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

Average Error: 7.2 → 3.4

Time: 24.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -7.150582175885458915277202527443364197416 \cdot 10^{54} \lor \neg \left(z \le 6.043915376694697440989411808244092375242 \cdot 10^{86}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{1}{\frac{t \cdot z - x}{y \cdot z - x}}}{x + 1}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r408990 = x;
        double r408991 = y;
        double r408992 = z;
        double r408993 = r408991 * r408992;
        double r408994 = r408993 - r408990;
        double r408995 = t;
        double r408996 = r408995 * r408992;
        double r408997 = r408996 - r408990;
        double r408998 = r408994 / r408997;
        double r408999 = r408990 + r408998;
        double r409000 = 1.0;
        double r409001 = r408990 + r409000;
        double r409002 = r408999 / r409001;
        return r409002;
}

↓

double f(double x, double y, double z, double t) {
        double r409003 = z;
        double r409004 = -7.150582175885459e+54;
        bool r409005 = r409003 <= r409004;
        double r409006 = 6.043915376694697e+86;
        bool r409007 = r409003 <= r409006;
        double r409008 = !r409007;
        bool r409009 = r409005 || r409008;
        double r409010 = x;
        double r409011 = y;
        double r409012 = t;
        double r409013 = r409011 / r409012;
        double r409014 = r409010 + r409013;
        double r409015 = 1.0;
        double r409016 = r409010 + r409015;
        double r409017 = r409014 / r409016;
        double r409018 = 1.0;
        double r409019 = r409012 * r409003;
        double r409020 = r409019 - r409010;
        double r409021 = r409011 * r409003;
        double r409022 = r409021 - r409010;
        double r409023 = r409020 / r409022;
        double r409024 = r409018 / r409023;
        double r409025 = r409010 + r409024;
        double r409026 = r409025 / r409016;
        double r409027 = r409009 ? r409017 : r409026;
        return r409027;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.2
Target	0.3
Herbie	3.4

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

Derivation

1. Split input into 2 regimes

2. if z < -7.150582175885459e+54 or 6.043915376694697e+86 < z

   1. Initial program 18.2

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

   2. Taylor expanded around inf 7.8

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

   if -7.150582175885459e+54 < z < 6.043915376694697e+86

   1. Initial program 0.7

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

   3. Applied clear-num 0.8

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

4. Final simplification 3.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -7.150582175885458915277202527443364197416 \cdot 10^{54} \lor \neg \left(z \le 6.043915376694697440989411808244092375242 \cdot 10^{86}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{1}{\frac{t \cdot z - x}{y \cdot z - x}}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1))

  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1)))