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

Average Error: 7.2 → 3.6

Time: 34.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -8.287165386060872715453133833603909602049 \cdot 10^{86}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{1 + x}\\ \mathbf{elif}\;z \le 3.348305293084260034623912739354303673528 \cdot 10^{56}:\\ \;\;\;\;\frac{x + \frac{1}{\frac{t \cdot z - x}{y \cdot z - x}}}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{1 + x}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r30571588 = x;
        double r30571589 = y;
        double r30571590 = z;
        double r30571591 = r30571589 * r30571590;
        double r30571592 = r30571591 - r30571588;
        double r30571593 = t;
        double r30571594 = r30571593 * r30571590;
        double r30571595 = r30571594 - r30571588;
        double r30571596 = r30571592 / r30571595;
        double r30571597 = r30571588 + r30571596;
        double r30571598 = 1.0;
        double r30571599 = r30571588 + r30571598;
        double r30571600 = r30571597 / r30571599;
        return r30571600;
}

↓

double f(double x, double y, double z, double t) {
        double r30571601 = z;
        double r30571602 = -8.287165386060873e+86;
        bool r30571603 = r30571601 <= r30571602;
        double r30571604 = x;
        double r30571605 = y;
        double r30571606 = t;
        double r30571607 = r30571605 / r30571606;
        double r30571608 = r30571604 + r30571607;
        double r30571609 = 1.0;
        double r30571610 = r30571609 + r30571604;
        double r30571611 = r30571608 / r30571610;
        double r30571612 = 3.34830529308426e+56;
        bool r30571613 = r30571601 <= r30571612;
        double r30571614 = 1.0;
        double r30571615 = r30571606 * r30571601;
        double r30571616 = r30571615 - r30571604;
        double r30571617 = r30571605 * r30571601;
        double r30571618 = r30571617 - r30571604;
        double r30571619 = r30571616 / r30571618;
        double r30571620 = r30571614 / r30571619;
        double r30571621 = r30571604 + r30571620;
        double r30571622 = r30571621 / r30571610;
        double r30571623 = r30571613 ? r30571622 : r30571611;
        double r30571624 = r30571603 ? r30571611 : r30571623;
        return r30571624;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.2
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 < -8.287165386060873e+86 or 3.34830529308426e+56 < z

   1. Initial program 18.0

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

   2. Taylor expanded around inf 8.3

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

   if -8.287165386060873e+86 < z < 3.34830529308426e+56

   1. Initial program 0.8

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
   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 3.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -8.287165386060872715453133833603909602049 \cdot 10^{86}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{1 + x}\\ \mathbf{elif}\;z \le 3.348305293084260034623912739354303673528 \cdot 10^{56}:\\ \;\;\;\;\frac{x + \frac{1}{\frac{t \cdot z - x}{y \cdot z - x}}}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{1 + x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +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)))