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

Average Error: 10.2 → 0.4

Time: 4.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.3843947922367833 \cdot 10^{104} \lor \neg \left(x \le 1.0079533659797822 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + 1 \cdot \frac{x}{z}\right) - x\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r706591 = x;
        double r706592 = y;
        double r706593 = z;
        double r706594 = r706592 - r706593;
        double r706595 = 1.0;
        double r706596 = r706594 + r706595;
        double r706597 = r706591 * r706596;
        double r706598 = r706597 / r706593;
        return r706598;
}

↓

double f(double x, double y, double z) {
        double r706599 = x;
        double r706600 = -1.3843947922367833e+104;
        bool r706601 = r706599 <= r706600;
        double r706602 = 1.0079533659797822e-09;
        bool r706603 = r706599 <= r706602;
        double r706604 = !r706603;
        bool r706605 = r706601 || r706604;
        double r706606 = z;
        double r706607 = r706599 / r706606;
        double r706608 = y;
        double r706609 = r706608 - r706606;
        double r706610 = 1.0;
        double r706611 = r706609 + r706610;
        double r706612 = r706607 * r706611;
        double r706613 = r706599 * r706608;
        double r706614 = r706613 / r706606;
        double r706615 = r706610 * r706607;
        double r706616 = r706614 + r706615;
        double r706617 = r706616 - r706599;
        double r706618 = r706605 ? r706612 : r706617;
        return r706618;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.2
Target	0.5
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if x < -1.3843947922367833e+104 or 1.0079533659797822e-09 < x

   1. Initial program 29.4

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

   3. Applied associate-/l* 0.1

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

   5. Applied associate-/r/ 0.1

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

   if -1.3843947922367833e+104 < x < 1.0079533659797822e-09

   1. Initial program 1.4

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

   2. Taylor expanded around 0 0.5

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.3843947922367833 \cdot 10^{104} \lor \neg \left(x \le 1.0079533659797822 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + 1 \cdot \frac{x}{z}\right) - x\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(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))