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

Average Error: 3.7 → 2.0

Time: 1.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.208395080102709112792128783245881603461 \cdot 10^{-212} \lor \neg \left(t \le 2.337263817047334211087764408008483118108 \cdot 10^{-101}\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{a + t}}{\sqrt[3]{t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{a + t}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - \left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right) \cdot \left(\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r372775 = x;
        double r372776 = y;
        double r372777 = 2.0;
        double r372778 = z;
        double r372779 = t;
        double r372780 = a;
        double r372781 = r372779 + r372780;
        double r372782 = sqrt(r372781);
        double r372783 = r372778 * r372782;
        double r372784 = r372783 / r372779;
        double r372785 = b;
        double r372786 = c;
        double r372787 = r372785 - r372786;
        double r372788 = 5.0;
        double r372789 = 6.0;
        double r372790 = r372788 / r372789;
        double r372791 = r372780 + r372790;
        double r372792 = 3.0;
        double r372793 = r372779 * r372792;
        double r372794 = r372777 / r372793;
        double r372795 = r372791 - r372794;
        double r372796 = r372787 * r372795;
        double r372797 = r372784 - r372796;
        double r372798 = r372777 * r372797;
        double r372799 = exp(r372798);
        double r372800 = r372776 * r372799;
        double r372801 = r372775 + r372800;
        double r372802 = r372775 / r372801;
        return r372802;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r372803 = t;
        double r372804 = -1.208395080102709e-212;
        bool r372805 = r372803 <= r372804;
        double r372806 = 2.3372638170473342e-101;
        bool r372807 = r372803 <= r372806;
        double r372808 = !r372807;
        bool r372809 = r372805 || r372808;
        double r372810 = x;
        double r372811 = y;
        double r372812 = 2.0;
        double r372813 = z;
        double r372814 = cbrt(r372803);
        double r372815 = r372814 * r372814;
        double r372816 = r372813 / r372815;
        double r372817 = a;
        double r372818 = r372817 + r372803;
        double r372819 = sqrt(r372818);
        double r372820 = r372819 / r372814;
        double r372821 = r372816 * r372820;
        double r372822 = b;
        double r372823 = c;
        double r372824 = r372822 - r372823;
        double r372825 = 5.0;
        double r372826 = 6.0;
        double r372827 = r372825 / r372826;
        double r372828 = r372817 + r372827;
        double r372829 = 3.0;
        double r372830 = r372803 * r372829;
        double r372831 = r372812 / r372830;
        double r372832 = r372828 - r372831;
        double r372833 = r372824 * r372832;
        double r372834 = r372821 - r372833;
        double r372835 = r372812 * r372834;
        double r372836 = exp(r372835);
        double r372837 = r372811 * r372836;
        double r372838 = r372810 + r372837;
        double r372839 = r372810 / r372838;
        double r372840 = r372813 * r372819;
        double r372841 = r372817 - r372827;
        double r372842 = r372841 * r372830;
        double r372843 = r372840 * r372842;
        double r372844 = r372815 * r372814;
        double r372845 = r372828 * r372842;
        double r372846 = r372841 * r372812;
        double r372847 = r372845 - r372846;
        double r372848 = r372824 * r372847;
        double r372849 = r372844 * r372848;
        double r372850 = r372843 - r372849;
        double r372851 = r372844 * r372842;
        double r372852 = r372850 / r372851;
        double r372853 = r372812 * r372852;
        double r372854 = exp(r372853);
        double r372855 = r372811 * r372854;
        double r372856 = r372810 + r372855;
        double r372857 = r372810 / r372856;
        double r372858 = r372809 ? r372839 : r372857;
        return r372858;
}

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	3.7
Target	3.1
Herbie	2.0

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

Derivation

1. Split input into 2 regimes

2. if t < -1.208395080102709e-212 or 2.3372638170473342e-101 < 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 times-frac 1.2

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

   5. Simplified 1.2

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

   if -1.208395080102709e-212 < t < 2.3372638170473342e-101

   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 add-cube-cbrt 6.5

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

   4. Applied times-frac 6.8

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

   5. Simplified 6.8

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

   7. Applied flip-+ 10.2

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

   8. Applied frac-sub 10.2

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

   9. Applied associate-*r/ 10.2

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

   10. Applied frac-times 9.9

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

   11. Applied frac-sub 7.4

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

   13. Applied difference-of-squares 7.4

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

   14. Applied associate-*l* 4.2

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

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.208395080102709112792128783245881603461 \cdot 10^{-212} \lor \neg \left(t \le 2.337263817047334211087764408008483118108 \cdot 10^{-101}\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{a + t}}{\sqrt[3]{t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{a + t}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - \left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right) \cdot \left(\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019291 
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, I"
  :precision binary64

  :herbie-target
  (if (< t -2.1183266448915811e-50) (/ x (+ x (* y (exp (* 2 (- (+ (* a c) (* 0.83333333333333337 c)) (* a b))))))) (if (< t 5.19658877065154709e-123) (/ x (+ x (* y (exp (* 2 (/ (- (* (* z (sqrt (+ t a))) (* (* 3 t) (- a (/ 5 6)))) (* (- (* (+ (/ 5 6) a) (* 3 t)) 2) (* (- a (/ 5 6)) (* (- b c) t)))) (* (* (* t t) 3) (- a (/ 5 6))))))))) (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3))))))))))))

  (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3)))))))))))