Average Error: 5.7 → 3.8
Time: 12.7s
Precision: 64
\[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 8078075.08265684452:\\ \;\;\;\;\left(\left(\left(\sqrt{x} + \sqrt{0.5}\right) \cdot \left(\left(\sqrt{x} - \sqrt{0.5}\right) \cdot \log x\right) - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\left(x - 0.5\right) \cdot \log \left({x}^{\frac{1}{3}}\right) - x\right)\right) + 0.91893853320467001\right) + \mathsf{fma}\left(\frac{z}{x}, -0.0027777777777778, \frac{{z}^{2}}{x} \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right)\right)\\ \end{array}\]
\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 8078075.08265684452:\\
\;\;\;\;\left(\left(\left(\sqrt{x} + \sqrt{0.5}\right) \cdot \left(\left(\sqrt{x} - \sqrt{0.5}\right) \cdot \log 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}\\

\mathbf{else}:\\
\;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\left(x - 0.5\right) \cdot \log \left({x}^{\frac{1}{3}}\right) - x\right)\right) + 0.91893853320467001\right) + \mathsf{fma}\left(\frac{z}{x}, -0.0027777777777778, \frac{{z}^{2}}{x} \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right)\right)\\

\end{array}
double f(double x, double y, double z) {
        double r444267 = x;
        double r444268 = 0.5;
        double r444269 = r444267 - r444268;
        double r444270 = log(r444267);
        double r444271 = r444269 * r444270;
        double r444272 = r444271 - r444267;
        double r444273 = 0.91893853320467;
        double r444274 = r444272 + r444273;
        double r444275 = y;
        double r444276 = 0.0007936500793651;
        double r444277 = r444275 + r444276;
        double r444278 = z;
        double r444279 = r444277 * r444278;
        double r444280 = 0.0027777777777778;
        double r444281 = r444279 - r444280;
        double r444282 = r444281 * r444278;
        double r444283 = 0.083333333333333;
        double r444284 = r444282 + r444283;
        double r444285 = r444284 / r444267;
        double r444286 = r444274 + r444285;
        return r444286;
}
↓
double f(double x, double y, double z) {
        double r444287 = x;
        double r444288 = 8078075.0826568445;
        bool r444289 = r444287 <= r444288;
        double r444290 = sqrt(r444287);
        double r444291 = 0.5;
        double r444292 = sqrt(r444291);
        double r444293 = r444290 + r444292;
        double r444294 = r444290 - r444292;
        double r444295 = log(r444287);
        double r444296 = r444294 * r444295;
        double r444297 = r444293 * r444296;
        double r444298 = r444297 - r444287;
        double r444299 = 0.91893853320467;
        double r444300 = r444298 + r444299;
        double r444301 = y;
        double r444302 = 0.0007936500793651;
        double r444303 = r444301 + r444302;
        double r444304 = z;
        double r444305 = r444303 * r444304;
        double r444306 = 0.0027777777777778;
        double r444307 = r444305 - r444306;
        double r444308 = r444307 * r444304;
        double r444309 = 0.083333333333333;
        double r444310 = r444308 + r444309;
        double r444311 = r444310 / r444287;
        double r444312 = r444300 + r444311;
        double r444313 = r444287 - r444291;
        double r444314 = cbrt(r444287);
        double r444315 = r444314 * r444314;
        double r444316 = log(r444315);
        double r444317 = r444313 * r444316;
        double r444318 = 0.3333333333333333;
        double r444319 = pow(r444287, r444318);
        double r444320 = log(r444319);
        double r444321 = r444313 * r444320;
        double r444322 = r444321 - r444287;
        double r444323 = r444317 + r444322;
        double r444324 = r444323 + r444299;
        double r444325 = r444304 / r444287;
        double r444326 = -r444306;
        double r444327 = 2.0;
        double r444328 = pow(r444304, r444327);
        double r444329 = r444328 / r444287;
        double r444330 = r444329 * r444303;
        double r444331 = fma(r444325, r444326, r444330);
        double r444332 = r444324 + r444331;
        double r444333 = r444289 ? r444312 : r444332;
        return r444333;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original5.7
Target1.2
Herbie3.8
\[\left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467001 - x\right)\right) + \frac{0.0833333333333329956}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) - 0.0027777777777778\right)\]

Derivation

  1. Split input into 2 regimes

  2. if x < 8078075.0826568445

    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-sqr-sqrt0.1

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

    4. Applied add-sqr-sqrt0.1

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

    5. Applied difference-of-squares0.1

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

    6. Applied associate-*l*0.1

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

    if 8078075.0826568445 < x

    1. Initial program 9.9

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

    3. Applied add-cube-cbrt9.9

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

    4. Applied log-prod10.0

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

    5. Applied distribute-lft-in10.0

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

    6. Applied associate--l+9.9

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

    8. Applied pow1/39.9

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\left(x - 0.5\right) \cdot \log \color{blue}{\left({x}^{\frac{1}{3}}\right)} - x\right)\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

    9. Taylor expanded around inf 9.9

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\left(x - 0.5\right) \cdot \log \left({x}^{\frac{1}{3}}\right) - x\right)\right) + 0.91893853320467001\right) + \color{blue}{\left(\left(7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} + \frac{{z}^{2} \cdot y}{x}\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)}\]

    10. Simplified6.4

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

  4. Final simplification3.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 8078075.08265684452:\\ \;\;\;\;\left(\left(\left(\sqrt{x} + \sqrt{0.5}\right) \cdot \left(\left(\sqrt{x} - \sqrt{0.5}\right) \cdot \log x\right) - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\left(x - 0.5\right) \cdot \log \left({x}^{\frac{1}{3}}\right) - x\right)\right) + 0.91893853320467001\right) + \mathsf{fma}\left(\frac{z}{x}, -0.0027777777777778, \frac{{z}^{2}}{x} \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(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)))