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

Average Error: 10.4 → 0.1

Time: 14.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -884510415733809645269240406496772096 \lor \neg \left(z \le 1.138503149755666810409392543726458946196 \cdot 10^{-7}\right):\\ \;\;\;\;x \cdot \frac{\left(y - z\right) + 1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r420104 = x;
        double r420105 = y;
        double r420106 = z;
        double r420107 = r420105 - r420106;
        double r420108 = 1.0;
        double r420109 = r420107 + r420108;
        double r420110 = r420104 * r420109;
        double r420111 = r420110 / r420106;
        return r420111;
}

↓

double f(double x, double y, double z) {
        double r420112 = z;
        double r420113 = -8.845104157338096e+35;
        bool r420114 = r420112 <= r420113;
        double r420115 = 1.1385031497556668e-07;
        bool r420116 = r420112 <= r420115;
        double r420117 = !r420116;
        bool r420118 = r420114 || r420117;
        double r420119 = x;
        double r420120 = y;
        double r420121 = r420120 - r420112;
        double r420122 = 1.0;
        double r420123 = r420121 + r420122;
        double r420124 = r420123 / r420112;
        double r420125 = r420119 * r420124;
        double r420126 = r420119 / r420112;
        double r420127 = r420126 * r420123;
        double r420128 = r420118 ? r420125 : r420127;
        return r420128;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.4
Target	0.5
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 z < -8.845104157338096e+35 or 1.1385031497556668e-07 < 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 -8.845104157338096e+35 < z < 1.1385031497556668e-07

   1. Initial program 0.3

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

   3. Applied associate-/l* 7.0

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1}}}\]
   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 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -884510415733809645269240406496772096 \lor \neg \left(z \le 1.138503149755666810409392543726458946196 \cdot 10^{-7}\right):\\ \;\;\;\;x \cdot \frac{\left(y - z\right) + 1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 +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))