Average Error: 2.0 → 1.0
Time: 25.7s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{\frac{\sqrt{{\left(\frac{1}{a}\right)}^{1}}}{\sqrt{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}{y} \cdot \left(\frac{\sqrt{{\left(\frac{1}{a}\right)}^{1}}}{\sqrt{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}} \cdot x\right)\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\frac{\frac{\sqrt{{\left(\frac{1}{a}\right)}^{1}}}{\sqrt{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} \cdot \left(\frac{\sqrt{{\left(\frac{1}{a}\right)}^{1}}}{\sqrt{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}} \cdot x\right)
double f(double x, double y, double z, double t, double a, double b) {
        double r139968 = x;
        double r139969 = y;
        double r139970 = z;
        double r139971 = log(r139970);
        double r139972 = r139969 * r139971;
        double r139973 = t;
        double r139974 = 1.0;
        double r139975 = r139973 - r139974;
        double r139976 = a;
        double r139977 = log(r139976);
        double r139978 = r139975 * r139977;
        double r139979 = r139972 + r139978;
        double r139980 = b;
        double r139981 = r139979 - r139980;
        double r139982 = exp(r139981);
        double r139983 = r139968 * r139982;
        double r139984 = r139983 / r139969;
        return r139984;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r139985 = 1.0;
        double r139986 = a;
        double r139987 = r139985 / r139986;
        double r139988 = 1.0;
        double r139989 = pow(r139987, r139988);
        double r139990 = sqrt(r139989);
        double r139991 = y;
        double r139992 = z;
        double r139993 = r139985 / r139992;
        double r139994 = log(r139993);
        double r139995 = log(r139987);
        double r139996 = t;
        double r139997 = b;
        double r139998 = fma(r139995, r139996, r139997);
        double r139999 = fma(r139991, r139994, r139998);
        double r140000 = exp(r139999);
        double r140001 = sqrt(r140000);
        double r140002 = r139990 / r140001;
        double r140003 = r140002 / r139991;
        double r140004 = x;
        double r140005 = r140002 * r140004;
        double r140006 = r140003 * r140005;
        return r140006;
}

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

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

  2. Taylor expanded around inf 2.0

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

  3. Simplified5.8

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

  5. Applied div-inv5.8

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

  6. Applied add-sqr-sqrt5.8

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

  7. Applied add-sqr-sqrt5.9

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

  8. Applied times-frac5.9

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

  9. Applied times-frac1.1

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

  10. Simplified1.0

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

  11. Final simplification1.0

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

Reproduce

herbie shell --seed 2020035 +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))