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

Average Error: 10.6 → 0.2

Time: 2.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -6.231177025534271 \cdot 10^{38} \lor \neg \left(z \le 1.30166855464399448 \cdot 10^{-27}\right):\\ \;\;\;\;x \cdot \frac{\left(y - z\right) + 1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r711342 = x;
        double r711343 = y;
        double r711344 = z;
        double r711345 = r711343 - r711344;
        double r711346 = 1.0;
        double r711347 = r711345 + r711346;
        double r711348 = r711342 * r711347;
        double r711349 = r711348 / r711344;
        return r711349;
}

↓

double f(double x, double y, double z) {
        double r711350 = z;
        double r711351 = -6.231177025534271e+38;
        bool r711352 = r711350 <= r711351;
        double r711353 = 1.3016685546439945e-27;
        bool r711354 = r711350 <= r711353;
        double r711355 = !r711354;
        bool r711356 = r711352 || r711355;
        double r711357 = x;
        double r711358 = y;
        double r711359 = r711358 - r711350;
        double r711360 = 1.0;
        double r711361 = r711359 + r711360;
        double r711362 = r711361 / r711350;
        double r711363 = r711357 * r711362;
        double r711364 = r711357 / r711350;
        double r711365 = r711364 * r711361;
        double r711366 = r711356 ? r711363 : r711365;
        return r711366;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.6
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 < -6.231177025534271e+38 or 1.3016685546439945e-27 < 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 -6.231177025534271e+38 < z < 1.3016685546439945e-27

   1. Initial program 0.2

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

   3. Applied associate-/l* 7.2

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1}}}\]
   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 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -6.231177025534271 \cdot 10^{38} \lor \neg \left(z \le 1.30166855464399448 \cdot 10^{-27}\right):\\ \;\;\;\;x \cdot \frac{\left(y - z\right) + 1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(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))