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

Average Error: 7.2 → 4.0

Time: 5.0s

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 r763144 = x;
        double r763145 = y;
        double r763146 = z;
        double r763147 = r763145 * r763146;
        double r763148 = r763147 - r763144;
        double r763149 = t;
        double r763150 = r763149 * r763146;
        double r763151 = r763150 - r763144;
        double r763152 = r763148 / r763151;
        double r763153 = r763144 + r763152;
        double r763154 = 1.0;
        double r763155 = r763144 + r763154;
        double r763156 = r763153 / r763155;
        return r763156;
}

↓

double f(double x, double y, double z, double t) {
        double r763157 = z;
        double r763158 = -4.575079891716448e+209;
        bool r763159 = r763157 <= r763158;
        double r763160 = 1.8378790055523983e+119;
        bool r763161 = r763157 <= r763160;
        double r763162 = !r763161;
        bool r763163 = r763159 || r763162;
        double r763164 = x;
        double r763165 = y;
        double r763166 = t;
        double r763167 = r763165 / r763166;
        double r763168 = r763164 + r763167;
        double r763169 = 1.0;
        double r763170 = r763164 + r763169;
        double r763171 = r763168 / r763170;
        double r763172 = r763165 * r763157;
        double r763173 = r763172 - r763164;
        double r763174 = r763166 * r763157;
        double r763175 = r763174 - r763164;
        double r763176 = r763173 / r763175;
        double r763177 = cbrt(r763176);
        double r763178 = r763177 * r763177;
        double r763179 = r763178 * r763177;
        double r763180 = r763164 + r763179;
        double r763181 = r763180 / r763170;
        double r763182 = r763163 ? r763171 : r763181;
        return r763182;
}

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)))