Average Error: 2.1 → 0.2
Time: 18.3s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot \left(\sqrt[3]{{\left(e^{2}\right)}^{\left(\mathsf{fma}\left(y, \log z - t, \left(\log 1 - \left(\mathsf{fma}\left(\frac{1}{2}, \frac{{z}^{2}}{{1}^{2}}, 1 \cdot z\right) + b\right)\right) \cdot a\right)\right)}} \cdot \sqrt[3]{{e}^{\left(\mathsf{fma}\left(y, \log z - t, \left(\log 1 - \left(\mathsf{fma}\left(\frac{1}{2}, \frac{{z}^{2}}{{1}^{2}}, 1 \cdot z\right) + b\right)\right) \cdot a\right)\right)}}\right)\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \cdot \left(\sqrt[3]{{\left(e^{2}\right)}^{\left(\mathsf{fma}\left(y, \log z - t, \left(\log 1 - \left(\mathsf{fma}\left(\frac{1}{2}, \frac{{z}^{2}}{{1}^{2}}, 1 \cdot z\right) + b\right)\right) \cdot a\right)\right)}} \cdot \sqrt[3]{{e}^{\left(\mathsf{fma}\left(y, \log z - t, \left(\log 1 - \left(\mathsf{fma}\left(\frac{1}{2}, \frac{{z}^{2}}{{1}^{2}}, 1 \cdot z\right) + b\right)\right) \cdot a\right)\right)}}\right)
double f(double x, double y, double z, double t, double a, double b) {
        double r120154 = x;
        double r120155 = y;
        double r120156 = z;
        double r120157 = log(r120156);
        double r120158 = t;
        double r120159 = r120157 - r120158;
        double r120160 = r120155 * r120159;
        double r120161 = a;
        double r120162 = 1.0;
        double r120163 = r120162 - r120156;
        double r120164 = log(r120163);
        double r120165 = b;
        double r120166 = r120164 - r120165;
        double r120167 = r120161 * r120166;
        double r120168 = r120160 + r120167;
        double r120169 = exp(r120168);
        double r120170 = r120154 * r120169;
        return r120170;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r120171 = x;
        double r120172 = 2.0;
        double r120173 = exp(r120172);
        double r120174 = y;
        double r120175 = z;
        double r120176 = log(r120175);
        double r120177 = t;
        double r120178 = r120176 - r120177;
        double r120179 = 1.0;
        double r120180 = log(r120179);
        double r120181 = 0.5;
        double r120182 = pow(r120175, r120172);
        double r120183 = pow(r120179, r120172);
        double r120184 = r120182 / r120183;
        double r120185 = r120179 * r120175;
        double r120186 = fma(r120181, r120184, r120185);
        double r120187 = b;
        double r120188 = r120186 + r120187;
        double r120189 = r120180 - r120188;
        double r120190 = a;
        double r120191 = r120189 * r120190;
        double r120192 = fma(r120174, r120178, r120191);
        double r120193 = pow(r120173, r120192);
        double r120194 = cbrt(r120193);
        double r120195 = exp(1.0);
        double r120196 = pow(r120195, r120192);
        double r120197 = cbrt(r120196);
        double r120198 = r120194 * r120197;
        double r120199 = r120171 * r120198;
        return r120199;
}

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 2.1

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

  2. Taylor expanded around 0 0.4

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

  4. Applied add-cbrt-cube0.4

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

  5. Simplified0.4

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

  7. Applied add-cube-cbrt0.5

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

  8. Applied unpow-prod-down0.5

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

  9. Simplified0.4

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

  10. Simplified0.2

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

  12. Applied cbrt-prod0.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2020035 +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))))))