Average Error: 2.0 → 0.5
Time: 27.3s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[\left(x \cdot \sqrt{e^{y \cdot \left(\log z - t\right) + \left(a \cdot \left(\log 1 - b\right) - 1 \cdot \left(a \cdot z\right)\right)}}\right) \cdot \sqrt{e^{y \cdot \left(\log z - t\right) + \left(a \cdot \left(\log 1 - b\right) - 1 \cdot \left(a \cdot z\right)\right)}}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
\left(x \cdot \sqrt{e^{y \cdot \left(\log z - t\right) + \left(a \cdot \left(\log 1 - b\right) - 1 \cdot \left(a \cdot z\right)\right)}}\right) \cdot \sqrt{e^{y \cdot \left(\log z - t\right) + \left(a \cdot \left(\log 1 - b\right) - 1 \cdot \left(a \cdot z\right)\right)}}
double f(double x, double y, double z, double t, double a, double b) {
        double r70067 = x;
        double r70068 = y;
        double r70069 = z;
        double r70070 = log(r70069);
        double r70071 = t;
        double r70072 = r70070 - r70071;
        double r70073 = r70068 * r70072;
        double r70074 = a;
        double r70075 = 1.0;
        double r70076 = r70075 - r70069;
        double r70077 = log(r70076);
        double r70078 = b;
        double r70079 = r70077 - r70078;
        double r70080 = r70074 * r70079;
        double r70081 = r70073 + r70080;
        double r70082 = exp(r70081);
        double r70083 = r70067 * r70082;
        return r70083;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r70084 = x;
        double r70085 = y;
        double r70086 = z;
        double r70087 = log(r70086);
        double r70088 = t;
        double r70089 = r70087 - r70088;
        double r70090 = r70085 * r70089;
        double r70091 = a;
        double r70092 = 1.0;
        double r70093 = log(r70092);
        double r70094 = b;
        double r70095 = r70093 - r70094;
        double r70096 = r70091 * r70095;
        double r70097 = r70091 * r70086;
        double r70098 = r70092 * r70097;
        double r70099 = r70096 - r70098;
        double r70100 = r70090 + r70099;
        double r70101 = exp(r70100);
        double r70102 = sqrt(r70101);
        double r70103 = r70084 * r70102;
        double r70104 = r70103 * r70102;
        return r70104;
}

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 0.5

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

  3. Simplified0.5

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

  5. Applied add-sqr-sqrt0.5

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

  6. Applied associate-*r*0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2019209 
(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))))))