497556668e-07 < z

   1. Initial program 17.9

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 17.9

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

   4. Applied times-frac 0.1

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

   5. Simplified 0.1

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

   if -8.845104157338096e+35 < z < 1.1385031497556668e-07

   1. Initial program 0.3

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
   2. Using strategy rm

   3. Applied associate-/l* 7.0

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

   5. Applied associate-/r/ 0.2

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -884510415733809645269240406496772096 \lor \neg \left(z \le 1.138503149755666810409392543726458946196 \cdot 10^{-7}\right):\\ \;\;\;\;x \cdot \frac{\left(y - z\right) + 1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< x -2.7148310671343599e-162) (- (* (+ 1 y) (/ x z)) x) (if (< x 3.87410881643954616e-197) (* (* x (+ (- y z) 1)) (/ 1 z)) (- (* (+ 1 y) (/ x z)) x)))

  (/ (* x (+ (- y z) 1)) z))