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

Average Error: 10.6 → 0.6

Time: 6.6s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -3.9021083944700275 \cdot 10^{-239} \lor \neg \left(x \le 1.5249655170051624 \cdot 10^{-193}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y + 1, -x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot \left(\left(y - z\right) + 1\right)}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r706278 = x;
        double r706279 = y;
        double r706280 = z;
        double r706281 = r706279 - r706280;
        double r706282 = 1.0;
        double r706283 = r706281 + r706282;
        double r706284 = r706278 * r706283;
        double r706285 = r706284 / r706280;
        return r706285;
}

↓

double f(double x, double y, double z) {
        double r706286 = x;
        double r706287 = -3.9021083944700275e-239;
        bool r706288 = r706286 <= r706287;
        double r706289 = 1.5249655170051624e-193;
        bool r706290 = r706286 <= r706289;
        double r706291 = !r706290;
        bool r706292 = r706288 || r706291;
        double r706293 = z;
        double r706294 = r706286 / r706293;
        double r706295 = y;
        double r706296 = 1.0;
        double r706297 = r706295 + r706296;
        double r706298 = -r706286;
        double r706299 = fma(r706294, r706297, r706298);
        double r706300 = 1.0;
        double r706301 = r706295 - r706293;
        double r706302 = r706301 + r706296;
        double r706303 = r706286 * r706302;
        double r706304 = r706293 / r706303;
        double r706305 = r706300 / r706304;
        double r706306 = r706292 ? r706299 : r706305;
        return r706306;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.6
Target	0.4
Herbie	0.6

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

Derivation

1. Split input into 2 regimes

2. if x < -3.9021083944700275e-239 or 1.5249655170051624e-193 < x

   1. Initial program 12.9

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

   2. Taylor expanded around 0 4.3

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

   3. Simplified 0.6

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

   5. Applied fma-neg 0.6

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

   if -3.9021083944700275e-239 < x < 1.5249655170051624e-193

   1. Initial program 0.2

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

   3. Applied clear-num 0.3

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.9021083944700275 \cdot 10^{-239} \lor \neg \left(x \le 1.5249655170051624 \cdot 10^{-193}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y + 1, -x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot \left(\left(y - z\right) + 1\right)}}\\ \end{array}\]

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))