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

Average Error: 6.3 → 0.9

Time: 4.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \le -10910987967306.1348 \lor \neg \left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \le -7.71869811703901377 \cdot 10^{-252}\right):\\ \;\;\;\;x + \frac{e^{y \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right) + y \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + e^{-1 \cdot z} \cdot \frac{1}{y}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r437575 = x;
        double r437576 = y;
        double r437577 = z;
        double r437578 = r437577 + r437576;
        double r437579 = r437576 / r437578;
        double r437580 = log(r437579);
        double r437581 = r437576 * r437580;
        double r437582 = exp(r437581);
        double r437583 = r437582 / r437576;
        double r437584 = r437575 + r437583;
        return r437584;
}

↓

double f(double x, double y, double z) {
        double r437585 = y;
        double r437586 = z;
        double r437587 = r437586 + r437585;
        double r437588 = r437585 / r437587;
        double r437589 = log(r437588);
        double r437590 = r437585 * r437589;
        double r437591 = exp(r437590);
        double r437592 = r437591 / r437585;
        double r437593 = -10910987967306.135;
        bool r437594 = r437592 <= r437593;
        double r437595 = -7.718698117039014e-252;
        bool r437596 = r437592 <= r437595;
        double r437597 = !r437596;
        bool r437598 = r437594 || r437597;
        double r437599 = x;
        double r437600 = 2.0;
        double r437601 = cbrt(r437585);
        double r437602 = cbrt(r437587);
        double r437603 = r437601 / r437602;
        double r437604 = log(r437603);
        double r437605 = r437600 * r437604;
        double r437606 = r437585 * r437605;
        double r437607 = r437585 * r437604;
        double r437608 = r437606 + r437607;
        double r437609 = exp(r437608);
        double r437610 = r437609 / r437585;
        double r437611 = r437599 + r437610;
        double r437612 = -1.0;
        double r437613 = r437612 * r437586;
        double r437614 = exp(r437613);
        double r437615 = 1.0;
        double r437616 = r437615 / r437585;
        double r437617 = r437614 * r437616;
        double r437618 = r437599 + r437617;
        double r437619 = r437598 ? r437611 : r437618;
        return r437619;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.3
Target	1.3
Herbie	0.9

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.1154157598 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (/ (exp (* y (log (/ y (+ z y))))) y) < -10910987967306.135 or -7.718698117039014e-252 < (/ (exp (* y (log (/ y (+ z y))))) y)

   1. Initial program 7.1

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

   3. Applied add-cube-cbrt 15.8

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

   4. Applied add-cube-cbrt 7.1

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

   5. Applied times-frac 7.1

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

   6. Applied log-prod 2.2

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

   7. Applied distribute-lft-in 2.2

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

   8. Simplified 0.8

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

   if -10910987967306.135 < (/ (exp (* y (log (/ y (+ z y))))) y) < -7.718698117039014e-252

   1. Initial program 3.0

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

   2. Taylor expanded around inf 1.1

      \[\leadsto \color{blue}{x + \frac{e^{-1 \cdot z}}{y}}\]
   3. Using strategy rm

   4. Applied div-inv 1.1

      \[\leadsto x + \color{blue}{e^{-1 \cdot z} \cdot \frac{1}{y}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \le -10910987967306.1348 \lor \neg \left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \le -7.71869811703901377 \cdot 10^{-252}\right):\\ \;\;\;\;x + \frac{e^{y \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right) + y \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + e^{-1 \cdot z} \cdot \frac{1}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, G"
  :precision binary64

  :herbie-target
  (if (< (/ y (+ z y)) 7.1154157597908e-315) (+ x (/ (exp (/ -1 z)) y)) (+ x (/ (exp (log (pow (/ y (+ y z)) y))) y)))

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