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

Average Error: 11.1 → 0.4

Time: 20.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -4.120156288663160099262452246870941235987 \cdot 10^{50} \lor \neg \left(x \le 1.115898472191414701666743120923343257633 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{1}{x \cdot e^{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) \cdot x + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\\ \end{array}\]

C

double f(double x, double y) {
        double r310646 = x;
        double r310647 = y;
        double r310648 = r310646 + r310647;
        double r310649 = r310646 / r310648;
        double r310650 = log(r310649);
        double r310651 = r310646 * r310650;
        double r310652 = exp(r310651);
        double r310653 = r310652 / r310646;
        return r310653;
}

↓

double f(double x, double y) {
        double r310654 = x;
        double r310655 = -4.12015628866316e+50;
        bool r310656 = r310654 <= r310655;
        double r310657 = 1.1158984721914147e-09;
        bool r310658 = r310654 <= r310657;
        double r310659 = !r310658;
        bool r310660 = r310656 || r310659;
        double r310661 = 1.0;
        double r310662 = y;
        double r310663 = exp(r310662);
        double r310664 = r310654 * r310663;
        double r310665 = r310661 / r310664;
        double r310666 = 2.0;
        double r310667 = cbrt(r310654);
        double r310668 = r310654 + r310662;
        double r310669 = cbrt(r310668);
        double r310670 = r310667 / r310669;
        double r310671 = log(r310670);
        double r310672 = r310666 * r310671;
        double r310673 = r310672 * r310654;
        double r310674 = r310654 * r310671;
        double r310675 = r310673 + r310674;
        double r310676 = exp(r310675);
        double r310677 = r310676 / r310654;
        double r310678 = r310660 ? r310665 : r310677;
        return r310678;
}

Error

Bits error versus x

Bits error versus y

Target

Original	11.1
Target	8.0
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if x < -4.12015628866316e+50 or 1.1158984721914147e-09 < x

   1. Initial program 11.7

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

   2. Taylor expanded around inf 0.5

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot y}}{x}}\]

   3. Simplified 0.5

      \[\leadsto \color{blue}{\frac{e^{-y}}{x}}\]
   4. Using strategy rm

   5. Applied neg-sub0 0.5

      \[\leadsto \frac{e^{\color{blue}{0 - y}}}{x}\]

   6. Applied exp-diff 0.5

      \[\leadsto \frac{\color{blue}{\frac{e^{0}}{e^{y}}}}{x}\]

   7. Applied associate-/l/ 0.5

      \[\leadsto \color{blue}{\frac{e^{0}}{x \cdot e^{y}}}\]

   if -4.12015628866316e+50 < x < 1.1158984721914147e-09

   1. Initial program 10.6

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

   3. Applied add-cube-cbrt 13.5

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

   4. Applied add-cube-cbrt 10.6

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

   5. Applied times-frac 10.6

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

   6. Applied log-prod 2.0

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

   7. Applied distribute-lft-in 2.0

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

   8. Simplified 0.2

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.120156288663160099262452246870941235987 \cdot 10^{50} \lor \neg \left(x \le 1.115898472191414701666743120923343257633 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{1}{x \cdot e^{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) \cdot x + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"
  :precision binary64

  :herbie-target
  (if (< y -3.73118442066479561e94) (/ (exp (/ -1 y)) x) (if (< y 2.81795924272828789e37) (/ (pow (/ x (+ y x)) x) x) (if (< y 2.347387415166998e178) (log (exp (/ (pow (/ x (+ y x)) x) x))) (/ (exp (/ -1 y)) x))))

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