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

Average Error: 7.6 → 3.8

Time: 22.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.160115839689408068409085936108477411806 \cdot 10^{172}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;z \le 3.641116230673848176898687453531338972974 \cdot 10^{130}:\\ \;\;\;\;\frac{\left(-x\right) - \frac{y \cdot z - x}{\mathsf{fma}\left(z, t, -x\right)}}{-\left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r27162248 = x;
        double r27162249 = y;
        double r27162250 = z;
        double r27162251 = r27162249 * r27162250;
        double r27162252 = r27162251 - r27162248;
        double r27162253 = t;
        double r27162254 = r27162253 * r27162250;
        double r27162255 = r27162254 - r27162248;
        double r27162256 = r27162252 / r27162255;
        double r27162257 = r27162248 + r27162256;
        double r27162258 = 1.0;
        double r27162259 = r27162248 + r27162258;
        double r27162260 = r27162257 / r27162259;
        return r27162260;
}

↓

double f(double x, double y, double z, double t) {
        double r27162261 = z;
        double r27162262 = -1.160115839689408e+172;
        bool r27162263 = r27162261 <= r27162262;
        double r27162264 = x;
        double r27162265 = y;
        double r27162266 = t;
        double r27162267 = r27162265 / r27162266;
        double r27162268 = r27162264 + r27162267;
        double r27162269 = 1.0;
        double r27162270 = r27162264 + r27162269;
        double r27162271 = r27162268 / r27162270;
        double r27162272 = 3.641116230673848e+130;
        bool r27162273 = r27162261 <= r27162272;
        double r27162274 = -r27162264;
        double r27162275 = r27162265 * r27162261;
        double r27162276 = r27162275 - r27162264;
        double r27162277 = fma(r27162261, r27162266, r27162274);
        double r27162278 = r27162276 / r27162277;
        double r27162279 = r27162274 - r27162278;
        double r27162280 = -r27162270;
        double r27162281 = r27162279 / r27162280;
        double r27162282 = r27162273 ? r27162281 : r27162271;
        double r27162283 = r27162263 ? r27162271 : r27162282;
        return r27162283;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.6
Target	0.4
Herbie	3.8

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

Derivation

1. Split input into 2 regimes

2. if z < -1.160115839689408e+172 or 3.641116230673848e+130 < z

   1. Initial program 22.4

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

   2. Taylor expanded around inf 7.4

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

   if -1.160115839689408e+172 < z < 3.641116230673848e+130

   1. Initial program 2.6

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

   3. Applied clear-num 2.6

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

   5. Applied frac-2neg 2.6

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

   6. Simplified 2.6

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

4. Final simplification 3.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.160115839689408068409085936108477411806 \cdot 10^{172}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;z \le 3.641116230673848176898687453531338972974 \cdot 10^{130}:\\ \;\;\;\;\frac{\left(-x\right) - \frac{y \cdot z - x}{\mathsf{fma}\left(z, t, -x\right)}}{-\left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +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)))