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

Average Error: 10.0 → 1.2

Time: 2.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -5.66746259480640952 \cdot 10^{118} \lor \neg \left(z \le 1.0502220249582006 \cdot 10^{104}\right):\\ \;\;\;\;x \cdot \frac{\left(y - z\right) + 1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r700333 = x;
        double r700334 = y;
        double r700335 = z;
        double r700336 = r700334 - r700335;
        double r700337 = 1.0;
        double r700338 = r700336 + r700337;
        double r700339 = r700333 * r700338;
        double r700340 = r700339 / r700335;
        return r700340;
}

↓

double f(double x, double y, double z) {
        double r700341 = z;
        double r700342 = -5.6674625948064095e+118;
        bool r700343 = r700341 <= r700342;
        double r700344 = 1.0502220249582006e+104;
        bool r700345 = r700341 <= r700344;
        double r700346 = !r700345;
        bool r700347 = r700343 || r700346;
        double r700348 = x;
        double r700349 = y;
        double r700350 = r700349 - r700341;
        double r700351 = 1.0;
        double r700352 = r700350 + r700351;
        double r700353 = r700352 / r700341;
        double r700354 = r700348 * r700353;
        double r700355 = r700348 / r700341;
        double r700356 = r700355 * r700352;
        double r700357 = r700347 ? r700354 : r700356;
        return r700357;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.0
Target	0.4
Herbie	1.2

\[\begin{array}{l} \mathbf{if}\;x \lt -2.7148310671343599 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.87410881643954616 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -5.6674625948064095e+118 or 1.0502220249582006e+104 < z

   1. Initial program 21.4

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

   3. Applied *-un-lft-identity 21.4

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

   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 -5.6674625948064095e+118 < z < 1.0502220249582006e+104

   1. Initial program 2.5

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

   3. Applied associate-/l* 5.1

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

   5. Applied div-inv 5.1

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

   7. Applied un-div-inv 5.1

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

   8. Applied associate-/r/ 1.9

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

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.66746259480640952 \cdot 10^{118} \lor \neg \left(z \le 1.0502220249582006 \cdot 10^{104}\right):\\ \;\;\;\;x \cdot \frac{\left(y - z\right) + 1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< x -2.71483106713436e-162) (- (* (+ 1 y) (/ x z)) x) (if (< x 3.874108816439546e-197) (* (* x (+ (- y z) 1)) (/ 1 z)) (- (* (+ 1 y) (/ x z)) x)))

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