Average Error: 2.0 → 0.5
Time: 17.5s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot \left(\left(\sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}} \cdot \sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}}\right) \cdot \sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}}\right)\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \cdot \left(\left(\sqrt[3]{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)}} \cdot \sqrt[3]{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)}}\right) \cdot \sqrt[3]{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)}}\right)
double f(double x, double y, double z, double t, double a, double b) {
        double r112214 = x;
        double r112215 = y;
        double r112216 = z;
        double r112217 = log(r112216);
        double r112218 = t;
        double r112219 = r112217 - r112218;
        double r112220 = r112215 * r112219;
        double r112221 = a;
        double r112222 = 1.0;
        double r112223 = r112222 - r112216;
        double r112224 = log(r112223);
        double r112225 = b;
        double r112226 = r112224 - r112225;
        double r112227 = r112221 * r112226;
        double r112228 = r112220 + r112227;
        double r112229 = exp(r112228);
        double r112230 = r112214 * r112229;
        return r112230;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r112231 = x;
        double r112232 = y;
        double r112233 = z;
        double r112234 = log(r112233);
        double r112235 = t;
        double r112236 = r112234 - r112235;
        double r112237 = r112232 * r112236;
        double r112238 = a;
        double r112239 = 1.0;
        double r112240 = log(r112239);
        double r112241 = 0.5;
        double r112242 = 2.0;
        double r112243 = pow(r112233, r112242);
        double r112244 = pow(r112239, r112242);
        double r112245 = r112243 / r112244;
        double r112246 = r112241 * r112245;
        double r112247 = r112239 * r112233;
        double r112248 = r112246 + r112247;
        double r112249 = r112240 - r112248;
        double r112250 = b;
        double r112251 = r112249 - r112250;
        double r112252 = r112238 * r112251;
        double r112253 = r112237 + r112252;
        double r112254 = exp(r112253);
        double r112255 = cbrt(r112254);
        double r112256 = r112255 * r112255;
        double r112257 = r112256 * r112255;
        double r112258 = r112231 * r112257;
        return r112258;
}

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 2.0

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

  2. Taylor expanded around 0 0.4

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

  4. Applied add-cube-cbrt0.5

    \[\leadsto x \cdot \color{blue}{\left(\left(\sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}} \cdot \sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}}\right) \cdot \sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}}\right)}\]

  5. Final simplification0.5

    \[\leadsto x \cdot \left(\left(\sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}} \cdot \sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}}\right) \cdot \sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}}\right)\]

Reproduce

herbie shell --seed 2020047 
(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))))))