Average Error: 1.9 → 0.5
Time: 11.8s
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 r160046 = x;
        double r160047 = y;
        double r160048 = z;
        double r160049 = log(r160048);
        double r160050 = t;
        double r160051 = r160049 - r160050;
        double r160052 = r160047 * r160051;
        double r160053 = a;
        double r160054 = 1.0;
        double r160055 = r160054 - r160048;
        double r160056 = log(r160055);
        double r160057 = b;
        double r160058 = r160056 - r160057;
        double r160059 = r160053 * r160058;
        double r160060 = r160052 + r160059;
        double r160061 = exp(r160060);
        double r160062 = r160046 * r160061;
        return r160062;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r160063 = x;
        double r160064 = y;
        double r160065 = z;
        double r160066 = log(r160065);
        double r160067 = t;
        double r160068 = r160066 - r160067;
        double r160069 = r160064 * r160068;
        double r160070 = a;
        double r160071 = 1.0;
        double r160072 = log(r160071);
        double r160073 = 0.5;
        double r160074 = 2.0;
        double r160075 = pow(r160065, r160074);
        double r160076 = pow(r160071, r160074);
        double r160077 = r160075 / r160076;
        double r160078 = r160073 * r160077;
        double r160079 = r160071 * r160065;
        double r160080 = r160078 + r160079;
        double r160081 = r160072 - r160080;
        double r160082 = b;
        double r160083 = r160081 - r160082;
        double r160084 = r160070 * r160083;
        double r160085 = r160069 + r160084;
        double r160086 = exp(r160085);
        double r160087 = r160063 * r160086;
        return r160087;
}

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 2020001 
(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))))))