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

Average Error: 7.0 → 3.8

Time: 21.5s

Precision: 64

Math

\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1.0}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -2.648599861110283 \cdot 10^{+100}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1.0}\\ \mathbf{elif}\;z \le 9.631568013141158 \cdot 10^{+215}:\\ \;\;\;\;\frac{x + \left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}{x + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1.0}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r34324931 = x;
        double r34324932 = y;
        double r34324933 = z;
        double r34324934 = r34324932 * r34324933;
        double r34324935 = r34324934 - r34324931;
        double r34324936 = t;
        double r34324937 = r34324936 * r34324933;
        double r34324938 = r34324937 - r34324931;
        double r34324939 = r34324935 / r34324938;
        double r34324940 = r34324931 + r34324939;
        double r34324941 = 1.0;
        double r34324942 = r34324931 + r34324941;
        double r34324943 = r34324940 / r34324942;
        return r34324943;
}

↓

double f(double x, double y, double z, double t) {
        double r34324944 = z;
        double r34324945 = -2.648599861110283e+100;
        bool r34324946 = r34324944 <= r34324945;
        double r34324947 = x;
        double r34324948 = y;
        double r34324949 = t;
        double r34324950 = r34324948 / r34324949;
        double r34324951 = r34324947 + r34324950;
        double r34324952 = 1.0;
        double r34324953 = r34324947 + r34324952;
        double r34324954 = r34324951 / r34324953;
        double r34324955 = 9.631568013141158e+215;
        bool r34324956 = r34324944 <= r34324955;
        double r34324957 = r34324948 * r34324944;
        double r34324958 = r34324957 - r34324947;
        double r34324959 = 1.0;
        double r34324960 = r34324949 * r34324944;
        double r34324961 = r34324960 - r34324947;
        double r34324962 = r34324959 / r34324961;
        double r34324963 = r34324958 * r34324962;
        double r34324964 = r34324947 + r34324963;
        double r34324965 = r34324964 / r34324953;
        double r34324966 = r34324956 ? r34324965 : r34324954;
        double r34324967 = r34324946 ? r34324954 : r34324966;
        return r34324967;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.0
Target	0.3
Herbie	3.8

\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1.0}\]

Derivation

1. Split input into 2 regimes

2. if z < -2.648599861110283e+100 or 9.631568013141158e+215 < z

   1. Initial program 20.9

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1.0}\]

   2. Taylor expanded around inf 6.8

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1.0}\]

   if -2.648599861110283e+100 < z < 9.631568013141158e+215

   1. Initial program 2.9

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1.0}\]
   2. Using strategy rm

   3. Applied div-inv 2.9

      \[\leadsto \frac{x + \color{blue}{\left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}}{x + 1.0}\]
3. Recombined 2 regimes into one program.

4. Final simplification 3.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.648599861110283 \cdot 10^{+100}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1.0}\\ \mathbf{elif}\;z \le 9.631568013141158 \cdot 10^{+215}:\\ \;\;\;\;\frac{x + \left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}{x + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +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)))