Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2

Average Error: 4.1 → 5.6

Time: 13.7s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -2.3617424863871238 \cdot 10^{-92}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + 0.83333333333333337\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \le 9.2554044737280137 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - \left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right) \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - {\left({\left(\log \left(e^{\frac{\frac{2}{t}}{3}}\right)\right)}^{3}\right)}^{\frac{1}{3}}\right)\right)}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r101928 = x;
        double r101929 = y;
        double r101930 = 2.0;
        double r101931 = z;
        double r101932 = t;
        double r101933 = a;
        double r101934 = r101932 + r101933;
        double r101935 = sqrt(r101934);
        double r101936 = r101931 * r101935;
        double r101937 = r101936 / r101932;
        double r101938 = b;
        double r101939 = c;
        double r101940 = r101938 - r101939;
        double r101941 = 5.0;
        double r101942 = 6.0;
        double r101943 = r101941 / r101942;
        double r101944 = r101933 + r101943;
        double r101945 = 3.0;
        double r101946 = r101932 * r101945;
        double r101947 = r101930 / r101946;
        double r101948 = r101944 - r101947;
        double r101949 = r101940 * r101948;
        double r101950 = r101937 - r101949;
        double r101951 = r101930 * r101950;
        double r101952 = exp(r101951);
        double r101953 = r101929 * r101952;
        double r101954 = r101928 + r101953;
        double r101955 = r101928 / r101954;
        return r101955;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r101956 = t;
        double r101957 = -2.3617424863871238e-92;
        bool r101958 = r101956 <= r101957;
        double r101959 = x;
        double r101960 = y;
        double r101961 = 2.0;
        double r101962 = c;
        double r101963 = a;
        double r101964 = 0.8333333333333334;
        double r101965 = r101963 + r101964;
        double r101966 = r101962 * r101965;
        double r101967 = b;
        double r101968 = r101963 * r101967;
        double r101969 = r101966 - r101968;
        double r101970 = r101961 * r101969;
        double r101971 = exp(r101970);
        double r101972 = r101960 * r101971;
        double r101973 = r101959 + r101972;
        double r101974 = r101959 / r101973;
        double r101975 = 9.255404473728014e-17;
        bool r101976 = r101956 <= r101975;
        double r101977 = z;
        double r101978 = r101956 + r101963;
        double r101979 = sqrt(r101978);
        double r101980 = r101977 * r101979;
        double r101981 = 5.0;
        double r101982 = 6.0;
        double r101983 = r101981 / r101982;
        double r101984 = r101963 - r101983;
        double r101985 = 3.0;
        double r101986 = r101956 * r101985;
        double r101987 = r101984 * r101986;
        double r101988 = r101980 * r101987;
        double r101989 = cbrt(r101956);
        double r101990 = r101989 * r101989;
        double r101991 = r101990 * r101989;
        double r101992 = r101967 - r101962;
        double r101993 = r101963 * r101963;
        double r101994 = r101983 * r101983;
        double r101995 = r101993 - r101994;
        double r101996 = r101995 * r101986;
        double r101997 = r101984 * r101961;
        double r101998 = r101996 - r101997;
        double r101999 = r101992 * r101998;
        double r102000 = r101991 * r101999;
        double r102001 = r101988 - r102000;
        double r102002 = r101991 * r101987;
        double r102003 = r102001 / r102002;
        double r102004 = r101961 * r102003;
        double r102005 = exp(r102004);
        double r102006 = r101960 * r102005;
        double r102007 = r101959 + r102006;
        double r102008 = r101959 / r102007;
        double r102009 = r101980 / r101956;
        double r102010 = r101963 + r101983;
        double r102011 = r101961 / r101956;
        double r102012 = r102011 / r101985;
        double r102013 = exp(r102012);
        double r102014 = log(r102013);
        double r102015 = 3.0;
        double r102016 = pow(r102014, r102015);
        double r102017 = 0.3333333333333333;
        double r102018 = pow(r102016, r102017);
        double r102019 = r102010 - r102018;
        double r102020 = r101992 * r102019;
        double r102021 = r102009 - r102020;
        double r102022 = r101961 * r102021;
        double r102023 = exp(r102022);
        double r102024 = r101960 * r102023;
        double r102025 = r101959 + r102024;
        double r102026 = r101959 / r102025;
        double r102027 = r101976 ? r102008 : r102026;
        double r102028 = r101958 ? r101974 : r102027;
        return r102028;
}

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

Derivation

1. Split input into 3 regimes

2. if t < -2.3617424863871238e-92

   1. Initial program 3.6

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

   2. Taylor expanded around inf 8.8

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\left(a \cdot c + 0.83333333333333337 \cdot c\right) - a \cdot b\right)}}}\]

   3. Simplified 8.8

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(a + 0.83333333333333337\right) - a \cdot b\right)}}}\]

   if -2.3617424863871238e-92 < t < 9.255404473728014e-17

   1. Initial program 5.4

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

   3. Applied add-cube-cbrt 5.4

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

   4. Applied times-frac 5.6

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

   6. Applied flip-+ 8.4

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

   7. Applied frac-sub 8.4

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

   8. Applied associate-*r/ 8.4

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

   9. Applied frac-times 8.2

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

   10. Applied frac-sub 6.7

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

   if 9.255404473728014e-17 < t

   1. Initial program 2.9

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

   3. Applied add-cbrt-cube 2.9

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

   4. Applied add-cbrt-cube 2.9

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

   5. Applied cbrt-unprod 2.9

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

   6. Applied add-cbrt-cube 2.9

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

   7. Applied cbrt-undiv 2.9

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

   8. Simplified 2.9

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

   10. Applied pow1/3 2.9

       \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \color{blue}{{\left({\left(\frac{2}{t \cdot 3}\right)}^{3}\right)}^{\frac{1}{3}}}\right)\right)}}\]
   11. Using strategy rm

   12. Applied add-log-exp 3.7

       \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - {\left({\color{blue}{\left(\log \left(e^{\frac{2}{t \cdot 3}}\right)\right)}}^{3}\right)}^{\frac{1}{3}}\right)\right)}}\]

   13. Simplified 3.7

       \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - {\left({\left(\log \color{blue}{\left(e^{\frac{\frac{2}{t}}{3}}\right)}\right)}^{3}\right)}^{\frac{1}{3}}\right)\right)}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 5.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.3617424863871238 \cdot 10^{-92}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + 0.83333333333333337\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \le 9.2554044737280137 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - \left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right) \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - {\left({\left(\log \left(e^{\frac{\frac{2}{t}}{3}}\right)\right)}^{3}\right)}^{\frac{1}{3}}\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2"
  :precision binary64
  (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3)))))))))))