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

Average Error: 10.6 → 0.4

Time: 2.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.4723758122722615 \cdot 10^{110} \lor \neg \left(x \le 1.8053743254521745 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, \frac{x \cdot y}{z}\right) - x\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r565686 = x;
        double r565687 = y;
        double r565688 = z;
        double r565689 = r565687 - r565688;
        double r565690 = 1.0;
        double r565691 = r565689 + r565690;
        double r565692 = r565686 * r565691;
        double r565693 = r565692 / r565688;
        return r565693;
}

↓

double f(double x, double y, double z) {
        double r565694 = x;
        double r565695 = -1.4723758122722615e+110;
        bool r565696 = r565694 <= r565695;
        double r565697 = 1.8053743254521745e-20;
        bool r565698 = r565694 <= r565697;
        double r565699 = !r565698;
        bool r565700 = r565696 || r565699;
        double r565701 = z;
        double r565702 = y;
        double r565703 = r565702 - r565701;
        double r565704 = 1.0;
        double r565705 = r565703 + r565704;
        double r565706 = r565701 / r565705;
        double r565707 = r565694 / r565706;
        double r565708 = r565694 / r565701;
        double r565709 = r565694 * r565702;
        double r565710 = r565709 / r565701;
        double r565711 = fma(r565704, r565708, r565710);
        double r565712 = r565711 - r565694;
        double r565713 = r565700 ? r565707 : r565712;
        return r565713;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.6
Target	0.4
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 < -1.4723758122722615e+110 or 1.8053743254521745e-20 < 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

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

   if -1.4723758122722615e+110 < x < 1.8053743254521745e-20

   1. Initial program 1.5

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

   3. Applied associate-/l* 4.4

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

   4. Taylor expanded around 0 0.5

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

   5. Simplified 0.5

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.4723758122722615 \cdot 10^{110} \lor \neg \left(x \le 1.8053743254521745 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, \frac{x \cdot y}{z}\right) - x\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 +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))