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

Average Error: 7.1 → 3.1

Time: 12.7s

Precision: 64

Math

\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1.0}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -4.397554981755772 \cdot 10^{+143}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1.0}\\ \mathbf{elif}\;z \le 1.2365924477248178 \cdot 10^{+144}:\\ \;\;\;\;\frac{x + \frac{\mathsf{fma}\left(y, z, -x\right)}{t \cdot z + \left(-x\right)}}{x + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1.0}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r11586796 = x;
        double r11586797 = y;
        double r11586798 = z;
        double r11586799 = r11586797 * r11586798;
        double r11586800 = r11586799 - r11586796;
        double r11586801 = t;
        double r11586802 = r11586801 * r11586798;
        double r11586803 = r11586802 - r11586796;
        double r11586804 = r11586800 / r11586803;
        double r11586805 = r11586796 + r11586804;
        double r11586806 = 1.0;
        double r11586807 = r11586796 + r11586806;
        double r11586808 = r11586805 / r11586807;
        return r11586808;
}

↓

double f(double x, double y, double z, double t) {
        double r11586809 = z;
        double r11586810 = -4.397554981755772e+143;
        bool r11586811 = r11586809 <= r11586810;
        double r11586812 = x;
        double r11586813 = y;
        double r11586814 = t;
        double r11586815 = r11586813 / r11586814;
        double r11586816 = r11586812 + r11586815;
        double r11586817 = 1.0;
        double r11586818 = r11586812 + r11586817;
        double r11586819 = r11586816 / r11586818;
        double r11586820 = 1.2365924477248178e+144;
        bool r11586821 = r11586809 <= r11586820;
        double r11586822 = -r11586812;
        double r11586823 = fma(r11586813, r11586809, r11586822);
        double r11586824 = r11586814 * r11586809;
        double r11586825 = r11586824 + r11586822;
        double r11586826 = r11586823 / r11586825;
        double r11586827 = r11586812 + r11586826;
        double r11586828 = r11586827 / r11586818;
        double r11586829 = r11586821 ? r11586828 : r11586819;
        double r11586830 = r11586811 ? r11586819 : r11586829;
        return r11586830;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.1
Target	0.3
Herbie	3.1

\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1.0}\]

Derivation

1. Split input into 2 regimes

2. if z < -4.397554981755772e+143 or 1.2365924477248178e+144 < z

   1. Initial program 21.9

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1.0}\]

   2. Taylor expanded around inf 6.5

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1.0}\]

   if -4.397554981755772e+143 < z < 1.2365924477248178e+144

   1. Initial program 2.0

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1.0}\]
   2. Using strategy rm

   3. Applied fma-neg 2.0

      \[\leadsto \frac{x + \frac{\color{blue}{\mathsf{fma}\left(y, z, -x\right)}}{t \cdot z - x}}{x + 1.0}\]
   4. Using strategy rm

   5. Applied sub-neg 2.0

      \[\leadsto \frac{x + \frac{\mathsf{fma}\left(y, z, -x\right)}{\color{blue}{t \cdot z + \left(-x\right)}}}{x + 1.0}\]
3. Recombined 2 regimes into one program.

4. Final simplification 3.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.397554981755772 \cdot 10^{+143}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1.0}\\ \mathbf{elif}\;z \le 1.2365924477248178 \cdot 10^{+144}:\\ \;\;\;\;\frac{x + \frac{\mathsf{fma}\left(y, z, -x\right)}{t \cdot z + \left(-x\right)}}{x + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +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)))