Average Error: 2.1 → 0.6
Time: 20.8s
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 r98015 = x;
        double r98016 = y;
        double r98017 = z;
        double r98018 = log(r98017);
        double r98019 = t;
        double r98020 = r98018 - r98019;
        double r98021 = r98016 * r98020;
        double r98022 = a;
        double r98023 = 1.0;
        double r98024 = r98023 - r98017;
        double r98025 = log(r98024);
        double r98026 = b;
        double r98027 = r98025 - r98026;
        double r98028 = r98022 * r98027;
        double r98029 = r98021 + r98028;
        double r98030 = exp(r98029);
        double r98031 = r98015 * r98030;
        return r98031;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r98032 = x;
        double r98033 = y;
        double r98034 = z;
        double r98035 = log(r98034);
        double r98036 = t;
        double r98037 = r98035 - r98036;
        double r98038 = r98033 * r98037;
        double r98039 = a;
        double r98040 = 1.0;
        double r98041 = log(r98040);
        double r98042 = 0.5;
        double r98043 = 2.0;
        double r98044 = pow(r98034, r98043);
        double r98045 = pow(r98040, r98043);
        double r98046 = r98044 / r98045;
        double r98047 = r98042 * r98046;
        double r98048 = r98040 * r98034;
        double r98049 = r98047 + r98048;
        double r98050 = r98041 - r98049;
        double r98051 = b;
        double r98052 = r98050 - r98051;
        double r98053 = r98039 * r98052;
        double r98054 = r98038 + r98053;
        double r98055 = exp(r98054);
        double r98056 = r98032 * r98055;
        return r98056;
}

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 0.6

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

  3. Final simplification0.6

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

Reproduce

herbie shell --seed 2019326 
(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))))))