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

Average Error: 10.4 → 0.5

Time: 3.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2.6583684505465714 \cdot 10^{34} \lor \neg \left(x \le 5.3429144278201139 \cdot 10^{-89}\right):\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, \frac{x \cdot y}{z}\right) - x\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r899843 = x;
        double r899844 = y;
        double r899845 = z;
        double r899846 = r899844 - r899845;
        double r899847 = 1.0;
        double r899848 = r899846 + r899847;
        double r899849 = r899843 * r899848;
        double r899850 = r899849 / r899845;
        return r899850;
}

↓

double f(double x, double y, double z) {
        double r899851 = x;
        double r899852 = -2.6583684505465714e+34;
        bool r899853 = r899851 <= r899852;
        double r899854 = 5.342914427820114e-89;
        bool r899855 = r899851 <= r899854;
        double r899856 = !r899855;
        bool r899857 = r899853 || r899856;
        double r899858 = z;
        double r899859 = r899851 / r899858;
        double r899860 = y;
        double r899861 = r899860 - r899858;
        double r899862 = 1.0;
        double r899863 = r899861 + r899862;
        double r899864 = r899859 * r899863;
        double r899865 = r899851 * r899860;
        double r899866 = r899865 / r899858;
        double r899867 = fma(r899862, r899859, r899866);
        double r899868 = r899867 - r899851;
        double r899869 = r899857 ? r899864 : r899868;
        return r899869;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.4
Target	0.4
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if x < -2.6583684505465714e+34 or 5.342914427820114e-89 < x

   1. Initial program 22.7

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

   3. Applied associate-/l* 0.4

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

   5. Applied associate-/r/ 0.9

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

   if -2.6583684505465714e+34 < x < 5.342914427820114e-89

   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

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.6583684505465714 \cdot 10^{34} \lor \neg \left(x \le 5.3429144278201139 \cdot 10^{-89}\right):\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, \frac{x \cdot y}{z}\right) - x\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +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))