Average Error: 2.1 → 0.7
Time: 9.5s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[\left(x \cdot \left(\sqrt[3]{e^{y \cdot \left(\log z - t\right) + \left(a \cdot \log 1 - \left(a \cdot b + 1 \cdot \left(a \cdot z\right)\right)\right)}} \cdot \sqrt[3]{e^{y \cdot \left(\log z - t\right) + \left(a \cdot \log 1 - \left(a \cdot b + 1 \cdot \left(a \cdot z\right)\right)\right)}}\right)\right) \cdot \sqrt[3]{e^{y \cdot \left(\log z - t\right) + \left(a \cdot \log 1 - \left(a \cdot b + 1 \cdot \left(a \cdot z\right)\right)\right)}}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
\left(x \cdot \left(\sqrt[3]{e^{y \cdot \left(\log z - t\right) + \left(a \cdot \log 1 - \left(a \cdot b + 1 \cdot \left(a \cdot z\right)\right)\right)}} \cdot \sqrt[3]{e^{y \cdot \left(\log z - t\right) + \left(a \cdot \log 1 - \left(a \cdot b + 1 \cdot \left(a \cdot z\right)\right)\right)}}\right)\right) \cdot \sqrt[3]{e^{y \cdot \left(\log z - t\right) + \left(a \cdot \log 1 - \left(a \cdot b + 1 \cdot \left(a \cdot z\right)\right)\right)}}
double f(double x, double y, double z, double t, double a, double b) {
        double r112021 = x;
        double r112022 = y;
        double r112023 = z;
        double r112024 = log(r112023);
        double r112025 = t;
        double r112026 = r112024 - r112025;
        double r112027 = r112022 * r112026;
        double r112028 = a;
        double r112029 = 1.0;
        double r112030 = r112029 - r112023;
        double r112031 = log(r112030);
        double r112032 = b;
        double r112033 = r112031 - r112032;
        double r112034 = r112028 * r112033;
        double r112035 = r112027 + r112034;
        double r112036 = exp(r112035);
        double r112037 = r112021 * r112036;
        return r112037;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r112038 = x;
        double r112039 = y;
        double r112040 = z;
        double r112041 = log(r112040);
        double r112042 = t;
        double r112043 = r112041 - r112042;
        double r112044 = r112039 * r112043;
        double r112045 = a;
        double r112046 = 1.0;
        double r112047 = log(r112046);
        double r112048 = r112045 * r112047;
        double r112049 = b;
        double r112050 = r112045 * r112049;
        double r112051 = r112045 * r112040;
        double r112052 = r112046 * r112051;
        double r112053 = r112050 + r112052;
        double r112054 = r112048 - r112053;
        double r112055 = r112044 + r112054;
        double r112056 = exp(r112055);
        double r112057 = cbrt(r112056);
        double r112058 = r112057 * r112057;
        double r112059 = r112038 * r112058;
        double r112060 = r112059 * r112057;
        return r112060;
}

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 2.1

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

  2. Taylor expanded around 0 0.6

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

  4. Applied add-cube-cbrt0.7

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

  5. Applied associate-*r*0.7

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

  6. Final simplification0.7

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

Reproduce

herbie shell --seed 2020056 
(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))))))