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

Average Error: 7.2 → 4.0

Time: 5.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -4.575079891716447743739485168859446908663 \cdot 10^{209} \lor \neg \left(z \le 1.837879005552398283274618224863548209635 \cdot 10^{119}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(\sqrt[3]{\frac{y \cdot z - x}{t \cdot z - x}} \cdot \sqrt[3]{\frac{y \cdot z - x}{t \cdot z - x}}\right) \cdot \sqrt[3]{\frac{y \cdot z - x}{t \cdot z - x}}}{x + 1}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r708990 = x;
        double r708991 = y;
        double r708992 = z;
        double r708993 = r708991 * r708992;
        double r708994 = r708993 - r708990;
        double r708995 = t;
        double r708996 = r708995 * r708992;
        double r708997 = r708996 - r708990;
        double r708998 = r708994 / r708997;
        double r708999 = r708990 + r708998;
        double r709000 = 1.0;
        double r709001 = r708990 + r709000;
        double r709002 = r708999 / r709001;
        return r709002;
}

↓

double f(double x, double y, double z, double t) {
        double r709003 = z;
        double r709004 = -4.575079891716448e+209;
        bool r709005 = r709003 <= r709004;
        double r709006 = 1.8378790055523983e+119;
        bool r709007 = r709003 <= r709006;
        double r709008 = !r709007;
        bool r709009 = r709005 || r709008;
        double r709010 = x;
        double r709011 = y;
        double r709012 = t;
        double r709013 = r709011 / r709012;
        double r709014 = r709010 + r709013;
        double r709015 = 1.0;
        double r709016 = r709010 + r709015;
        double r709017 = r709014 / r709016;
        double r709018 = r709011 * r709003;
        double r709019 = r709018 - r709010;
        double r709020 = r709012 * r709003;
        double r709021 = r709020 - r709010;
        double r709022 = r709019 / r709021;
        double r709023 = cbrt(r709022);
        double r709024 = r709023 * r709023;
        double r709025 = r709024 * r709023;
        double r709026 = r709010 + r709025;
        double r709027 = r709026 / r709016;
        double r709028 = r709009 ? r709017 : r709027;
        return r709028;
}

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	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 < -4.575079891716448e+209 or 1.8378790055523983e+119 < z

   1. Initial program 23.1

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

   2. Taylor expanded around inf 7.1

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

   if -4.575079891716448e+209 < z < 1.8378790055523983e+119

   1. Initial program 2.8

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

   3. Applied add-cube-cbrt 3.1

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

4. Final simplification 4.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.575079891716447743739485168859446908663 \cdot 10^{209} \lor \neg \left(z \le 1.837879005552398283274618224863548209635 \cdot 10^{119}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(\sqrt[3]{\frac{y \cdot z - x}{t \cdot z - x}} \cdot \sqrt[3]{\frac{y \cdot z - x}{t \cdot z - x}}\right) \cdot \sqrt[3]{\frac{y \cdot z - x}{t \cdot z - x}}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(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)))