Average Error: 2.0 → 0.7
Time: 35.6s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le 2.425819830564266198213947300688724832234 \cdot 10^{164}:\\ \;\;\;\;{\left(\frac{1}{{a}^{1}}\right)}^{1} \cdot \frac{x}{e^{\log \left(\frac{1}{z}\right) \cdot y + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}} \cdot \frac{\frac{{\left(\frac{1}{a}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{y}\\ \end{array}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\begin{array}{l}
\mathbf{if}\;a \le 2.425819830564266198213947300688724832234 \cdot 10^{164}:\\
\;\;\;\;{\left(\frac{1}{{a}^{1}}\right)}^{1} \cdot \frac{x}{e^{\log \left(\frac{1}{z}\right) \cdot y + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)} \cdot y}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r74548 = x;
        double r74549 = y;
        double r74550 = z;
        double r74551 = log(r74550);
        double r74552 = r74549 * r74551;
        double r74553 = t;
        double r74554 = 1.0;
        double r74555 = r74553 - r74554;
        double r74556 = a;
        double r74557 = log(r74556);
        double r74558 = r74555 * r74557;
        double r74559 = r74552 + r74558;
        double r74560 = b;
        double r74561 = r74559 - r74560;
        double r74562 = exp(r74561);
        double r74563 = r74548 * r74562;
        double r74564 = r74563 / r74549;
        return r74564;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r74565 = a;
        double r74566 = 2.425819830564266e+164;
        bool r74567 = r74565 <= r74566;
        double r74568 = 1.0;
        double r74569 = 1.0;
        double r74570 = pow(r74565, r74569);
        double r74571 = r74568 / r74570;
        double r74572 = pow(r74571, r74569);
        double r74573 = x;
        double r74574 = z;
        double r74575 = r74568 / r74574;
        double r74576 = log(r74575);
        double r74577 = y;
        double r74578 = r74576 * r74577;
        double r74579 = r74568 / r74565;
        double r74580 = log(r74579);
        double r74581 = t;
        double r74582 = r74580 * r74581;
        double r74583 = b;
        double r74584 = r74582 + r74583;
        double r74585 = r74578 + r74584;
        double r74586 = exp(r74585);
        double r74587 = r74586 * r74577;
        double r74588 = r74573 / r74587;
        double r74589 = r74572 * r74588;
        double r74590 = r74577 * r74576;
        double r74591 = r74590 + r74584;
        double r74592 = exp(r74591);
        double r74593 = sqrt(r74592);
        double r74594 = r74573 / r74593;
        double r74595 = pow(r74579, r74569);
        double r74596 = r74595 / r74593;
        double r74597 = r74596 / r74577;
        double r74598 = r74594 * r74597;
        double r74599 = r74567 ? r74589 : r74598;
        return r74599;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if a < 2.425819830564266e+164

    1. Initial program 1.2

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

    2. Taylor expanded around inf 1.2

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

    3. Simplified9.3

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

    5. Applied *-un-lft-identity9.3

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

    6. Applied add-sqr-sqrt9.3

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

    7. Applied times-frac0.5

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

    8. Applied times-frac2.0

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

    9. Simplified2.0

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

    10. Taylor expanded around inf 0.9

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

    if 2.425819830564266e+164 < a

    1. Initial program 4.3

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

    2. Taylor expanded around inf 4.2

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

    3. Simplified3.3

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

    5. Applied *-un-lft-identity3.3

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

    6. Applied add-sqr-sqrt3.3

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

    7. Applied times-frac3.3

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

    8. Applied times-frac0.1

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

    9. Simplified0.1

      \[\leadsto \color{blue}{\frac{x}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}} \cdot \frac{\frac{{\left(\frac{1}{a}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{y}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le 2.425819830564266198213947300688724832234 \cdot 10^{164}:\\ \;\;\;\;{\left(\frac{1}{{a}^{1}}\right)}^{1} \cdot \frac{x}{e^{\log \left(\frac{1}{z}\right) \cdot y + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}} \cdot \frac{\frac{{\left(\frac{1}{a}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019291 
(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))