Average Error: 1.9 → 0.6
Time: 37.4s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.87699396231292923 \cdot 10^{-107}:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(-1\right)}}{e^{b - \mathsf{fma}\left(\log z, y, \log a \cdot t\right)}}}{y}\\ \mathbf{elif}\;x \le 5.6413362742687982 \cdot 10^{-108}:\\ \;\;\;\;\frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\frac{{a}^{\left(-1\right)}}{e^{b - \mathsf{fma}\left(\log z, y, \log a \cdot t\right)}}}{\sqrt[3]{y}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{\sqrt{e^{b - \mathsf{fma}\left(\log z, y, \log a \cdot t\right)}}} \cdot \frac{{a}^{\left(-1\right)}}{\sqrt{e^{b - \mathsf{fma}\left(\log z, y, \log a \cdot t\right)}}}\right) \cdot \frac{1}{y}\\ \end{array}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3.87699396231292923 \cdot 10^{-107}:\\
\;\;\;\;\frac{x \cdot \frac{{a}^{\left(-1\right)}}{e^{b - \mathsf{fma}\left(\log z, y, \log a \cdot t\right)}}}{y}\\

\mathbf{elif}\;x \le 5.6413362742687982 \cdot 10^{-108}:\\
\;\;\;\;\frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\frac{{a}^{\left(-1\right)}}{e^{b - \mathsf{fma}\left(\log z, y, \log a \cdot t\right)}}}{\sqrt[3]{y}}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{x}{\sqrt{e^{b - \mathsf{fma}\left(\log z, y, \log a \cdot t\right)}}} \cdot \frac{{a}^{\left(-1\right)}}{\sqrt{e^{b - \mathsf{fma}\left(\log z, y, \log a \cdot t\right)}}}\right) \cdot \frac{1}{y}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r65317 = x;
        double r65318 = y;
        double r65319 = z;
        double r65320 = log(r65319);
        double r65321 = r65318 * r65320;
        double r65322 = t;
        double r65323 = 1.0;
        double r65324 = r65322 - r65323;
        double r65325 = a;
        double r65326 = log(r65325);
        double r65327 = r65324 * r65326;
        double r65328 = r65321 + r65327;
        double r65329 = b;
        double r65330 = r65328 - r65329;
        double r65331 = exp(r65330);
        double r65332 = r65317 * r65331;
        double r65333 = r65332 / r65318;
        return r65333;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r65334 = x;
        double r65335 = -3.876993962312929e-107;
        bool r65336 = r65334 <= r65335;
        double r65337 = a;
        double r65338 = 1.0;
        double r65339 = -r65338;
        double r65340 = pow(r65337, r65339);
        double r65341 = b;
        double r65342 = z;
        double r65343 = log(r65342);
        double r65344 = y;
        double r65345 = log(r65337);
        double r65346 = t;
        double r65347 = r65345 * r65346;
        double r65348 = fma(r65343, r65344, r65347);
        double r65349 = r65341 - r65348;
        double r65350 = exp(r65349);
        double r65351 = r65340 / r65350;
        double r65352 = r65334 * r65351;
        double r65353 = r65352 / r65344;
        double r65354 = 5.641336274268798e-108;
        bool r65355 = r65334 <= r65354;
        double r65356 = cbrt(r65344);
        double r65357 = r65356 * r65356;
        double r65358 = r65334 / r65357;
        double r65359 = r65351 / r65356;
        double r65360 = r65358 * r65359;
        double r65361 = sqrt(r65350);
        double r65362 = r65334 / r65361;
        double r65363 = r65340 / r65361;
        double r65364 = r65362 * r65363;
        double r65365 = 1.0;
        double r65366 = r65365 / r65344;
        double r65367 = r65364 * r65366;
        double r65368 = r65355 ? r65360 : r65367;
        double r65369 = r65336 ? r65353 : r65368;
        return r65369;
}

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. Split input into 3 regimes

  2. if x < -3.876993962312929e-107

    1. Initial program 1.0

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

    2. Taylor expanded around inf 1.0

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

    3. Simplified0.3

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

    4. Taylor expanded around 0 0.3

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

    5. Simplified0.3

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

    if -3.876993962312929e-107 < x < 5.641336274268798e-108

    1. Initial program 3.3

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

    2. Taylor expanded around inf 3.3

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

    3. Simplified2.6

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

    4. Taylor expanded around 0 2.6

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

    5. Simplified2.6

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

    7. Applied add-cube-cbrt2.8

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

    8. Applied times-frac1.1

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

    if 5.641336274268798e-108 < x

    1. Initial program 1.1

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

    2. Taylor expanded around inf 1.2

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

    3. Simplified0.3

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

    4. Taylor expanded around 0 0.3

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

    5. Simplified0.3

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

    7. Applied add-sqr-sqrt0.3

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

    8. Applied *-un-lft-identity0.3

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

    9. Applied unpow-prod-down0.3

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

    10. Applied times-frac0.3

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

    11. Applied associate-*r*0.3

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

    12. Simplified0.3

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

    14. Applied div-inv0.3

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.87699396231292923 \cdot 10^{-107}:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(-1\right)}}{e^{b - \mathsf{fma}\left(\log z, y, \log a \cdot t\right)}}}{y}\\ \mathbf{elif}\;x \le 5.6413362742687982 \cdot 10^{-108}:\\ \;\;\;\;\frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\frac{{a}^{\left(-1\right)}}{e^{b - \mathsf{fma}\left(\log z, y, \log a \cdot t\right)}}}{\sqrt[3]{y}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{\sqrt{e^{b - \mathsf{fma}\left(\log z, y, \log a \cdot t\right)}}} \cdot \frac{{a}^{\left(-1\right)}}{\sqrt{e^{b - \mathsf{fma}\left(\log z, y, \log a \cdot t\right)}}}\right) \cdot \frac{1}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 +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))