Average Error: 2.2 → 0.5
Time: 30.1s
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 r104177 = x;
        double r104178 = y;
        double r104179 = z;
        double r104180 = log(r104179);
        double r104181 = t;
        double r104182 = r104180 - r104181;
        double r104183 = r104178 * r104182;
        double r104184 = a;
        double r104185 = 1.0;
        double r104186 = r104185 - r104179;
        double r104187 = log(r104186);
        double r104188 = b;
        double r104189 = r104187 - r104188;
        double r104190 = r104184 * r104189;
        double r104191 = r104183 + r104190;
        double r104192 = exp(r104191);
        double r104193 = r104177 * r104192;
        return r104193;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r104194 = x;
        double r104195 = y;
        double r104196 = z;
        double r104197 = log(r104196);
        double r104198 = t;
        double r104199 = r104197 - r104198;
        double r104200 = r104195 * r104199;
        double r104201 = a;
        double r104202 = 1.0;
        double r104203 = log(r104202);
        double r104204 = 0.5;
        double r104205 = 2.0;
        double r104206 = pow(r104196, r104205);
        double r104207 = pow(r104202, r104205);
        double r104208 = r104206 / r104207;
        double r104209 = r104204 * r104208;
        double r104210 = r104202 * r104196;
        double r104211 = r104209 + r104210;
        double r104212 = r104203 - r104211;
        double r104213 = b;
        double r104214 = r104212 - r104213;
        double r104215 = r104201 * r104214;
        double r104216 = r104200 + r104215;
        double r104217 = exp(r104216);
        double r104218 = r104194 * r104217;
        return r104218;
}

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 2.2

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

  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 2019208 
(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))))))