Average Error: 2.1 → 7.6
Time: 5.4m
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1.0 - z\right) - b\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -2.7635926318262873 \cdot 10^{+131}:\\ \;\;\;\;e^{\log 1.0 \cdot a + \left(\log z - t\right) \cdot y} \cdot x\\ \mathbf{elif}\;y \le -3.3939892220623794 \cdot 10^{+73}:\\ \;\;\;\;x \cdot e^{-\left(1.0 \cdot \left(a \cdot z\right) + \left(a \cdot b + 0.5 \cdot \left(a \cdot \left(z \cdot z\right)\right)\right)\right)}\\ \mathbf{elif}\;y \le -8.816833859851974 \cdot 10^{-47}:\\ \;\;\;\;e^{\log 1.0 \cdot a + \left(\log z - t\right) \cdot y} \cdot x\\ \mathbf{elif}\;y \le 9.077191310208735 \cdot 10^{-77}:\\ \;\;\;\;x \cdot e^{-\left(1.0 \cdot \left(a \cdot z\right) + \left(a \cdot b + 0.5 \cdot \left(a \cdot \left(z \cdot z\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\log 1.0 \cdot a + \left(\log z - t\right) \cdot y} \cdot x\\ \end{array}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1.0 - z\right) - b\right)}
↓
\begin{array}{l}
\mathbf{if}\;y \le -2.7635926318262873 \cdot 10^{+131}:\\
\;\;\;\;e^{\log 1.0 \cdot a + \left(\log z - t\right) \cdot y} \cdot x\\

\mathbf{elif}\;y \le -3.3939892220623794 \cdot 10^{+73}:\\
\;\;\;\;x \cdot e^{-\left(1.0 \cdot \left(a \cdot z\right) + \left(a \cdot b + 0.5 \cdot \left(a \cdot \left(z \cdot z\right)\right)\right)\right)}\\

\mathbf{elif}\;y \le -8.816833859851974 \cdot 10^{-47}:\\
\;\;\;\;e^{\log 1.0 \cdot a + \left(\log z - t\right) \cdot y} \cdot x\\

\mathbf{elif}\;y \le 9.077191310208735 \cdot 10^{-77}:\\
\;\;\;\;x \cdot e^{-\left(1.0 \cdot \left(a \cdot z\right) + \left(a \cdot b + 0.5 \cdot \left(a \cdot \left(z \cdot z\right)\right)\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;e^{\log 1.0 \cdot a + \left(\log z - t\right) \cdot y} \cdot x\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r7266308 = x;
        double r7266309 = y;
        double r7266310 = z;
        double r7266311 = log(r7266310);
        double r7266312 = t;
        double r7266313 = r7266311 - r7266312;
        double r7266314 = r7266309 * r7266313;
        double r7266315 = a;
        double r7266316 = 1.0;
        double r7266317 = r7266316 - r7266310;
        double r7266318 = log(r7266317);
        double r7266319 = b;
        double r7266320 = r7266318 - r7266319;
        double r7266321 = r7266315 * r7266320;
        double r7266322 = r7266314 + r7266321;
        double r7266323 = exp(r7266322);
        double r7266324 = r7266308 * r7266323;
        return r7266324;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r7266325 = y;
        double r7266326 = -2.7635926318262873e+131;
        bool r7266327 = r7266325 <= r7266326;
        double r7266328 = 1.0;
        double r7266329 = log(r7266328);
        double r7266330 = a;
        double r7266331 = r7266329 * r7266330;
        double r7266332 = z;
        double r7266333 = log(r7266332);
        double r7266334 = t;
        double r7266335 = r7266333 - r7266334;
        double r7266336 = r7266335 * r7266325;
        double r7266337 = r7266331 + r7266336;
        double r7266338 = exp(r7266337);
        double r7266339 = x;
        double r7266340 = r7266338 * r7266339;
        double r7266341 = -3.3939892220623794e+73;
        bool r7266342 = r7266325 <= r7266341;
        double r7266343 = r7266330 * r7266332;
        double r7266344 = r7266328 * r7266343;
        double r7266345 = b;
        double r7266346 = r7266330 * r7266345;
        double r7266347 = 0.5;
        double r7266348 = r7266332 * r7266332;
        double r7266349 = r7266330 * r7266348;
        double r7266350 = r7266347 * r7266349;
        double r7266351 = r7266346 + r7266350;
        double r7266352 = r7266344 + r7266351;
        double r7266353 = -r7266352;
        double r7266354 = exp(r7266353);
        double r7266355 = r7266339 * r7266354;
        double r7266356 = -8.816833859851974e-47;
        bool r7266357 = r7266325 <= r7266356;
        double r7266358 = 9.077191310208735e-77;
        bool r7266359 = r7266325 <= r7266358;
        double r7266360 = r7266359 ? r7266355 : r7266340;
        double r7266361 = r7266357 ? r7266340 : r7266360;
        double r7266362 = r7266342 ? r7266355 : r7266361;
        double r7266363 = r7266327 ? r7266340 : r7266362;
        return r7266363;
}

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. Split input into 2 regimes

  2. if y < -2.7635926318262873e+131 or -3.3939892220623794e+73 < y < -8.816833859851974e-47 or 9.077191310208735e-77 < y

    1. Initial program 1.7

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1.0 - z\right) - b\right)}\]

    2. Taylor expanded around 0 8.7

      \[\leadsto x \cdot e^{\color{blue}{\left(\log z \cdot y + a \cdot \log 1.0\right) - t \cdot y}}\]

    3. Simplified8.7

      \[\leadsto x \cdot e^{\color{blue}{\log 1.0 \cdot a + \left(\log z - t\right) \cdot y}}\]

    if -2.7635926318262873e+131 < y < -3.3939892220623794e+73 or -8.816833859851974e-47 < y < 9.077191310208735e-77

    1. Initial program 2.5

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1.0 - z\right) - b\right)}\]

    2. Taylor expanded around 0 0.1

      \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\color{blue}{\left(\log 1.0 - \left(1.0 \cdot z + \frac{1}{2} \cdot \frac{{z}^{2}}{{1.0}^{2}}\right)\right)} - b\right)}\]

    3. Simplified0.1

      \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\color{blue}{\left(\left(\log 1.0 - 1.0 \cdot z\right) - \frac{\frac{1}{2}}{\frac{1.0}{z} \cdot \frac{1.0}{z}}\right)} - b\right)}\]

    4. Taylor expanded around inf 6.6

      \[\leadsto x \cdot e^{\color{blue}{-\left(1.0 \cdot \left(a \cdot z\right) + \left(a \cdot b + 0.5 \cdot \left(a \cdot {z}^{2}\right)\right)\right)}}\]

    5. Simplified6.6

      \[\leadsto x \cdot e^{\color{blue}{-\left(\left(a \cdot z\right) \cdot 1.0 + \left(a \cdot b + \left(a \cdot \left(z \cdot z\right)\right) \cdot 0.5\right)\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification7.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.7635926318262873 \cdot 10^{+131}:\\ \;\;\;\;e^{\log 1.0 \cdot a + \left(\log z - t\right) \cdot y} \cdot x\\ \mathbf{elif}\;y \le -3.3939892220623794 \cdot 10^{+73}:\\ \;\;\;\;x \cdot e^{-\left(1.0 \cdot \left(a \cdot z\right) + \left(a \cdot b + 0.5 \cdot \left(a \cdot \left(z \cdot z\right)\right)\right)\right)}\\ \mathbf{elif}\;y \le -8.816833859851974 \cdot 10^{-47}:\\ \;\;\;\;e^{\log 1.0 \cdot a + \left(\log z - t\right) \cdot y} \cdot x\\ \mathbf{elif}\;y \le 9.077191310208735 \cdot 10^{-77}:\\ \;\;\;\;x \cdot e^{-\left(1.0 \cdot \left(a \cdot z\right) + \left(a \cdot b + 0.5 \cdot \left(a \cdot \left(z \cdot z\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\log 1.0 \cdot a + \left(\log z - t\right) \cdot y} \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, B"
  (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))