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

Average Error: 7.4 → 4.0

Time: 24.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -4.898559820385924943383218878329955525278 \cdot 10^{110}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;z \le 4.989921171097169236254159654900191065389 \cdot 10^{193}:\\ \;\;\;\;\frac{x + \frac{1}{\frac{t \cdot z - x}{y \cdot z - x}}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r27485733 = x;
        double r27485734 = y;
        double r27485735 = z;
        double r27485736 = r27485734 * r27485735;
        double r27485737 = r27485736 - r27485733;
        double r27485738 = t;
        double r27485739 = r27485738 * r27485735;
        double r27485740 = r27485739 - r27485733;
        double r27485741 = r27485737 / r27485740;
        double r27485742 = r27485733 + r27485741;
        double r27485743 = 1.0;
        double r27485744 = r27485733 + r27485743;
        double r27485745 = r27485742 / r27485744;
        return r27485745;
}

↓

double f(double x, double y, double z, double t) {
        double r27485746 = z;
        double r27485747 = -4.898559820385925e+110;
        bool r27485748 = r27485746 <= r27485747;
        double r27485749 = x;
        double r27485750 = y;
        double r27485751 = t;
        double r27485752 = r27485750 / r27485751;
        double r27485753 = r27485749 + r27485752;
        double r27485754 = 1.0;
        double r27485755 = r27485749 + r27485754;
        double r27485756 = r27485753 / r27485755;
        double r27485757 = 4.989921171097169e+193;
        bool r27485758 = r27485746 <= r27485757;
        double r27485759 = 1.0;
        double r27485760 = r27485751 * r27485746;
        double r27485761 = r27485760 - r27485749;
        double r27485762 = r27485750 * r27485746;
        double r27485763 = r27485762 - r27485749;
        double r27485764 = r27485761 / r27485763;
        double r27485765 = r27485759 / r27485764;
        double r27485766 = r27485749 + r27485765;
        double r27485767 = r27485766 / r27485755;
        double r27485768 = r27485758 ? r27485767 : r27485756;
        double r27485769 = r27485748 ? r27485756 : r27485768;
        return r27485769;
}

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	4.0

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

Derivation

1. Split input into 2 regimes

2. if z < -4.898559820385925e+110 or 4.989921171097169e+193 < z

   1. Initial program 22.1

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

   2. Taylor expanded around inf 8.1

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

   if -4.898559820385925e+110 < z < 4.989921171097169e+193

   1. Initial program 2.7

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

   3. Applied clear-num 2.7

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

   4. Taylor expanded around 0 2.7

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

4. Final simplification 4.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.898559820385924943383218878329955525278 \cdot 10^{110}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;z \le 4.989921171097169236254159654900191065389 \cdot 10^{193}:\\ \;\;\;\;\frac{x + \frac{1}{\frac{t \cdot z - x}{y \cdot z - x}}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +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)))