Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3

Average Error: 31.8 → 13.4

Time: 4.0s

Precision: 64

Math

\[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.4667592323261061 \cdot 10^{131}:\\ \;\;\;\;\sqrt[3]{1}\\ \mathbf{elif}\;x \le -8.0114334775173874 \cdot 10^{-117}:\\ \;\;\;\;\left(\sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\right) \cdot \sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\\ \mathbf{elif}\;x \le 1.9644387956096076 \cdot 10^{-169}:\\ \;\;\;\;\sqrt[3]{-1}\\ \mathbf{elif}\;x \le 1.06815270095652453 \cdot 10^{-15}:\\ \;\;\;\;\left(\sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\right) \cdot \sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\\ \mathbf{elif}\;x \le 2.0459804393047752 \cdot 10^{46}:\\ \;\;\;\;\sqrt[3]{-1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{1}\\ \end{array}\]

C

double f(double x, double y) {
        double r604323 = x;
        double r604324 = r604323 * r604323;
        double r604325 = y;
        double r604326 = 4.0;
        double r604327 = r604325 * r604326;
        double r604328 = r604327 * r604325;
        double r604329 = r604324 - r604328;
        double r604330 = r604324 + r604328;
        double r604331 = r604329 / r604330;
        return r604331;
}

↓

double f(double x, double y) {
        double r604332 = x;
        double r604333 = -1.4667592323261061e+131;
        bool r604334 = r604332 <= r604333;
        double r604335 = 1.0;
        double r604336 = cbrt(r604335);
        double r604337 = -8.011433477517387e-117;
        bool r604338 = r604332 <= r604337;
        double r604339 = r604332 * r604332;
        double r604340 = y;
        double r604341 = 4.0;
        double r604342 = r604340 * r604341;
        double r604343 = r604342 * r604340;
        double r604344 = r604339 - r604343;
        double r604345 = r604339 + r604343;
        double r604346 = r604344 / r604345;
        double r604347 = cbrt(r604346);
        double r604348 = r604347 * r604347;
        double r604349 = r604348 * r604347;
        double r604350 = 1.9644387956096076e-169;
        bool r604351 = r604332 <= r604350;
        double r604352 = -1.0;
        double r604353 = cbrt(r604352);
        double r604354 = 1.0681527009565245e-15;
        bool r604355 = r604332 <= r604354;
        double r604356 = 2.0459804393047752e+46;
        bool r604357 = r604332 <= r604356;
        double r604358 = r604357 ? r604353 : r604336;
        double r604359 = r604355 ? r604349 : r604358;
        double r604360 = r604351 ? r604353 : r604359;
        double r604361 = r604338 ? r604349 : r604360;
        double r604362 = r604334 ? r604336 : r604361;
        return r604362;
}

Error

Bits error versus x

Bits error versus y

Target

Original	31.8
Target	31.5
Herbie	13.4

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \lt 0.974323384962678118:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot y\right) \cdot 4} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{x \cdot x + \left(y \cdot y\right) \cdot 4}}\right)}^{2} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if x < -1.4667592323261061e+131 or 2.0459804393047752e+46 < x

   1. Initial program 51.0

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
   2. Using strategy rm

   3. Applied add-cbrt-cube 63.2

      \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{\sqrt[3]{\left(\left(x \cdot x + \left(y \cdot 4\right) \cdot y\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)}}}\]

   4. Applied add-cbrt-cube 63.6

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(x \cdot x - \left(y \cdot 4\right) \cdot y\right) \cdot \left(x \cdot x - \left(y \cdot 4\right) \cdot y\right)\right) \cdot \left(x \cdot x - \left(y \cdot 4\right) \cdot y\right)}}}{\sqrt[3]{\left(\left(x \cdot x + \left(y \cdot 4\right) \cdot y\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)}}\]

   5. Applied cbrt-undiv 63.6

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\left(x \cdot x - \left(y \cdot 4\right) \cdot y\right) \cdot \left(x \cdot x - \left(y \cdot 4\right) \cdot y\right)\right) \cdot \left(x \cdot x - \left(y \cdot 4\right) \cdot y\right)}{\left(\left(x \cdot x + \left(y \cdot 4\right) \cdot y\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)}}}\]

   6. Simplified 51.0

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)}^{3}}}\]

   7. Taylor expanded around inf 12.1

      \[\leadsto \sqrt[3]{\color{blue}{1}}\]

   if -1.4667592323261061e+131 < x < -8.011433477517387e-117 or 1.9644387956096076e-169 < x < 1.0681527009565245e-15

   1. Initial program 15.6

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 15.6

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\right) \cdot \sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}}\]

   if -8.011433477517387e-117 < x < 1.9644387956096076e-169 or 1.0681527009565245e-15 < x < 2.0459804393047752e+46

   1. Initial program 27.4

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
   2. Using strategy rm

   3. Applied add-cbrt-cube 48.4

      \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{\sqrt[3]{\left(\left(x \cdot x + \left(y \cdot 4\right) \cdot y\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)}}}\]

   4. Applied add-cbrt-cube 48.4

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(x \cdot x - \left(y \cdot 4\right) \cdot y\right) \cdot \left(x \cdot x - \left(y \cdot 4\right) \cdot y\right)\right) \cdot \left(x \cdot x - \left(y \cdot 4\right) \cdot y\right)}}}{\sqrt[3]{\left(\left(x \cdot x + \left(y \cdot 4\right) \cdot y\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)}}\]

   5. Applied cbrt-undiv 48.4

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\left(x \cdot x - \left(y \cdot 4\right) \cdot y\right) \cdot \left(x \cdot x - \left(y \cdot 4\right) \cdot y\right)\right) \cdot \left(x \cdot x - \left(y \cdot 4\right) \cdot y\right)}{\left(\left(x \cdot x + \left(y \cdot 4\right) \cdot y\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)}}}\]

   6. Simplified 27.4

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)}^{3}}}\]

   7. Taylor expanded around 0 12.6

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

4. Final simplification 13.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.4667592323261061 \cdot 10^{131}:\\ \;\;\;\;\sqrt[3]{1}\\ \mathbf{elif}\;x \le -8.0114334775173874 \cdot 10^{-117}:\\ \;\;\;\;\left(\sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\right) \cdot \sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\\ \mathbf{elif}\;x \le 1.9644387956096076 \cdot 10^{-169}:\\ \;\;\;\;\sqrt[3]{-1}\\ \mathbf{elif}\;x \le 1.06815270095652453 \cdot 10^{-15}:\\ \;\;\;\;\left(\sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\right) \cdot \sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\\ \mathbf{elif}\;x \le 2.0459804393047752 \cdot 10^{46}:\\ \;\;\;\;\sqrt[3]{-1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< (/ (- (* x x) (* (* y 4) y)) (+ (* x x) (* (* y 4) y))) 0.9743233849626781) (- (/ (* x x) (+ (* x x) (* (* y y) 4))) (/ (* (* y y) 4) (+ (* x x) (* (* y y) 4)))) (- (pow (/ x (sqrt (+ (* x x) (* (* y y) 4)))) 2) (/ (* (* y y) 4) (+ (* x x) (* (* y y) 4)))))

  (/ (- (* x x) (* (* y 4) y)) (+ (* x x) (* (* y 4) y))))