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

Average Error: 7.3 → 3.8

Time: 19.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.783955773783118498253832197380730306883 \cdot 10^{82} \lor \neg \left(z \le 2.58937550673558283876381176295060269569 \cdot 10^{197}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}{x + 1}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r475030 = x;
        double r475031 = y;
        double r475032 = z;
        double r475033 = r475031 * r475032;
        double r475034 = r475033 - r475030;
        double r475035 = t;
        double r475036 = r475035 * r475032;
        double r475037 = r475036 - r475030;
        double r475038 = r475034 / r475037;
        double r475039 = r475030 + r475038;
        double r475040 = 1.0;
        double r475041 = r475030 + r475040;
        double r475042 = r475039 / r475041;
        return r475042;
}

↓

double f(double x, double y, double z, double t) {
        double r475043 = z;
        double r475044 = -1.7839557737831185e+82;
        bool r475045 = r475043 <= r475044;
        double r475046 = 2.589375506735583e+197;
        bool r475047 = r475043 <= r475046;
        double r475048 = !r475047;
        bool r475049 = r475045 || r475048;
        double r475050 = x;
        double r475051 = y;
        double r475052 = t;
        double r475053 = r475051 / r475052;
        double r475054 = r475050 + r475053;
        double r475055 = 1.0;
        double r475056 = r475050 + r475055;
        double r475057 = r475054 / r475056;
        double r475058 = r475051 * r475043;
        double r475059 = r475058 - r475050;
        double r475060 = 1.0;
        double r475061 = r475052 * r475043;
        double r475062 = r475061 - r475050;
        double r475063 = r475060 / r475062;
        double r475064 = r475059 * r475063;
        double r475065 = r475050 + r475064;
        double r475066 = r475065 / r475056;
        double r475067 = r475049 ? r475057 : r475066;
        return r475067;
}

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	3.8

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

Derivation

1. Split input into 2 regimes

2. if z < -1.7839557737831185e+82 or 2.589375506735583e+197 < z

   1. Initial program 20.6

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

   2. Taylor expanded around inf 7.3

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

   if -1.7839557737831185e+82 < z < 2.589375506735583e+197

   1. Initial program 2.5

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

   3. Applied div-inv 2.5

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

4. Final simplification 3.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.783955773783118498253832197380730306883 \cdot 10^{82} \lor \neg \left(z \le 2.58937550673558283876381176295060269569 \cdot 10^{197}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(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)))