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

Average Error: 10.6 → 0.6

Time: 13.4s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -3.9021083944700275 \cdot 10^{-239} \lor \neg \left(x \le 1.5249655170051624 \cdot 10^{-193}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1 + y, -x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot \left(\left(y - z\right) + 1\right)}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r577879 = x;
        double r577880 = y;
        double r577881 = z;
        double r577882 = r577880 - r577881;
        double r577883 = 1.0;
        double r577884 = r577882 + r577883;
        double r577885 = r577879 * r577884;
        double r577886 = r577885 / r577881;
        return r577886;
}

↓

double f(double x, double y, double z) {
        double r577887 = x;
        double r577888 = -3.9021083944700275e-239;
        bool r577889 = r577887 <= r577888;
        double r577890 = 1.5249655170051624e-193;
        bool r577891 = r577887 <= r577890;
        double r577892 = !r577891;
        bool r577893 = r577889 || r577892;
        double r577894 = z;
        double r577895 = r577887 / r577894;
        double r577896 = 1.0;
        double r577897 = y;
        double r577898 = r577896 + r577897;
        double r577899 = -r577887;
        double r577900 = fma(r577895, r577898, r577899);
        double r577901 = 1.0;
        double r577902 = r577897 - r577894;
        double r577903 = r577902 + r577896;
        double r577904 = r577887 * r577903;
        double r577905 = r577894 / r577904;
        double r577906 = r577901 / r577905;
        double r577907 = r577893 ? r577900 : r577906;
        return r577907;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.6
Target	0.4
Herbie	0.6

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

Derivation

1. Split input into 2 regimes

2. if x < -3.9021083944700275e-239 or 1.5249655170051624e-193 < x

   1. Initial program 12.9

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

   2. Taylor expanded around 0 4.3

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

   3. Simplified 0.6

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

   5. Applied fma-neg 0.6

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

   if -3.9021083944700275e-239 < x < 1.5249655170051624e-193

   1. Initial program 0.2

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

   3. Applied clear-num 0.3

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.9021083944700275 \cdot 10^{-239} \lor \neg \left(x \le 1.5249655170051624 \cdot 10^{-193}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1 + y, -x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot \left(\left(y - z\right) + 1\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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))