Average Error: 1.9 → 1.0
Time: 43.4s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{\frac{x}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt{e^{\mathsf{fma}\left(t - 1, \log a, \log z \cdot y - b\right)}}}}}{\frac{\sqrt[3]{\sqrt[3]{y}} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)}{\sqrt{{e}^{\left(\mathsf{fma}\left(t - 1, \log a, \log z \cdot y - b\right)\right)}}}}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\frac{\frac{x}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt{e^{\mathsf{fma}\left(t - 1, \log a, \log z \cdot y - b\right)}}}}}{\frac{\sqrt[3]{\sqrt[3]{y}} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)}{\sqrt{{e}^{\left(\mathsf{fma}\left(t - 1, \log a, \log z \cdot y - b\right)\right)}}}}
double f(double x, double y, double z, double t, double a, double b) {
        double r3302021 = x;
        double r3302022 = y;
        double r3302023 = z;
        double r3302024 = log(r3302023);
        double r3302025 = r3302022 * r3302024;
        double r3302026 = t;
        double r3302027 = 1.0;
        double r3302028 = r3302026 - r3302027;
        double r3302029 = a;
        double r3302030 = log(r3302029);
        double r3302031 = r3302028 * r3302030;
        double r3302032 = r3302025 + r3302031;
        double r3302033 = b;
        double r3302034 = r3302032 - r3302033;
        double r3302035 = exp(r3302034);
        double r3302036 = r3302021 * r3302035;
        double r3302037 = r3302036 / r3302022;
        return r3302037;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r3302038 = x;
        double r3302039 = y;
        double r3302040 = cbrt(r3302039);
        double r3302041 = r3302040 * r3302040;
        double r3302042 = t;
        double r3302043 = 1.0;
        double r3302044 = r3302042 - r3302043;
        double r3302045 = a;
        double r3302046 = log(r3302045);
        double r3302047 = z;
        double r3302048 = log(r3302047);
        double r3302049 = r3302048 * r3302039;
        double r3302050 = b;
        double r3302051 = r3302049 - r3302050;
        double r3302052 = fma(r3302044, r3302046, r3302051);
        double r3302053 = exp(r3302052);
        double r3302054 = sqrt(r3302053);
        double r3302055 = r3302041 / r3302054;
        double r3302056 = r3302038 / r3302055;
        double r3302057 = cbrt(r3302040);
        double r3302058 = r3302057 * r3302057;
        double r3302059 = r3302057 * r3302058;
        double r3302060 = exp(1.0);
        double r3302061 = pow(r3302060, r3302052);
        double r3302062 = sqrt(r3302061);
        double r3302063 = r3302059 / r3302062;
        double r3302064 = r3302056 / r3302063;
        return r3302064;
}

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. Using strategy rm

  3. Applied associate-/l*2.0

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

  4. Simplified2.0

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

  6. Applied add-sqr-sqrt2.0

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

  7. Applied add-cube-cbrt2.0

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

  8. Applied times-frac2.0

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

  9. Applied associate-/r*1.0

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

  11. Applied *-un-lft-identity1.0

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

  12. Applied exp-prod1.0

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

  13. Simplified1.0

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

  15. Applied add-cube-cbrt1.0

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

  16. Final simplification1.0

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

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2"
  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))