Average Error: 1.9 → 0.5
Time: 17.8s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \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)}
double f(double x, double y, double z, double t, double a, double b) {
        double r127930 = x;
        double r127931 = y;
        double r127932 = z;
        double r127933 = log(r127932);
        double r127934 = t;
        double r127935 = r127933 - r127934;
        double r127936 = r127931 * r127935;
        double r127937 = a;
        double r127938 = 1.0;
        double r127939 = r127938 - r127932;
        double r127940 = log(r127939);
        double r127941 = b;
        double r127942 = r127940 - r127941;
        double r127943 = r127937 * r127942;
        double r127944 = r127936 + r127943;
        double r127945 = exp(r127944);
        double r127946 = r127930 * r127945;
        return r127946;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r127947 = x;
        double r127948 = y;
        double r127949 = z;
        double r127950 = log(r127949);
        double r127951 = t;
        double r127952 = r127950 - r127951;
        double r127953 = r127948 * r127952;
        double r127954 = a;
        double r127955 = 1.0;
        double r127956 = log(r127955);
        double r127957 = 0.5;
        double r127958 = 2.0;
        double r127959 = pow(r127949, r127958);
        double r127960 = pow(r127955, r127958);
        double r127961 = r127959 / r127960;
        double r127962 = r127957 * r127961;
        double r127963 = r127955 * r127949;
        double r127964 = r127962 + r127963;
        double r127965 = r127956 - r127964;
        double r127966 = b;
        double r127967 = r127965 - r127966;
        double r127968 = r127954 * r127967;
        double r127969 = r127953 + r127968;
        double r127970 = exp(r127969);
        double r127971 = r127947 * r127970;
        return r127971;
}

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 1.9

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

  2. Taylor expanded around 0 0.5

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

  3. Final simplification0.5

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

Reproduce

herbie shell --seed 2020046 
(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))))))