Average Error: 2.1 → 0.4
Time: 15.7s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot \sqrt[3]{{\left(e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\right)}^{3}}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \cdot \sqrt[3]{{\left(e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\right)}^{3}}
double f(double x, double y, double z, double t, double a, double b) {
        double r141929 = x;
        double r141930 = y;
        double r141931 = z;
        double r141932 = log(r141931);
        double r141933 = t;
        double r141934 = r141932 - r141933;
        double r141935 = r141930 * r141934;
        double r141936 = a;
        double r141937 = 1.0;
        double r141938 = r141937 - r141931;
        double r141939 = log(r141938);
        double r141940 = b;
        double r141941 = r141939 - r141940;
        double r141942 = r141936 * r141941;
        double r141943 = r141935 + r141942;
        double r141944 = exp(r141943);
        double r141945 = r141929 * r141944;
        return r141945;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r141946 = x;
        double r141947 = y;
        double r141948 = z;
        double r141949 = log(r141948);
        double r141950 = t;
        double r141951 = r141949 - r141950;
        double r141952 = r141947 * r141951;
        double r141953 = a;
        double r141954 = 1.0;
        double r141955 = log(r141954);
        double r141956 = 0.5;
        double r141957 = 2.0;
        double r141958 = pow(r141948, r141957);
        double r141959 = pow(r141954, r141957);
        double r141960 = r141958 / r141959;
        double r141961 = r141956 * r141960;
        double r141962 = r141954 * r141948;
        double r141963 = r141961 + r141962;
        double r141964 = r141955 - r141963;
        double r141965 = b;
        double r141966 = r141964 - r141965;
        double r141967 = r141953 * r141966;
        double r141968 = r141952 + r141967;
        double r141969 = exp(r141968);
        double r141970 = 3.0;
        double r141971 = pow(r141969, r141970);
        double r141972 = cbrt(r141971);
        double r141973 = r141946 * r141972;
        return r141973;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 2.1

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

  2. Taylor expanded around 0 0.4

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

  4. Applied add-cbrt-cube0.4

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

  5. Simplified0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2020035 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, B"
  :precision binary64
  (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1 z)) b))))))