Average Error: 6.1 → 4.3
Time: 51.9s
Precision: 64
\[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 2.858940889989538 \cdot 10^{+33}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} + \left(\left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right) + \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right)\right) - x\right) + 0.91893853320467\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \left(\frac{z \cdot z}{x} \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)\\ \end{array}\]
\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le 2.858940889989538 \cdot 10^{+33}:\\
\;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} + \left(\left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right) + \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right)\right) - x\right) + 0.91893853320467\right)\\

\mathbf{else}:\\
\;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \left(\frac{z \cdot z}{x} \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)\\

\end{array}
double f(double x, double y, double z) {
        double r23146294 = x;
        double r23146295 = 0.5;
        double r23146296 = r23146294 - r23146295;
        double r23146297 = log(r23146294);
        double r23146298 = r23146296 * r23146297;
        double r23146299 = r23146298 - r23146294;
        double r23146300 = 0.91893853320467;
        double r23146301 = r23146299 + r23146300;
        double r23146302 = y;
        double r23146303 = 0.0007936500793651;
        double r23146304 = r23146302 + r23146303;
        double r23146305 = z;
        double r23146306 = r23146304 * r23146305;
        double r23146307 = 0.0027777777777778;
        double r23146308 = r23146306 - r23146307;
        double r23146309 = r23146308 * r23146305;
        double r23146310 = 0.083333333333333;
        double r23146311 = r23146309 + r23146310;
        double r23146312 = r23146311 / r23146294;
        double r23146313 = r23146301 + r23146312;
        return r23146313;
}
↓
double f(double x, double y, double z) {
        double r23146314 = x;
        double r23146315 = 2.858940889989538e+33;
        bool r23146316 = r23146314 <= r23146315;
        double r23146317 = 0.083333333333333;
        double r23146318 = z;
        double r23146319 = y;
        double r23146320 = 0.0007936500793651;
        double r23146321 = r23146319 + r23146320;
        double r23146322 = r23146318 * r23146321;
        double r23146323 = 0.0027777777777778;
        double r23146324 = r23146322 - r23146323;
        double r23146325 = r23146318 * r23146324;
        double r23146326 = r23146317 + r23146325;
        double r23146327 = r23146326 / r23146314;
        double r23146328 = 0.5;
        double r23146329 = r23146314 - r23146328;
        double r23146330 = cbrt(r23146314);
        double r23146331 = log(r23146330);
        double r23146332 = r23146329 * r23146331;
        double r23146333 = r23146330 * r23146330;
        double r23146334 = log(r23146333);
        double r23146335 = r23146334 * r23146329;
        double r23146336 = r23146332 + r23146335;
        double r23146337 = r23146336 - r23146314;
        double r23146338 = 0.91893853320467;
        double r23146339 = r23146337 + r23146338;
        double r23146340 = r23146327 + r23146339;
        double r23146341 = log(r23146314);
        double r23146342 = r23146341 * r23146329;
        double r23146343 = r23146342 - r23146314;
        double r23146344 = r23146338 + r23146343;
        double r23146345 = r23146318 * r23146318;
        double r23146346 = r23146345 / r23146314;
        double r23146347 = r23146346 * r23146321;
        double r23146348 = r23146318 / r23146314;
        double r23146349 = r23146323 * r23146348;
        double r23146350 = r23146347 - r23146349;
        double r23146351 = r23146344 + r23146350;
        double r23146352 = r23146316 ? r23146340 : r23146351;
        return r23146352;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.1
Target1.2
Herbie4.3
\[\left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467 - x\right)\right) + \frac{0.083333333333333}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)\]

Derivation

  1. Split input into 2 regimes

  2. if x < 2.858940889989538e+33

    1. Initial program 0.2

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt0.2

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

    4. Applied log-prod0.2

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \log \left(\sqrt[3]{x}\right)\right)} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

    5. Applied distribute-lft-in0.2

      \[\leadsto \left(\left(\color{blue}{\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right)} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

    if 2.858940889989538e+33 < x

    1. Initial program 11.2

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

    2. Taylor expanded around inf 11.3

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\left(\left(0.0007936500793651 \cdot \frac{{z}^{2}}{x} + \frac{{z}^{2} \cdot y}{x}\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)}\]

    3. Simplified7.8

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\left(\frac{z \cdot z}{x} \cdot \left(0.0007936500793651 + y\right) - \frac{z}{x} \cdot 0.0027777777777778\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification4.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 2.858940889989538 \cdot 10^{+33}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} + \left(\left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right) + \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right)\right) - x\right) + 0.91893853320467\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \left(\frac{z \cdot z}{x} \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"

  :herbie-target
  (+ (+ (+ (* (- x 0.5) (log x)) (- 0.91893853320467 x)) (/ 0.083333333333333 x)) (* (/ z x) (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))

  (+ (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467) (/ (+ (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 0.083333333333333) x)))