Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F

Average Error: 10.9 → 4.6

Time: 6.0s

Precision: 64

Math

\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]

↓

\[\begin{array}{l} \mathbf{if}\;y \le 227.4569151677616218876210041344165802002 \lor \neg \left(y \le 1.480529159109445059080956924684942271604 \cdot 10^{146}\right):\\ \;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \log \left(\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}} \cdot \frac{x}{\sqrt[3]{x + y}}\right)}}{x}\\ \end{array}\]

C

double f(double x, double y) {
        double r488257 = x;
        double r488258 = y;
        double r488259 = r488257 + r488258;
        double r488260 = r488257 / r488259;
        double r488261 = log(r488260);
        double r488262 = r488257 * r488261;
        double r488263 = exp(r488262);
        double r488264 = r488263 / r488257;
        return r488264;
}

↓

double f(double x, double y) {
        double r488265 = y;
        double r488266 = 227.45691516776162;
        bool r488267 = r488265 <= r488266;
        double r488268 = 1.480529159109445e+146;
        bool r488269 = r488265 <= r488268;
        double r488270 = !r488269;
        bool r488271 = r488267 || r488270;
        double r488272 = x;
        double r488273 = 2.0;
        double r488274 = cbrt(r488272);
        double r488275 = r488272 + r488265;
        double r488276 = cbrt(r488275);
        double r488277 = r488274 / r488276;
        double r488278 = log(r488277);
        double r488279 = r488273 * r488278;
        double r488280 = r488279 + r488278;
        double r488281 = r488272 * r488280;
        double r488282 = exp(r488281);
        double r488283 = r488282 / r488272;
        double r488284 = 1.0;
        double r488285 = r488276 * r488276;
        double r488286 = r488284 / r488285;
        double r488287 = r488272 / r488276;
        double r488288 = r488286 * r488287;
        double r488289 = log(r488288);
        double r488290 = r488272 * r488289;
        double r488291 = exp(r488290);
        double r488292 = r488291 / r488272;
        double r488293 = r488271 ? r488283 : r488292;
        return r488293;
}

Error

Bits error versus x

Bits error versus y

Target

Original	10.9
Target	7.7
Herbie	4.6

\[\begin{array}{l} \mathbf{if}\;y \lt -3.73118442066479561492798134439269393419 \cdot 10^{94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y \lt 28179592427282878868860376020282245120:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y \lt 2.347387415166997963747840232163110922613 \cdot 10^{178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < 227.45691516776162 or 1.480529159109445e+146 < y

   1. Initial program 7.8

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 28.1

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

   4. Applied add-cube-cbrt 7.8

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

   5. Applied times-frac 7.8

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

   6. Applied log-prod 3.5

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

   7. Simplified 2.1

      \[\leadsto \frac{e^{x \cdot \left(\color{blue}{2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)} + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\]

   if 227.45691516776162 < y < 1.480529159109445e+146

   1. Initial program 34.5

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 23.2

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

   4. Applied *-un-lft-identity 23.2

      \[\leadsto \frac{e^{x \cdot \log \left(\frac{\color{blue}{1 \cdot x}}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) \cdot \sqrt[3]{x + y}}\right)}}{x}\]

   5. Applied times-frac 23.2

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

4. Final simplification 4.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 227.4569151677616218876210041344165802002 \lor \neg \left(y \le 1.480529159109445059080956924684942271604 \cdot 10^{146}\right):\\ \;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \log \left(\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}} \cdot \frac{x}{\sqrt[3]{x + y}}\right)}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"
  :precision binary64

  :herbie-target
  (if (< y -3.7311844206647956e+94) (/ (exp (/ -1 y)) x) (if (< y 2.817959242728288e+37) (/ (pow (/ x (+ y x)) x) x) (if (< y 2.347387415166998e+178) (log (exp (/ (pow (/ x (+ y x)) x) x))) (/ (exp (/ -1 y)) x))))

  (/ (exp (* x (log (/ x (+ x y))))) x))