Average Error: 6.2 → 5.3
Time: 8.5s
Precision: 64
\[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.55935694262055325 \cdot 10^{121}:\\ \;\;\;\;\left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \left(\left(\log \left(\sqrt{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467001\right)\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \left(\left(7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} + 0.0833333333333329956 \cdot \frac{1}{x}\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)\\ \end{array}\]
\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 1.55935694262055325 \cdot 10^{121}:\\
\;\;\;\;\left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \left(\left(\log \left(\sqrt{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467001\right)\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 r427416 = x;
        double r427417 = 0.5;
        double r427418 = r427416 - r427417;
        double r427419 = log(r427416);
        double r427420 = r427418 * r427419;
        double r427421 = r427420 - r427416;
        double r427422 = 0.91893853320467;
        double r427423 = r427421 + r427422;
        double r427424 = y;
        double r427425 = 0.0007936500793651;
        double r427426 = r427424 + r427425;
        double r427427 = z;
        double r427428 = r427426 * r427427;
        double r427429 = 0.0027777777777778;
        double r427430 = r427428 - r427429;
        double r427431 = r427430 * r427427;
        double r427432 = 0.083333333333333;
        double r427433 = r427431 + r427432;
        double r427434 = r427433 / r427416;
        double r427435 = r427423 + r427434;
        return r427435;
}
↓
double f(double x, double y, double z) {
        double r427436 = x;
        double r427437 = 1.5593569426205532e+121;
        bool r427438 = r427436 <= r427437;
        double r427439 = 0.5;
        double r427440 = r427436 - r427439;
        double r427441 = sqrt(r427436);
        double r427442 = log(r427441);
        double r427443 = r427440 * r427442;
        double r427444 = r427442 * r427440;
        double r427445 = r427444 - r427436;
        double r427446 = 0.91893853320467;
        double r427447 = r427445 + r427446;
        double r427448 = r427443 + r427447;
        double r427449 = y;
        double r427450 = 0.0007936500793651;
        double r427451 = r427449 + r427450;
        double r427452 = z;
        double r427453 = r427451 * r427452;
        double r427454 = 0.0027777777777778;
        double r427455 = r427453 - r427454;
        double r427456 = r427455 * r427452;
        double r427457 = 0.083333333333333;
        double r427458 = r427456 + r427457;
        double r427459 = r427458 / r427436;
        double r427460 = r427448 + r427459;
        double r427461 = log(r427436);
        double r427462 = r427440 * r427461;
        double r427463 = r427462 - r427436;
        double r427464 = r427463 + r427446;
        double r427465 = 2.0;
        double r427466 = pow(r427452, r427465);
        double r427467 = r427466 / r427436;
        double r427468 = r427450 * r427467;
        double r427469 = 1.0;
        double r427470 = r427469 / r427436;
        double r427471 = r427457 * r427470;
        double r427472 = r427468 + r427471;
        double r427473 = r427452 / r427436;
        double r427474 = r427454 * r427473;
        double r427475 = r427472 - r427474;
        double r427476 = r427464 + r427475;
        double r427477 = r427438 ? r427460 : r427476;
        return r427477;
}

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

Original6.2
Target1.2
Herbie5.3
\[\left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467001 - x\right)\right) + \frac{0.0833333333333329956}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) - 0.0027777777777778\right)\]

Derivation

  1. Split input into 2 regimes

  2. if x < 1.5593569426205532e+121

    1. Initial program 1.7

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

    3. Applied add-sqr-sqrt1.7

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

    4. Applied log-prod1.7

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

    5. Applied distribute-lft-in1.7

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

    6. Applied associate--l+1.7

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

    7. Applied associate-+l+1.7

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

    8. Simplified1.7

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

    if 1.5593569426205532e+121 < x

    1. Initial program 13.8

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

    2. Taylor expanded around 0 11.4

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

  4. Final simplification5.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.55935694262055325 \cdot 10^{121}:\\ \;\;\;\;\left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \left(\left(\log \left(\sqrt{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467001\right)\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \left(\left(7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} + 0.0833333333333329956 \cdot \frac{1}{x}\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(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)))