Average Error: 2.1 → 1.9
Time: 35.6s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(t - 1\right) \cdot \log a \le -702.6583505807568599266232922673225402832 \lor \neg \left(\left(t - 1\right) \cdot \log a \le -106.4073004455099749065993819385766983032\right):\\ \;\;\;\;\frac{x \cdot e^{\left(\left(t - 1\right) \cdot \log a + \log z \cdot y\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left({z}^{y}\right) + t \cdot \log a}}{y \cdot \left(a \cdot e^{b}\right)} \cdot x\\ \end{array}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\begin{array}{l}
\mathbf{if}\;\left(t - 1\right) \cdot \log a \le -702.6583505807568599266232922673225402832 \lor \neg \left(\left(t - 1\right) \cdot \log a \le -106.4073004455099749065993819385766983032\right):\\
\;\;\;\;\frac{x \cdot e^{\left(\left(t - 1\right) \cdot \log a + \log z \cdot y\right) - b}}{y}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r375169 = x;
        double r375170 = y;
        double r375171 = z;
        double r375172 = log(r375171);
        double r375173 = r375170 * r375172;
        double r375174 = t;
        double r375175 = 1.0;
        double r375176 = r375174 - r375175;
        double r375177 = a;
        double r375178 = log(r375177);
        double r375179 = r375176 * r375178;
        double r375180 = r375173 + r375179;
        double r375181 = b;
        double r375182 = r375180 - r375181;
        double r375183 = exp(r375182);
        double r375184 = r375169 * r375183;
        double r375185 = r375184 / r375170;
        return r375185;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r375186 = t;
        double r375187 = 1.0;
        double r375188 = r375186 - r375187;
        double r375189 = a;
        double r375190 = log(r375189);
        double r375191 = r375188 * r375190;
        double r375192 = -702.6583505807569;
        bool r375193 = r375191 <= r375192;
        double r375194 = -106.40730044550997;
        bool r375195 = r375191 <= r375194;
        double r375196 = !r375195;
        bool r375197 = r375193 || r375196;
        double r375198 = x;
        double r375199 = z;
        double r375200 = log(r375199);
        double r375201 = y;
        double r375202 = r375200 * r375201;
        double r375203 = r375191 + r375202;
        double r375204 = b;
        double r375205 = r375203 - r375204;
        double r375206 = exp(r375205);
        double r375207 = r375198 * r375206;
        double r375208 = r375207 / r375201;
        double r375209 = pow(r375199, r375201);
        double r375210 = log(r375209);
        double r375211 = r375186 * r375190;
        double r375212 = r375210 + r375211;
        double r375213 = exp(r375212);
        double r375214 = exp(r375204);
        double r375215 = r375189 * r375214;
        double r375216 = r375201 * r375215;
        double r375217 = r375213 / r375216;
        double r375218 = r375217 * r375198;
        double r375219 = r375197 ? r375208 : r375218;
        return r375219;
}

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

Target

Original2.1
Target11.6
Herbie1.9
\[\begin{array}{l} \mathbf{if}\;t \lt -0.8845848504127471478852839936735108494759:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \mathbf{elif}\;t \lt 852031.228837407310493290424346923828125:\\ \;\;\;\;\frac{\frac{x}{y} \cdot {a}^{\left(t - 1\right)}}{e^{b - \log z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* (- t 1.0) (log a)) < -702.6583505807569 or -106.40730044550997 < (* (- t 1.0) (log a))

    1. Initial program 0.6

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

    if -702.6583505807569 < (* (- t 1.0) (log a)) < -106.40730044550997

    1. Initial program 6.9

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

    3. Applied associate-/l*2.3

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

    4. Simplified6.1

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

    6. Applied *-un-lft-identity6.1

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

    7. Applied sub-neg6.1

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

    8. Applied unpow-prod-up6.0

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

    9. Applied times-frac6.0

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

    10. Applied associate-*r*6.0

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

    11. Simplified6.0

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

    13. Applied div-inv6.0

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

    14. Simplified6.0

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

    15. Taylor expanded around inf 6.0

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

    16. Simplified6.0

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

  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t - 1\right) \cdot \log a \le -702.6583505807568599266232922673225402832 \lor \neg \left(\left(t - 1\right) \cdot \log a \le -106.4073004455099749065993819385766983032\right):\\ \;\;\;\;\frac{x \cdot e^{\left(\left(t - 1\right) \cdot \log a + \log z \cdot y\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left({z}^{y}\right) + t \cdot \log a}}{y \cdot \left(a \cdot e^{b}\right)} \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A"

  :herbie-target
  (if (< t -0.8845848504127471) (/ (* x (/ (pow a (- t 1.0)) y)) (- (+ b 1.0) (* y (log z)))) (if (< t 852031.2288374073) (/ (* (/ x y) (pow a (- t 1.0))) (exp (- b (* (log z) y)))) (/ (* x (/ (pow a (- t 1.0)) y)) (- (+ b 1.0) (* y (log z))))))

  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))