Average Error: 1.9 → 0.5
Time: 20.6s
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 r72206 = x;
        double r72207 = y;
        double r72208 = z;
        double r72209 = log(r72208);
        double r72210 = t;
        double r72211 = r72209 - r72210;
        double r72212 = r72207 * r72211;
        double r72213 = a;
        double r72214 = 1.0;
        double r72215 = r72214 - r72208;
        double r72216 = log(r72215);
        double r72217 = b;
        double r72218 = r72216 - r72217;
        double r72219 = r72213 * r72218;
        double r72220 = r72212 + r72219;
        double r72221 = exp(r72220);
        double r72222 = r72206 * r72221;
        return r72222;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r72223 = x;
        double r72224 = y;
        double r72225 = z;
        double r72226 = log(r72225);
        double r72227 = t;
        double r72228 = r72226 - r72227;
        double r72229 = r72224 * r72228;
        double r72230 = a;
        double r72231 = 1.0;
        double r72232 = log(r72231);
        double r72233 = 0.5;
        double r72234 = 2.0;
        double r72235 = pow(r72225, r72234);
        double r72236 = pow(r72231, r72234);
        double r72237 = r72235 / r72236;
        double r72238 = r72233 * r72237;
        double r72239 = r72231 * r72225;
        double r72240 = r72238 + r72239;
        double r72241 = r72232 - r72240;
        double r72242 = b;
        double r72243 = r72241 - r72242;
        double r72244 = r72230 * r72243;
        double r72245 = r72229 + r72244;
        double r72246 = exp(r72245);
        double r72247 = r72223 * r72246;
        return r72247;
}

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 1978988140 
(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))))))