Average Error: 1.9 → 1.2
Time: 13.8s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{\left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}\right)}^{1} \cdot x\right) \cdot \frac{{\left(\frac{\sqrt[3]{1}}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}{y}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\frac{\left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}\right)}^{1} \cdot x\right) \cdot \frac{{\left(\frac{\sqrt[3]{1}}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}{y}
double f(double x, double y, double z, double t, double a, double b) {
        double r107995 = x;
        double r107996 = y;
        double r107997 = z;
        double r107998 = log(r107997);
        double r107999 = r107996 * r107998;
        double r108000 = t;
        double r108001 = 1.0;
        double r108002 = r108000 - r108001;
        double r108003 = a;
        double r108004 = log(r108003);
        double r108005 = r108002 * r108004;
        double r108006 = r107999 + r108005;
        double r108007 = b;
        double r108008 = r108006 - r108007;
        double r108009 = exp(r108008);
        double r108010 = r107995 * r108009;
        double r108011 = r108010 / r107996;
        return r108011;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r108012 = 1.0;
        double r108013 = cbrt(r108012);
        double r108014 = r108013 * r108013;
        double r108015 = r108014 / r108012;
        double r108016 = 1.0;
        double r108017 = pow(r108015, r108016);
        double r108018 = x;
        double r108019 = r108017 * r108018;
        double r108020 = a;
        double r108021 = r108013 / r108020;
        double r108022 = pow(r108021, r108016);
        double r108023 = y;
        double r108024 = z;
        double r108025 = r108012 / r108024;
        double r108026 = log(r108025);
        double r108027 = r108012 / r108020;
        double r108028 = log(r108027);
        double r108029 = t;
        double r108030 = b;
        double r108031 = fma(r108028, r108029, r108030);
        double r108032 = fma(r108023, r108026, r108031);
        double r108033 = exp(r108032);
        double r108034 = r108022 / r108033;
        double r108035 = r108019 * r108034;
        double r108036 = r108035 / r108023;
        return r108036;
}

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

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

  2. Taylor expanded around inf 1.9

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

  3. Simplified1.2

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

  5. Applied *-un-lft-identity1.2

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

  6. Applied *-un-lft-identity1.2

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

  7. Applied add-cube-cbrt1.2

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

  8. Applied times-frac1.2

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

  9. Applied unpow-prod-down1.2

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

  10. Applied times-frac1.2

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

  11. Applied associate-*r*1.2

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

  12. Simplified1.2

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

  13. Final simplification1.2

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

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2"
  :precision binary64
  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1) (log a))) b))) y))