Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, A

Average Error: 7.3 → 3.7

Time: 21.8s

Precision: 64

Math

\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -5.0213726882368712246896067821148316655 \cdot 10^{81}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;z \le 2.653747744881045351380565842467581880116 \cdot 10^{131}:\\ \;\;\;\;\frac{x + \left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r27832927 = x;
        double r27832928 = y;
        double r27832929 = z;
        double r27832930 = r27832928 * r27832929;
        double r27832931 = r27832930 - r27832927;
        double r27832932 = t;
        double r27832933 = r27832932 * r27832929;
        double r27832934 = r27832933 - r27832927;
        double r27832935 = r27832931 / r27832934;
        double r27832936 = r27832927 + r27832935;
        double r27832937 = 1.0;
        double r27832938 = r27832927 + r27832937;
        double r27832939 = r27832936 / r27832938;
        return r27832939;
}

↓

double f(double x, double y, double z, double t) {
        double r27832940 = z;
        double r27832941 = -5.021372688236871e+81;
        bool r27832942 = r27832940 <= r27832941;
        double r27832943 = x;
        double r27832944 = y;
        double r27832945 = t;
        double r27832946 = r27832944 / r27832945;
        double r27832947 = r27832943 + r27832946;
        double r27832948 = 1.0;
        double r27832949 = r27832943 + r27832948;
        double r27832950 = r27832947 / r27832949;
        double r27832951 = 2.6537477448810454e+131;
        bool r27832952 = r27832940 <= r27832951;
        double r27832953 = r27832944 * r27832940;
        double r27832954 = r27832953 - r27832943;
        double r27832955 = 1.0;
        double r27832956 = r27832945 * r27832940;
        double r27832957 = r27832956 - r27832943;
        double r27832958 = r27832955 / r27832957;
        double r27832959 = r27832954 * r27832958;
        double r27832960 = r27832943 + r27832959;
        double r27832961 = r27832960 / r27832949;
        double r27832962 = r27832952 ? r27832961 : r27832950;
        double r27832963 = r27832942 ? r27832950 : r27832962;
        return r27832963;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.3
Target	0.4
Herbie	3.7

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

Derivation

1. Split input into 2 regimes

2. if z < -5.021372688236871e+81 or 2.6537477448810454e+131 < z

   1. Initial program 19.7

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]

   2. Taylor expanded around inf 8.1

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1}\]

   if -5.021372688236871e+81 < z < 2.6537477448810454e+131

   1. Initial program 1.6

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
   2. Using strategy rm

   3. Applied div-inv 1.7

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

4. Final simplification 3.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.0213726882368712246896067821148316655 \cdot 10^{81}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;z \le 2.653747744881045351380565842467581880116 \cdot 10^{131}:\\ \;\;\;\;\frac{x + \left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"

  :herbie-target
  (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1.0))

  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))