Average Error: 2.0 → 0.2
Time: 17.6s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot e^{\sqrt[3]{{\left(\mathsf{fma}\left(\log 1 - \left(\mathsf{fma}\left(\frac{1}{2}, \frac{{z}^{2}}{{1}^{2}}, 1 \cdot z\right) + b\right), a, y \cdot \left(\log z - t\right)\right)\right)}^{3}}}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \cdot e^{\sqrt[3]{{\left(\mathsf{fma}\left(\log 1 - \left(\mathsf{fma}\left(\frac{1}{2}, \frac{{z}^{2}}{{1}^{2}}, 1 \cdot z\right) + b\right), a, y \cdot \left(\log z - t\right)\right)\right)}^{3}}}
double f(double x, double y, double z, double t, double a, double b) {
        double r141044 = x;
        double r141045 = y;
        double r141046 = z;
        double r141047 = log(r141046);
        double r141048 = t;
        double r141049 = r141047 - r141048;
        double r141050 = r141045 * r141049;
        double r141051 = a;
        double r141052 = 1.0;
        double r141053 = r141052 - r141046;
        double r141054 = log(r141053);
        double r141055 = b;
        double r141056 = r141054 - r141055;
        double r141057 = r141051 * r141056;
        double r141058 = r141050 + r141057;
        double r141059 = exp(r141058);
        double r141060 = r141044 * r141059;
        return r141060;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r141061 = x;
        double r141062 = 1.0;
        double r141063 = log(r141062);
        double r141064 = 0.5;
        double r141065 = z;
        double r141066 = 2.0;
        double r141067 = pow(r141065, r141066);
        double r141068 = pow(r141062, r141066);
        double r141069 = r141067 / r141068;
        double r141070 = r141062 * r141065;
        double r141071 = fma(r141064, r141069, r141070);
        double r141072 = b;
        double r141073 = r141071 + r141072;
        double r141074 = r141063 - r141073;
        double r141075 = a;
        double r141076 = y;
        double r141077 = log(r141065);
        double r141078 = t;
        double r141079 = r141077 - r141078;
        double r141080 = r141076 * r141079;
        double r141081 = fma(r141074, r141075, r141080);
        double r141082 = 3.0;
        double r141083 = pow(r141081, r141082);
        double r141084 = cbrt(r141083);
        double r141085 = exp(r141084);
        double r141086 = r141061 * r141085;
        return r141086;
}

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

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

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

  4. Applied add-cbrt-cube0.5

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(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))))))