Average Error: 6.1 → 5.8
Time: 10.1s
Precision: 64
\[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.02457834646528565 \cdot 10^{73}:\\ \;\;\;\;\left(\left(\log \left({x}^{\frac{1}{3}}\right) \cdot \left(3 \cdot x - 1.5\right) - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \left(\left(7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} + 0.0833333333333329956 \cdot \frac{1}{x}\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)\\ \end{array}\]
\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le 1.02457834646528565 \cdot 10^{73}:\\
\;\;\;\;\left(\left(\log \left({x}^{\frac{1}{3}}\right) \cdot \left(3 \cdot x - 1.5\right) - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\

\mathbf{else}:\\
\;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \left(\left(7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} + 0.0833333333333329956 \cdot \frac{1}{x}\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)\\

\end{array}
double f(double x, double y, double z) {
        double r504400 = x;
        double r504401 = 0.5;
        double r504402 = r504400 - r504401;
        double r504403 = log(r504400);
        double r504404 = r504402 * r504403;
        double r504405 = r504404 - r504400;
        double r504406 = 0.91893853320467;
        double r504407 = r504405 + r504406;
        double r504408 = y;
        double r504409 = 0.0007936500793651;
        double r504410 = r504408 + r504409;
        double r504411 = z;
        double r504412 = r504410 * r504411;
        double r504413 = 0.0027777777777778;
        double r504414 = r504412 - r504413;
        double r504415 = r504414 * r504411;
        double r504416 = 0.083333333333333;
        double r504417 = r504415 + r504416;
        double r504418 = r504417 / r504400;
        double r504419 = r504407 + r504418;
        return r504419;
}
↓
double f(double x, double y, double z) {
        double r504420 = x;
        double r504421 = 1.0245783464652857e+73;
        bool r504422 = r504420 <= r504421;
        double r504423 = 0.3333333333333333;
        double r504424 = pow(r504420, r504423);
        double r504425 = log(r504424);
        double r504426 = 3.0;
        double r504427 = r504426 * r504420;
        double r504428 = 1.5;
        double r504429 = r504427 - r504428;
        double r504430 = r504425 * r504429;
        double r504431 = r504430 - r504420;
        double r504432 = 0.91893853320467;
        double r504433 = r504431 + r504432;
        double r504434 = y;
        double r504435 = 0.0007936500793651;
        double r504436 = r504434 + r504435;
        double r504437 = z;
        double r504438 = r504436 * r504437;
        double r504439 = 0.0027777777777778;
        double r504440 = r504438 - r504439;
        double r504441 = r504440 * r504437;
        double r504442 = 0.083333333333333;
        double r504443 = r504441 + r504442;
        double r504444 = r504443 / r504420;
        double r504445 = r504433 + r504444;
        double r504446 = 0.5;
        double r504447 = r504420 - r504446;
        double r504448 = log(r504420);
        double r504449 = r504447 * r504448;
        double r504450 = r504449 - r504420;
        double r504451 = r504450 + r504432;
        double r504452 = 2.0;
        double r504453 = pow(r504437, r504452);
        double r504454 = r504453 / r504420;
        double r504455 = r504435 * r504454;
        double r504456 = 1.0;
        double r504457 = r504456 / r504420;
        double r504458 = r504442 * r504457;
        double r504459 = r504455 + r504458;
        double r504460 = r504437 / r504420;
        double r504461 = r504439 * r504460;
        double r504462 = r504459 - r504461;
        double r504463 = r504451 + r504462;
        double r504464 = r504422 ? r504445 : r504463;
        return r504464;
}

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
Target1.1
Herbie5.8
\[\left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467001 - x\right)\right) + \frac{0.0833333333333329956}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) - 0.0027777777777778\right)\]

Derivation

  1. Split input into 2 regimes

  2. if x < 1.0245783464652857e+73

    1. Initial program 0.8

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt0.8

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

    4. Applied log-prod0.8

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \log \left(\sqrt[3]{x}\right)\right)} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

    5. Applied distribute-lft-in0.8

      \[\leadsto \left(\left(\color{blue}{\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right)} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

    6. Simplified0.8

      \[\leadsto \left(\left(\left(\color{blue}{\left(x - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right)\right)} + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

    7. Taylor expanded around 0 0.8

      \[\leadsto \left(\left(\color{blue}{\left(3 \cdot \left(x \cdot \log \left({x}^{\frac{1}{3}}\right)\right) - 1.5 \cdot \log \left({x}^{\frac{1}{3}}\right)\right)} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

    8. Simplified0.8

      \[\leadsto \left(\left(\color{blue}{\log \left({x}^{\frac{1}{3}}\right) \cdot \left(3 \cdot x - 1.5\right)} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

    if 1.0245783464652857e+73 < x

    1. Initial program 12.1

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

    2. Taylor expanded around 0 11.3

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \color{blue}{\left(\left(7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} + 0.0833333333333329956 \cdot \frac{1}{x}\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification5.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.02457834646528565 \cdot 10^{73}:\\ \;\;\;\;\left(\left(\log \left({x}^{\frac{1}{3}}\right) \cdot \left(3 \cdot x - 1.5\right) - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \left(\left(7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} + 0.0833333333333329956 \cdot \frac{1}{x}\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"
  :precision binary64

  :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)))