Average Error: 1.9 → 0.2
Time: 14.0s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[\left(x \cdot {\left(e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)\right)}\right)}^{\frac{2}{3}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)\right)}}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
\left(x \cdot {\left(e^{\mathsf{fma}\left(y, \log z - t, 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)}^{\frac{2}{3}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)\right)}}
double f(double x, double y, double z, double t, double a, double b) {
        double r129208 = x;
        double r129209 = y;
        double r129210 = z;
        double r129211 = log(r129210);
        double r129212 = t;
        double r129213 = r129211 - r129212;
        double r129214 = r129209 * r129213;
        double r129215 = a;
        double r129216 = 1.0;
        double r129217 = r129216 - r129210;
        double r129218 = log(r129217);
        double r129219 = b;
        double r129220 = r129218 - r129219;
        double r129221 = r129215 * r129220;
        double r129222 = r129214 + r129221;
        double r129223 = exp(r129222);
        double r129224 = r129208 * r129223;
        return r129224;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r129225 = x;
        double r129226 = y;
        double r129227 = z;
        double r129228 = log(r129227);
        double r129229 = t;
        double r129230 = r129228 - r129229;
        double r129231 = a;
        double r129232 = 1.0;
        double r129233 = log(r129232);
        double r129234 = 0.5;
        double r129235 = 2.0;
        double r129236 = pow(r129227, r129235);
        double r129237 = pow(r129232, r129235);
        double r129238 = r129236 / r129237;
        double r129239 = r129234 * r129238;
        double r129240 = r129232 * r129227;
        double r129241 = r129239 + r129240;
        double r129242 = r129233 - r129241;
        double r129243 = b;
        double r129244 = r129242 - r129243;
        double r129245 = r129231 * r129244;
        double r129246 = fma(r129226, r129230, r129245);
        double r129247 = exp(r129246);
        double r129248 = 0.6666666666666666;
        double r129249 = pow(r129247, r129248);
        double r129250 = r129225 * r129249;
        double r129251 = cbrt(r129247);
        double r129252 = r129250 * r129251;
        return r129252;
}

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 1.9

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

  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 fma-def0.2

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

  6. Applied add-cube-cbrt0.3

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

  7. Applied associate-*r*0.3

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

  9. Applied pow1/30.3

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

  10. Applied pow1/30.3

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

  11. Applied pow-prod-up0.2

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

  12. Simplified0.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2020025 +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))))))