Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2

Average Error: 2.0 → 0.7

Time: 31.1s

Precision: 64

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

Math

\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]

↓

\[\begin{array}{l} \mathbf{if}\;\left(t - 1\right) \cdot \log a \le 52.279119054873007:\\ \;\;\;\;\frac{x}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}} \cdot \frac{\frac{\frac{1}{{a}^{1}}}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{y}\\ \mathbf{elif}\;\left(t - 1\right) \cdot \log a \le 567.847783482109776:\\ \;\;\;\;\frac{\frac{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{{a}^{1}} \cdot \sqrt[3]{{a}^{1}}}}{\frac{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}{\frac{\sqrt[3]{x}}{\sqrt[3]{{a}^{1}}}}}}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}} \cdot \left(\sqrt[3]{\frac{\frac{\frac{1}{{a}^{1}}}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{y}} \cdot \sqrt[3]{\frac{\frac{\frac{1}{{a}^{1}}}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{y}}\right)\right) \cdot \sqrt[3]{\frac{\frac{\frac{1}{{a}^{1}}}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{y}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r146929 = x;
        double r146930 = y;
        double r146931 = z;
        double r146932 = log(r146931);
        double r146933 = r146930 * r146932;
        double r146934 = t;
        double r146935 = 1.0;
        double r146936 = r146934 - r146935;
        double r146937 = a;
        double r146938 = log(r146937);
        double r146939 = r146936 * r146938;
        double r146940 = r146933 + r146939;
        double r146941 = b;
        double r146942 = r146940 - r146941;
        double r146943 = exp(r146942);
        double r146944 = r146929 * r146943;
        double r146945 = r146944 / r146930;
        return r146945;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r146946 = t;
        double r146947 = 1.0;
        double r146948 = r146946 - r146947;
        double r146949 = a;
        double r146950 = log(r146949);
        double r146951 = r146948 * r146950;
        double r146952 = 52.27911905487301;
        bool r146953 = r146951 <= r146952;
        double r146954 = x;
        double r146955 = y;
        double r146956 = z;
        double r146957 = log(r146956);
        double r146958 = -r146957;
        double r146959 = -r146950;
        double r146960 = b;
        double r146961 = fma(r146959, r146946, r146960);
        double r146962 = fma(r146955, r146958, r146961);
        double r146963 = exp(r146962);
        double r146964 = sqrt(r146963);
        double r146965 = r146954 / r146964;
        double r146966 = 1.0;
        double r146967 = pow(r146949, r146947);
        double r146968 = r146966 / r146967;
        double r146969 = r146968 / r146964;
        double r146970 = r146969 / r146955;
        double r146971 = r146965 * r146970;
        double r146972 = 567.8477834821098;
        bool r146973 = r146951 <= r146972;
        double r146974 = cbrt(r146954);
        double r146975 = r146974 * r146974;
        double r146976 = cbrt(r146967);
        double r146977 = r146976 * r146976;
        double r146978 = r146975 / r146977;
        double r146979 = r146974 / r146976;
        double r146980 = r146963 / r146979;
        double r146981 = r146978 / r146980;
        double r146982 = r146981 / r146955;
        double r146983 = cbrt(r146970);
        double r146984 = r146983 * r146983;
        double r146985 = r146965 * r146984;
        double r146986 = r146985 * r146983;
        double r146987 = r146973 ? r146982 : r146986;
        double r146988 = r146953 ? r146971 : r146987;
        return r146988;
}

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

Derivation

1. Split input into 3 regimes

2. if (* (- t 1.0) (log a)) < 52.27911905487301

   1. Initial program 2.5

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]

   2. Taylor expanded around inf 2.5

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

   3. Simplified 6.1

      \[\leadsto \color{blue}{\frac{\frac{\frac{x}{{a}^{1}}}{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}{y}}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity 6.1

      \[\leadsto \frac{\frac{\frac{x}{{a}^{1}}}{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}{\color{blue}{1 \cdot y}}\]

   6. Applied add-sqr-sqrt 6.1

      \[\leadsto \frac{\frac{\frac{x}{{a}^{1}}}{\color{blue}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}} \cdot \sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}}{1 \cdot y}\]

   7. Applied div-inv 6.1

      \[\leadsto \frac{\frac{\color{blue}{x \cdot \frac{1}{{a}^{1}}}}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}} \cdot \sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{1 \cdot y}\]

   8. Applied times-frac 1.9

      \[\leadsto \frac{\color{blue}{\frac{x}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}} \cdot \frac{\frac{1}{{a}^{1}}}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}}{1 \cdot y}\]

   9. Applied times-frac 0.1

      \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{1} \cdot \frac{\frac{\frac{1}{{a}^{1}}}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{y}}\]

   10. Simplified 0.1

       \[\leadsto \color{blue}{\frac{x}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}} \cdot \frac{\frac{\frac{1}{{a}^{1}}}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{y}\]

   if 52.27911905487301 < (* (- t 1.0) (log a)) < 567.8477834821098

   1. Initial program 1.5

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]

   2. Taylor expanded around inf 1.5

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

   3. Simplified 11.2

      \[\leadsto \color{blue}{\frac{\frac{\frac{x}{{a}^{1}}}{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}{y}}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 11.5

      \[\leadsto \frac{\frac{\frac{x}{\color{blue}{\left(\sqrt[3]{{a}^{1}} \cdot \sqrt[3]{{a}^{1}}\right) \cdot \sqrt[3]{{a}^{1}}}}}{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}{y}\]

   6. Applied add-cube-cbrt 11.6

      \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\left(\sqrt[3]{{a}^{1}} \cdot \sqrt[3]{{a}^{1}}\right) \cdot \sqrt[3]{{a}^{1}}}}{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}{y}\]

   7. Applied times-frac 11.6

      \[\leadsto \frac{\frac{\color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{{a}^{1}} \cdot \sqrt[3]{{a}^{1}}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{{a}^{1}}}}}{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}{y}\]

   8. Applied associate-/l* 1.7

      \[\leadsto \frac{\color{blue}{\frac{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{{a}^{1}} \cdot \sqrt[3]{{a}^{1}}}}{\frac{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}{\frac{\sqrt[3]{x}}{\sqrt[3]{{a}^{1}}}}}}}{y}\]

   if 567.8477834821098 < (* (- t 1.0) (log a))

   1. Initial program 0.3

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]

   2. Taylor expanded around inf 0.3

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

   3. Simplified 14.2

      \[\leadsto \color{blue}{\frac{\frac{\frac{x}{{a}^{1}}}{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}{y}}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity 14.2

      \[\leadsto \frac{\frac{\frac{x}{{a}^{1}}}{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}{\color{blue}{1 \cdot y}}\]

   6. Applied add-sqr-sqrt 14.2

      \[\leadsto \frac{\frac{\frac{x}{{a}^{1}}}{\color{blue}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}} \cdot \sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}}{1 \cdot y}\]

   7. Applied div-inv 14.2

      \[\leadsto \frac{\frac{\color{blue}{x \cdot \frac{1}{{a}^{1}}}}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}} \cdot \sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{1 \cdot y}\]

   8. Applied times-frac 0.0

      \[\leadsto \frac{\color{blue}{\frac{x}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}} \cdot \frac{\frac{1}{{a}^{1}}}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}}{1 \cdot y}\]

   9. Applied times-frac 2.2

      \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{1} \cdot \frac{\frac{\frac{1}{{a}^{1}}}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{y}}\]

   10. Simplified 2.2

       \[\leadsto \color{blue}{\frac{x}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}} \cdot \frac{\frac{\frac{1}{{a}^{1}}}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{y}\]
   11. Using strategy rm

   12. Applied add-cube-cbrt 2.2

       \[\leadsto \frac{x}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}} \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{\frac{\frac{1}{{a}^{1}}}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{y}} \cdot \sqrt[3]{\frac{\frac{\frac{1}{{a}^{1}}}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{y}}\right) \cdot \sqrt[3]{\frac{\frac{\frac{1}{{a}^{1}}}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{y}}\right)}\]

   13. Applied associate-*r* 2.2

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t - 1\right) \cdot \log a \le 52.279119054873007:\\ \;\;\;\;\frac{x}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}} \cdot \frac{\frac{\frac{1}{{a}^{1}}}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{y}\\ \mathbf{elif}\;\left(t - 1\right) \cdot \log a \le 567.847783482109776:\\ \;\;\;\;\frac{\frac{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{{a}^{1}} \cdot \sqrt[3]{{a}^{1}}}}{\frac{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}{\frac{\sqrt[3]{x}}{\sqrt[3]{{a}^{1}}}}}}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}} \cdot \left(\sqrt[3]{\frac{\frac{\frac{1}{{a}^{1}}}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{y}} \cdot \sqrt[3]{\frac{\frac{\frac{1}{{a}^{1}}}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{y}}\right)\right) \cdot \sqrt[3]{\frac{\frac{\frac{1}{{a}^{1}}}{\sqrt{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2"
  :precision binary64
  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1) (log a))) b))) y))