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

Average Error: 10.2 → 0.1

Time: 8.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -32557379355462839325163520 \lor \neg \left(x \le 7.170043320155678175136459501169904402577 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, \frac{x \cdot y}{z}\right) - x\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r425249 = x;
        double r425250 = y;
        double r425251 = z;
        double r425252 = r425250 - r425251;
        double r425253 = 1.0;
        double r425254 = r425252 + r425253;
        double r425255 = r425249 * r425254;
        double r425256 = r425255 / r425251;
        return r425256;
}

↓

double f(double x, double y, double z) {
        double r425257 = x;
        double r425258 = -3.255737935546284e+25;
        bool r425259 = r425257 <= r425258;
        double r425260 = 7.170043320155678e-21;
        bool r425261 = r425257 <= r425260;
        double r425262 = !r425261;
        bool r425263 = r425259 || r425262;
        double r425264 = z;
        double r425265 = y;
        double r425266 = r425265 - r425264;
        double r425267 = 1.0;
        double r425268 = r425266 + r425267;
        double r425269 = r425264 / r425268;
        double r425270 = r425257 / r425269;
        double r425271 = r425257 / r425264;
        double r425272 = r425257 * r425265;
        double r425273 = r425272 / r425264;
        double r425274 = fma(r425271, r425267, r425273);
        double r425275 = r425274 - r425257;
        double r425276 = r425263 ? r425270 : r425275;
        return r425276;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.2
Target	0.4
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 x < -3.255737935546284e+25 or 7.170043320155678e-21 < x

   1. Initial program 25.9

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

   3. Applied associate-/l* 0.1

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

   if -3.255737935546284e+25 < x < 7.170043320155678e-21

   1. Initial program 0.3

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

   2. Taylor expanded around 0 0.2

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

   3. Simplified 0.2

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -32557379355462839325163520 \lor \neg \left(x \le 7.170043320155678175136459501169904402577 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, \frac{x \cdot y}{z}\right) - x\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 +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 -2.7148310671343599e-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))