Average Error: 2.0 → 1.1
Time: 44.4s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{x}{\frac{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}}{\sqrt[3]{{e}^{\left(\mathsf{fma}\left(t - 1, \log a, \log z \cdot y - b\right)\right)}}}} \cdot \frac{1}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{e^{\mathsf{fma}\left(t - 1, \log a, \log z \cdot y - b\right)}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(t - 1, \log a, \log z \cdot y - b\right)}}}}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\frac{x}{\frac{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}}{\sqrt[3]{{e}^{\left(\mathsf{fma}\left(t - 1, \log a, \log z \cdot y - b\right)\right)}}}} \cdot \frac{1}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{e^{\mathsf{fma}\left(t - 1, \log a, \log z \cdot y - b\right)}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(t - 1, \log a, \log z \cdot y - b\right)}}}}
double f(double x, double y, double z, double t, double a, double b) {
        double r3025195 = x;
        double r3025196 = y;
        double r3025197 = z;
        double r3025198 = log(r3025197);
        double r3025199 = r3025196 * r3025198;
        double r3025200 = t;
        double r3025201 = 1.0;
        double r3025202 = r3025200 - r3025201;
        double r3025203 = a;
        double r3025204 = log(r3025203);
        double r3025205 = r3025202 * r3025204;
        double r3025206 = r3025199 + r3025205;
        double r3025207 = b;
        double r3025208 = r3025206 - r3025207;
        double r3025209 = exp(r3025208);
        double r3025210 = r3025195 * r3025209;
        double r3025211 = r3025210 / r3025196;
        return r3025211;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r3025212 = x;
        double r3025213 = y;
        double r3025214 = cbrt(r3025213);
        double r3025215 = r3025214 * r3025214;
        double r3025216 = cbrt(r3025215);
        double r3025217 = cbrt(r3025214);
        double r3025218 = r3025216 * r3025217;
        double r3025219 = exp(1.0);
        double r3025220 = t;
        double r3025221 = 1.0;
        double r3025222 = r3025220 - r3025221;
        double r3025223 = a;
        double r3025224 = log(r3025223);
        double r3025225 = z;
        double r3025226 = log(r3025225);
        double r3025227 = r3025226 * r3025213;
        double r3025228 = b;
        double r3025229 = r3025227 - r3025228;
        double r3025230 = fma(r3025222, r3025224, r3025229);
        double r3025231 = pow(r3025219, r3025230);
        double r3025232 = cbrt(r3025231);
        double r3025233 = r3025218 / r3025232;
        double r3025234 = r3025212 / r3025233;
        double r3025235 = 1.0;
        double r3025236 = exp(r3025230);
        double r3025237 = cbrt(r3025236);
        double r3025238 = r3025237 * r3025237;
        double r3025239 = r3025215 / r3025238;
        double r3025240 = r3025235 / r3025239;
        double r3025241 = r3025234 * r3025240;
        return r3025241;
}

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. Using strategy rm

  3. Applied associate-/l*1.9

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

  4. Simplified1.9

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

  6. Applied add-cube-cbrt1.9

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

  7. Applied add-cube-cbrt1.9

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

  8. Applied times-frac1.9

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

  9. Applied *-un-lft-identity1.9

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

  10. Applied times-frac1.1

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

  12. Applied *-un-lft-identity1.1

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

  13. Applied exp-prod1.1

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

  14. Simplified1.1

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

  16. Applied add-cube-cbrt1.1

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

  17. Applied cbrt-prod1.1

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

  18. Final simplification1.1

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

Reproduce

herbie shell --seed 2019171 +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))