Average Error: 6.1 → 4.2
Time: 8.8s
Precision: 64
\[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 2147002421983032.5:\\ \;\;\;\;\left(\left(\left(\sqrt{x} + \sqrt{0.5}\right) \cdot \left(\left(\sqrt{x} - \sqrt{0.5}\right) \cdot \log x\right) - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \mathsf{fma}\left(\log \left(\sqrt[3]{x}\right), x - 0.5, 0.91893853320467001 - x\right)\right) + \mathsf{fma}\left(\frac{{z}^{2}}{x}, y + 7.93650079365100015 \cdot 10^{-4}, -0.0027777777777778 \cdot \frac{z}{x}\right)\\ \end{array}\]
\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 2147002421983032.5:\\
\;\;\;\;\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(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \mathsf{fma}\left(\log \left(\sqrt[3]{x}\right), x - 0.5, 0.91893853320467001 - x\right)\right) + \mathsf{fma}\left(\frac{{z}^{2}}{x}, y + 7.93650079365100015 \cdot 10^{-4}, -0.0027777777777778 \cdot \frac{z}{x}\right)\\

\end{array}
double f(double x, double y, double z) {
        double r415308 = x;
        double r415309 = 0.5;
        double r415310 = r415308 - r415309;
        double r415311 = log(r415308);
        double r415312 = r415310 * r415311;
        double r415313 = r415312 - r415308;
        double r415314 = 0.91893853320467;
        double r415315 = r415313 + r415314;
        double r415316 = y;
        double r415317 = 0.0007936500793651;
        double r415318 = r415316 + r415317;
        double r415319 = z;
        double r415320 = r415318 * r415319;
        double r415321 = 0.0027777777777778;
        double r415322 = r415320 - r415321;
        double r415323 = r415322 * r415319;
        double r415324 = 0.083333333333333;
        double r415325 = r415323 + r415324;
        double r415326 = r415325 / r415308;
        double r415327 = r415315 + r415326;
        return r415327;
}
↓
double f(double x, double y, double z) {
        double r415328 = x;
        double r415329 = 2147002421983032.5;
        bool r415330 = r415328 <= r415329;
        double r415331 = sqrt(r415328);
        double r415332 = 0.5;
        double r415333 = sqrt(r415332);
        double r415334 = r415331 + r415333;
        double r415335 = r415331 - r415333;
        double r415336 = log(r415328);
        double r415337 = r415335 * r415336;
        double r415338 = r415334 * r415337;
        double r415339 = r415338 - r415328;
        double r415340 = 0.91893853320467;
        double r415341 = r415339 + r415340;
        double r415342 = y;
        double r415343 = 0.0007936500793651;
        double r415344 = r415342 + r415343;
        double r415345 = z;
        double r415346 = r415344 * r415345;
        double r415347 = 0.0027777777777778;
        double r415348 = r415346 - r415347;
        double r415349 = r415348 * r415345;
        double r415350 = 0.083333333333333;
        double r415351 = r415349 + r415350;
        double r415352 = r415351 / r415328;
        double r415353 = r415341 + r415352;
        double r415354 = cbrt(r415328);
        double r415355 = r415354 * r415354;
        double r415356 = log(r415355);
        double r415357 = r415328 - r415332;
        double r415358 = r415356 * r415357;
        double r415359 = log(r415354);
        double r415360 = r415340 - r415328;
        double r415361 = fma(r415359, r415357, r415360);
        double r415362 = r415358 + r415361;
        double r415363 = 2.0;
        double r415364 = pow(r415345, r415363);
        double r415365 = r415364 / r415328;
        double r415366 = r415345 / r415328;
        double r415367 = r415347 * r415366;
        double r415368 = -r415367;
        double r415369 = fma(r415365, r415344, r415368);
        double r415370 = r415362 + r415369;
        double r415371 = r415330 ? r415353 : r415370;
        return r415371;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.1
Target1.3
Herbie4.2
\[\left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467001 - x\right)\right) + \frac{0.0833333333333329956}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) - 0.0027777777777778\right)\]

Derivation

  1. Split input into 2 regimes

  2. if x < 2147002421983032.5

    1. Initial program 0.2

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

    3. Applied add-sqr-sqrt0.2

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

    4. Applied add-sqr-sqrt0.2

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

    5. Applied difference-of-squares0.2

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

    6. Applied associate-*l*0.2

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

    if 2147002421983032.5 < x

    1. Initial program 10.7

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

    3. Applied add-cube-cbrt10.7

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

    4. Applied log-prod10.8

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

    5. Applied distribute-rgt-in10.8

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

    6. Applied associate--l+10.8

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

    7. Applied associate-+l+10.8

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

    8. Simplified10.7

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

    9. Taylor expanded around inf 10.8

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

    10. Simplified7.3

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

  4. Final simplification4.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 2147002421983032.5:\\ \;\;\;\;\left(\left(\left(\sqrt{x} + \sqrt{0.5}\right) \cdot \left(\left(\sqrt{x} - \sqrt{0.5}\right) \cdot \log x\right) - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \mathsf{fma}\left(\log \left(\sqrt[3]{x}\right), x - 0.5, 0.91893853320467001 - x\right)\right) + \mathsf{fma}\left(\frac{{z}^{2}}{x}, y + 7.93650079365100015 \cdot 10^{-4}, -0.0027777777777778 \cdot \frac{z}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 +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)))