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

Average Error: 10.5 → 0.6

Time: 10.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -24086198947376115712 \lor \neg \left(x \le 9.896449769877687409365259539005345521731 \cdot 10^{-188}\right):\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, \frac{x \cdot y}{z}\right) - x\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r445891 = x;
        double r445892 = y;
        double r445893 = z;
        double r445894 = r445892 - r445893;
        double r445895 = 1.0;
        double r445896 = r445894 + r445895;
        double r445897 = r445891 * r445896;
        double r445898 = r445897 / r445893;
        return r445898;
}

↓

double f(double x, double y, double z) {
        double r445899 = x;
        double r445900 = -2.4086198947376116e+19;
        bool r445901 = r445899 <= r445900;
        double r445902 = 9.896449769877687e-188;
        bool r445903 = r445899 <= r445902;
        double r445904 = !r445903;
        bool r445905 = r445901 || r445904;
        double r445906 = z;
        double r445907 = y;
        double r445908 = r445907 - r445906;
        double r445909 = 1.0;
        double r445910 = r445908 + r445909;
        double r445911 = r445906 / r445910;
        double r445912 = r445899 / r445911;
        double r445913 = r445899 / r445906;
        double r445914 = r445899 * r445907;
        double r445915 = r445914 / r445906;
        double r445916 = fma(r445913, r445909, r445915);
        double r445917 = r445916 - r445899;
        double r445918 = r445905 ? r445912 : r445917;
        return r445918;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.5
Target	0.4
Herbie	0.6

\[\begin{array}{l} \mathbf{if}\;x \lt -2.714831067134359919650240696134672137284 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.874108816439546156869494499878029491333 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if x < -2.4086198947376116e+19 or 9.896449769877687e-188 < x

   1. Initial program 18.5

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

   3. Applied associate-/l* 0.9

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

   if -2.4086198947376116e+19 < x < 9.896449769877687e-188

   1. Initial program 0.2

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

   2. Taylor expanded around 0 0.1

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

   3. Simplified 0.1

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -24086198947376115712 \lor \neg \left(x \le 9.896449769877687409365259539005345521731 \cdot 10^{-188}\right):\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, \frac{x \cdot y}{z}\right) - x\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 +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))