Kahan p9 Example

Average Error: 21.0 → 5.3

Time: 6.9s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

\[0.0 \lt x \lt 1 \land y \lt 1\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -2.17401776624398403 \cdot 10^{153}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -2.03359079106827272 \cdot 10^{-162} \lor \neg \left(y \le 1.5900983176733966 \cdot 10^{-155}\right):\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

C

double f(double x, double y) {
        double r84134 = x;
        double r84135 = y;
        double r84136 = r84134 - r84135;
        double r84137 = r84134 + r84135;
        double r84138 = r84136 * r84137;
        double r84139 = r84134 * r84134;
        double r84140 = r84135 * r84135;
        double r84141 = r84139 + r84140;
        double r84142 = r84138 / r84141;
        return r84142;
}

↓

double f(double x, double y) {
        double r84143 = y;
        double r84144 = -2.174017766243984e+153;
        bool r84145 = r84143 <= r84144;
        double r84146 = -1.0;
        double r84147 = -2.0335907910682727e-162;
        bool r84148 = r84143 <= r84147;
        double r84149 = 1.5900983176733966e-155;
        bool r84150 = r84143 <= r84149;
        double r84151 = !r84150;
        bool r84152 = r84148 || r84151;
        double r84153 = x;
        double r84154 = r84153 - r84143;
        double r84155 = r84153 + r84143;
        double r84156 = r84154 * r84155;
        double r84157 = r84153 * r84153;
        double r84158 = r84143 * r84143;
        double r84159 = r84157 + r84158;
        double r84160 = r84156 / r84159;
        double r84161 = 1.0;
        double r84162 = r84152 ? r84160 : r84161;
        double r84163 = r84145 ? r84146 : r84162;
        return r84163;
}

Error

Bits error versus x

Bits error versus y

Target

Original	21.0
Target	0.1
Herbie	5.3

\[\begin{array}{l} \mathbf{if}\;0.5 \lt \left|\frac{x}{y}\right| \lt 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if y < -2.174017766243984e+153

   1. Initial program 63.6

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

   2. Taylor expanded around 0 0

      \[\leadsto \color{blue}{-1}\]

   if -2.174017766243984e+153 < y < -2.0335907910682727e-162 or 1.5900983176733966e-155 < y

   1. Initial program 0.0

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

   if -2.0335907910682727e-162 < y < 1.5900983176733966e-155

   1. Initial program 30.7

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

   2. Taylor expanded around inf 16.6

      \[\leadsto \color{blue}{1}\]
3. Recombined 3 regimes into one program.

4. Final simplification 5.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.17401776624398403 \cdot 10^{153}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -2.03359079106827272 \cdot 10^{-162} \lor \neg \left(y \le 1.5900983176733966 \cdot 10^{-155}\right):\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (< 0.0 x 1) (< y 1))

  :herbie-target
  (if (< 0.5 (fabs (/ x y)) 2) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1 (/ 2 (+ 1 (* (/ x y) (/ x y))))))

  (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))