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

Average Error: 10.2 → 0.2

Time: 2.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -9.33691623494502822 \cdot 10^{51} \lor \neg \left(z \le 3.0566564905939977 \cdot 10^{-28}\right):\\ \;\;\;\;x \cdot \frac{1}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r648451 = x;
        double r648452 = y;
        double r648453 = z;
        double r648454 = r648452 - r648453;
        double r648455 = 1.0;
        double r648456 = r648454 + r648455;
        double r648457 = r648451 * r648456;
        double r648458 = r648457 / r648453;
        return r648458;
}

↓

double f(double x, double y, double z) {
        double r648459 = z;
        double r648460 = -9.336916234945028e+51;
        bool r648461 = r648459 <= r648460;
        double r648462 = 3.0566564905939977e-28;
        bool r648463 = r648459 <= r648462;
        double r648464 = !r648463;
        bool r648465 = r648461 || r648464;
        double r648466 = x;
        double r648467 = 1.0;
        double r648468 = y;
        double r648469 = r648468 - r648459;
        double r648470 = 1.0;
        double r648471 = r648469 + r648470;
        double r648472 = r648459 / r648471;
        double r648473 = r648467 / r648472;
        double r648474 = r648466 * r648473;
        double r648475 = r648466 / r648459;
        double r648476 = r648475 * r648471;
        double r648477 = r648465 ? r648474 : r648476;
        return r648477;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.2
Target	0.4
Herbie	0.2

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

Derivation

1. Split input into 2 regimes

2. if z < -9.336916234945028e+51 or 3.0566564905939977e-28 < z

   1. Initial program 17.1

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

   3. Applied *-un-lft-identity 17.1

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

   4. Applied times-frac 0.1

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

   5. Simplified 0.1

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

   7. Applied clear-num 0.1

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

   if -9.336916234945028e+51 < z < 3.0566564905939977e-28

   1. Initial program 0.4

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

   3. Applied associate-/l* 7.5

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

   5. Applied associate-/r/ 0.4

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -9.33691623494502822 \cdot 10^{51} \lor \neg \left(z \le 3.0566564905939977 \cdot 10^{-28}\right):\\ \;\;\;\;x \cdot \frac{1}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 
(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))