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

Average Error: 3.9 → 1.7

Time: 55.8s

Precision: 64

Math

\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.863752365364698602827271216718095908021 \cdot 10^{-196} \lor \neg \left(t \le 1.377649032039963197338721484104213014264 \cdot 10^{-82}\right):\\ \;\;\;\;\frac{x}{e^{\left(\frac{\frac{\sqrt{t + a}}{\sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{\sqrt[3]{t}} - \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(b - c\right)\right) \cdot 2} \cdot y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + e^{2 \cdot \frac{\left(\left(t \cdot 3\right) \cdot \left(\sqrt{t + a} \cdot z\right)\right) \cdot \left(a - \frac{5}{6}\right) - \left(t \cdot \left(b - c\right)\right) \cdot \left(\left(t \cdot \left(\left(a - \frac{5}{6}\right) \cdot 3\right)\right) \cdot \left(a + \frac{5}{6}\right) - 2 \cdot \left(a - \frac{5}{6}\right)\right)}{\left(t \cdot 3\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot t\right)}} \cdot y}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r630797 = x;
        double r630798 = y;
        double r630799 = 2.0;
        double r630800 = z;
        double r630801 = t;
        double r630802 = a;
        double r630803 = r630801 + r630802;
        double r630804 = sqrt(r630803);
        double r630805 = r630800 * r630804;
        double r630806 = r630805 / r630801;
        double r630807 = b;
        double r630808 = c;
        double r630809 = r630807 - r630808;
        double r630810 = 5.0;
        double r630811 = 6.0;
        double r630812 = r630810 / r630811;
        double r630813 = r630802 + r630812;
        double r630814 = 3.0;
        double r630815 = r630801 * r630814;
        double r630816 = r630799 / r630815;
        double r630817 = r630813 - r630816;
        double r630818 = r630809 * r630817;
        double r630819 = r630806 - r630818;
        double r630820 = r630799 * r630819;
        double r630821 = exp(r630820);
        double r630822 = r630798 * r630821;
        double r630823 = r630797 + r630822;
        double r630824 = r630797 / r630823;
        return r630824;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r630825 = t;
        double r630826 = -1.8637523653646986e-196;
        bool r630827 = r630825 <= r630826;
        double r630828 = 1.3776490320399632e-82;
        bool r630829 = r630825 <= r630828;
        double r630830 = !r630829;
        bool r630831 = r630827 || r630830;
        double r630832 = x;
        double r630833 = a;
        double r630834 = r630825 + r630833;
        double r630835 = sqrt(r630834);
        double r630836 = cbrt(r630825);
        double r630837 = r630835 / r630836;
        double r630838 = z;
        double r630839 = r630838 / r630836;
        double r630840 = r630837 * r630839;
        double r630841 = r630840 / r630836;
        double r630842 = 5.0;
        double r630843 = 6.0;
        double r630844 = r630842 / r630843;
        double r630845 = r630833 + r630844;
        double r630846 = 2.0;
        double r630847 = 3.0;
        double r630848 = r630825 * r630847;
        double r630849 = r630846 / r630848;
        double r630850 = r630845 - r630849;
        double r630851 = b;
        double r630852 = c;
        double r630853 = r630851 - r630852;
        double r630854 = r630850 * r630853;
        double r630855 = r630841 - r630854;
        double r630856 = r630855 * r630846;
        double r630857 = exp(r630856);
        double r630858 = y;
        double r630859 = r630857 * r630858;
        double r630860 = r630859 + r630832;
        double r630861 = r630832 / r630860;
        double r630862 = r630835 * r630838;
        double r630863 = r630848 * r630862;
        double r630864 = r630833 - r630844;
        double r630865 = r630863 * r630864;
        double r630866 = r630825 * r630853;
        double r630867 = r630864 * r630847;
        double r630868 = r630825 * r630867;
        double r630869 = r630868 * r630845;
        double r630870 = r630846 * r630864;
        double r630871 = r630869 - r630870;
        double r630872 = r630866 * r630871;
        double r630873 = r630865 - r630872;
        double r630874 = r630864 * r630825;
        double r630875 = r630848 * r630874;
        double r630876 = r630873 / r630875;
        double r630877 = r630846 * r630876;
        double r630878 = exp(r630877);
        double r630879 = r630878 * r630858;
        double r630880 = r630832 + r630879;
        double r630881 = r630832 / r630880;
        double r630882 = r630831 ? r630861 : r630881;
        return r630882;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	3.9
Target	2.9
Herbie	1.7

\[\begin{array}{l} \mathbf{if}\;t \lt -2.118326644891581057561884576920117070548 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot c + 0.8333333333333333703407674875052180141211 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \lt 5.196588770651547088010424937268931048836 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3 \cdot t\right) \cdot \left(a - \frac{5}{6}\right)\right) - \left(\left(\frac{5}{6} + a\right) \cdot \left(3 \cdot t\right) - 2\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if t < -1.8637523653646986e-196 or 1.3776490320399632e-82 < t

   1. Initial program 2.7

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 2.7

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

   4. Applied associate-/r* 2.7

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\frac{\frac{z \cdot \sqrt{t + a}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{t}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

   5. Simplified 0.9

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{\color{blue}{\frac{z}{\sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}}}{\sqrt[3]{t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

   if -1.8637523653646986e-196 < t < 1.3776490320399632e-82

   1. Initial program 6.5

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
   2. Using strategy rm

   3. Applied flip-+ 10.0

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\color{blue}{\frac{a \cdot a - \frac{5}{6} \cdot \frac{5}{6}}{a - \frac{5}{6}}} - \frac{2}{t \cdot 3}\right)\right)}}\]

   4. Applied frac-sub 10.0

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \color{blue}{\frac{\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2}{\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)}}\right)}}\]

   5. Applied associate-*r/ 10.0

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \color{blue}{\frac{\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)}{\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)}}\right)}}\]

   6. Applied frac-sub 7.1

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{t \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}}\]

   7. Simplified 3.4

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \frac{\color{blue}{\left(\left(\sqrt{t + a} \cdot z\right) \cdot \left(t \cdot 3\right)\right) \cdot \left(a - \frac{5}{6}\right) - \left(t \cdot \left(b - c\right)\right) \cdot \left(\left(a + \frac{5}{6}\right) \cdot \left(t \cdot \left(3 \cdot \left(a - \frac{5}{6}\right)\right)\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)}}{t \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\]

   8. Simplified 3.4

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \frac{\left(\left(\sqrt{t + a} \cdot z\right) \cdot \left(t \cdot 3\right)\right) \cdot \left(a - \frac{5}{6}\right) - \left(t \cdot \left(b - c\right)\right) \cdot \left(\left(a + \frac{5}{6}\right) \cdot \left(t \cdot \left(3 \cdot \left(a - \frac{5}{6}\right)\right)\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)}{\color{blue}{\left(t \cdot 3\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot t\right)}}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.863752365364698602827271216718095908021 \cdot 10^{-196} \lor \neg \left(t \le 1.377649032039963197338721484104213014264 \cdot 10^{-82}\right):\\ \;\;\;\;\frac{x}{e^{\left(\frac{\frac{\sqrt{t + a}}{\sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{\sqrt[3]{t}} - \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(b - c\right)\right) \cdot 2} \cdot y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + e^{2 \cdot \frac{\left(\left(t \cdot 3\right) \cdot \left(\sqrt{t + a} \cdot z\right)\right) \cdot \left(a - \frac{5}{6}\right) - \left(t \cdot \left(b - c\right)\right) \cdot \left(\left(t \cdot \left(\left(a - \frac{5}{6}\right) \cdot 3\right)\right) \cdot \left(a + \frac{5}{6}\right) - 2 \cdot \left(a - \frac{5}{6}\right)\right)}{\left(t \cdot 3\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot t\right)}} \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, I"

  :herbie-target
  (if (< t -2.118326644891581e-50) (/ x (+ x (* y (exp (* 2.0 (- (+ (* a c) (* 0.8333333333333334 c)) (* a b))))))) (if (< t 5.196588770651547e-123) (/ x (+ x (* y (exp (* 2.0 (/ (- (* (* z (sqrt (+ t a))) (* (* 3.0 t) (- a (/ 5.0 6.0)))) (* (- (* (+ (/ 5.0 6.0) a) (* 3.0 t)) 2.0) (* (- a (/ 5.0 6.0)) (* (- b c) t)))) (* (* (* t t) 3.0) (- a (/ 5.0 6.0))))))))) (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))))))))))

  (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))))