Average Error: 6.0 → 4.3
Time: 14.7s
Precision: 64
\[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 38583220048788.3828:\\ \;\;\;\;\left(\left(\left(\left(2 \cdot \log \left(\sqrt[3]{x}\right)\right) \cdot \left(x - 0.5\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \left(\frac{{z}^{2}}{x} \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)\\ \end{array}\]
\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 38583220048788.3828:\\
\;\;\;\;\left(\left(\left(\left(2 \cdot \log \left(\sqrt[3]{x}\right)\right) \cdot \left(x - 0.5\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}\\

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

\end{array}
double f(double x, double y, double z) {
        double r458530 = x;
        double r458531 = 0.5;
        double r458532 = r458530 - r458531;
        double r458533 = log(r458530);
        double r458534 = r458532 * r458533;
        double r458535 = r458534 - r458530;
        double r458536 = 0.91893853320467;
        double r458537 = r458535 + r458536;
        double r458538 = y;
        double r458539 = 0.0007936500793651;
        double r458540 = r458538 + r458539;
        double r458541 = z;
        double r458542 = r458540 * r458541;
        double r458543 = 0.0027777777777778;
        double r458544 = r458542 - r458543;
        double r458545 = r458544 * r458541;
        double r458546 = 0.083333333333333;
        double r458547 = r458545 + r458546;
        double r458548 = r458547 / r458530;
        double r458549 = r458537 + r458548;
        return r458549;
}
↓
double f(double x, double y, double z) {
        double r458550 = x;
        double r458551 = 38583220048788.38;
        bool r458552 = r458550 <= r458551;
        double r458553 = 2.0;
        double r458554 = cbrt(r458550);
        double r458555 = log(r458554);
        double r458556 = r458553 * r458555;
        double r458557 = 0.5;
        double r458558 = r458550 - r458557;
        double r458559 = r458556 * r458558;
        double r458560 = r458558 * r458555;
        double r458561 = r458559 + r458560;
        double r458562 = r458561 - r458550;
        double r458563 = 0.91893853320467;
        double r458564 = r458562 + r458563;
        double r458565 = y;
        double r458566 = 0.0007936500793651;
        double r458567 = r458565 + r458566;
        double r458568 = z;
        double r458569 = r458567 * r458568;
        double r458570 = 0.0027777777777778;
        double r458571 = r458569 - r458570;
        double r458572 = r458571 * r458568;
        double r458573 = 0.083333333333333;
        double r458574 = r458572 + r458573;
        double r458575 = r458574 / r458550;
        double r458576 = r458564 + r458575;
        double r458577 = log(r458550);
        double r458578 = r458558 * r458577;
        double r458579 = r458578 - r458550;
        double r458580 = r458579 + r458563;
        double r458581 = pow(r458568, r458553);
        double r458582 = r458581 / r458550;
        double r458583 = r458582 * r458567;
        double r458584 = r458568 / r458550;
        double r458585 = r458570 * r458584;
        double r458586 = r458583 - r458585;
        double r458587 = r458580 + r458586;
        double r458588 = r458552 ? r458576 : r458587;
        return r458588;
}

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

Original6.0
Target1.1
Herbie4.3
\[\left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467001 - x\right)\right) + \frac{0.0833333333333329956}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) - 0.0027777777777778\right)\]

Derivation

  1. Split input into 2 regimes

  2. if x < 38583220048788.38

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt0.1

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

    4. Applied log-prod0.1

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

    5. Applied distribute-lft-in0.1

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

    6. Simplified0.1

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

    if 38583220048788.38 < x

    1. Initial program 10.5

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

    2. Taylor expanded around inf 10.6

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

    3. Simplified7.5

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

  4. Final simplification4.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 38583220048788.3828:\\ \;\;\;\;\left(\left(\left(\left(2 \cdot \log \left(\sqrt[3]{x}\right)\right) \cdot \left(x - 0.5\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \left(\frac{{z}^{2}}{x} \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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)))