Average Error: 1.9 → 1.4
Time: 44.6s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{1}{\sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}} \cdot \frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}\right)}{\frac{y}{\sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}
↓
\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{1}{\sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}} \cdot \frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}\right)}{\frac{y}{\sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}}
double f(double x, double y, double z, double t, double a, double b) {
        double r3190127 = x;
        double r3190128 = y;
        double r3190129 = z;
        double r3190130 = log(r3190129);
        double r3190131 = r3190128 * r3190130;
        double r3190132 = t;
        double r3190133 = 1.0;
        double r3190134 = r3190132 - r3190133;
        double r3190135 = a;
        double r3190136 = log(r3190135);
        double r3190137 = r3190134 * r3190136;
        double r3190138 = r3190131 + r3190137;
        double r3190139 = b;
        double r3190140 = r3190138 - r3190139;
        double r3190141 = exp(r3190140);
        double r3190142 = r3190127 * r3190141;
        double r3190143 = r3190142 / r3190128;
        return r3190143;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r3190144 = x;
        double r3190145 = cbrt(r3190144);
        double r3190146 = r3190145 * r3190145;
        double r3190147 = 1.0;
        double r3190148 = t;
        double r3190149 = 1.0;
        double r3190150 = r3190148 - r3190149;
        double r3190151 = a;
        double r3190152 = log(r3190151);
        double r3190153 = z;
        double r3190154 = log(r3190153);
        double r3190155 = y;
        double r3190156 = r3190154 * r3190155;
        double r3190157 = fma(r3190150, r3190152, r3190156);
        double r3190158 = b;
        double r3190159 = r3190157 - r3190158;
        double r3190160 = exp(r3190159);
        double r3190161 = cbrt(r3190160);
        double r3190162 = r3190161 * r3190161;
        double r3190163 = r3190147 / r3190162;
        double r3190164 = r3190146 / r3190163;
        double r3190165 = cbrt(r3190146);
        double r3190166 = cbrt(r3190145);
        double r3190167 = r3190166 * r3190166;
        double r3190168 = cbrt(r3190167);
        double r3190169 = cbrt(r3190166);
        double r3190170 = r3190168 * r3190169;
        double r3190171 = r3190165 * r3190170;
        double r3190172 = r3190155 / r3190161;
        double r3190173 = r3190171 / r3190172;
        double r3190174 = r3190164 * r3190173;
        return r3190174;
}

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

    \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
  2. Using strategy rm

  3. Applied associate-/l*2.0

    \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}}}\]

  4. Simplified2.0

    \[\leadsto \frac{x}{\color{blue}{\frac{y}{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt2.0

    \[\leadsto \frac{x}{\frac{y}{\color{blue}{\left(\sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}}}\]

  7. Applied *-un-lft-identity2.0

    \[\leadsto \frac{x}{\frac{\color{blue}{1 \cdot y}}{\left(\sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}}\]

  8. Applied times-frac2.0

    \[\leadsto \frac{x}{\color{blue}{\frac{1}{\sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}} \cdot \frac{y}{\sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}}}\]

  9. Applied add-cube-cbrt2.0

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\frac{1}{\sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}} \cdot \frac{y}{\sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}}\]

  10. Applied times-frac1.4

    \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{1}{\sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}} \cdot \frac{\sqrt[3]{x}}{\frac{y}{\sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}}}\]
  11. Using strategy rm

  12. Applied add-cube-cbrt1.4

    \[\leadsto \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{1}{\sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}} \cdot \frac{\sqrt[3]{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}}{\frac{y}{\sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}}\]

  13. Applied cbrt-prod1.4

    \[\leadsto \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{1}{\sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}} \cdot \frac{\color{blue}{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}}}{\frac{y}{\sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}}\]
  14. Using strategy rm

  15. Applied add-cube-cbrt1.4

    \[\leadsto \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{1}{\sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}} \cdot \frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}}}{\frac{y}{\sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}}\]

  16. Applied cbrt-prod1.4

    \[\leadsto \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{1}{\sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}} \cdot \frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \color{blue}{\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}\right)}}{\frac{y}{\sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}}\]

  17. Final simplification1.4

    \[\leadsto \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{1}{\sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}} \cdot \frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}\right)}{\frac{y}{\sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}}\]

Reproduce

herbie shell --seed 2019158 +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))