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

Average Error: 7.4 → 3.6

Time: 5.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.58916669710278921 \cdot 10^{134} \lor \neg \left(z \le 3.3811815738088196 \cdot 10^{134}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x + 1}{x + \frac{1}{\frac{t \cdot z - x}{y \cdot z - x}}}}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r1223145 = x;
        double r1223146 = y;
        double r1223147 = z;
        double r1223148 = r1223146 * r1223147;
        double r1223149 = r1223148 - r1223145;
        double r1223150 = t;
        double r1223151 = r1223150 * r1223147;
        double r1223152 = r1223151 - r1223145;
        double r1223153 = r1223149 / r1223152;
        double r1223154 = r1223145 + r1223153;
        double r1223155 = 1.0;
        double r1223156 = r1223145 + r1223155;
        double r1223157 = r1223154 / r1223156;
        return r1223157;
}

↓

double f(double x, double y, double z, double t) {
        double r1223158 = z;
        double r1223159 = -1.5891666971027892e+134;
        bool r1223160 = r1223158 <= r1223159;
        double r1223161 = 3.3811815738088196e+134;
        bool r1223162 = r1223158 <= r1223161;
        double r1223163 = !r1223162;
        bool r1223164 = r1223160 || r1223163;
        double r1223165 = x;
        double r1223166 = y;
        double r1223167 = t;
        double r1223168 = r1223166 / r1223167;
        double r1223169 = r1223165 + r1223168;
        double r1223170 = 1.0;
        double r1223171 = r1223165 + r1223170;
        double r1223172 = r1223169 / r1223171;
        double r1223173 = 1.0;
        double r1223174 = r1223167 * r1223158;
        double r1223175 = r1223174 - r1223165;
        double r1223176 = r1223166 * r1223158;
        double r1223177 = r1223176 - r1223165;
        double r1223178 = r1223175 / r1223177;
        double r1223179 = r1223173 / r1223178;
        double r1223180 = r1223165 + r1223179;
        double r1223181 = r1223171 / r1223180;
        double r1223182 = r1223173 / r1223181;
        double r1223183 = r1223164 ? r1223172 : r1223182;
        return r1223183;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.4
Target	0.4
Herbie	3.6

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

Derivation

1. Split input into 2 regimes

2. if z < -1.5891666971027892e+134 or 3.3811815738088196e+134 < z

   1. Initial program 21.2

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

   2. Taylor expanded around inf 7.2

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

   if -1.5891666971027892e+134 < z < 3.3811815738088196e+134

   1. Initial program 2.2

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

   3. Applied clear-num 2.2

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

   5. Applied clear-num 2.3

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

4. Final simplification 3.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.58916669710278921 \cdot 10^{134} \lor \neg \left(z \le 3.3811815738088196 \cdot 10^{134}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x + 1}{x + \frac{1}{\frac{t \cdot z - x}{y \cdot z - x}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 
(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)))