Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B

Average Error: 5.7 → 0.6

Time: 31.2s

Precision: 64

Math

\[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le 2849501.9306197646:\\ \;\;\;\;\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 + \left(\sqrt[3]{\log x} \cdot \left(\left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}\right) \cdot \left(x - 0.5\right)\right) - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{\frac{x}{z}} \cdot \left(y + 0.0007936500793651\right) - \frac{z \cdot 0.0027777777777778}{x}\right) + \left(\log \left({\left({x}^{\left(\sqrt[3]{\frac{2}{3}} \cdot \sqrt[3]{\frac{2}{3}}\right)}\right)}^{\left(\sqrt[3]{\frac{2}{3}}\right)}\right) \cdot \left(x - 0.5\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r25501254 = x;
        double r25501255 = 0.5;
        double r25501256 = r25501254 - r25501255;
        double r25501257 = log(r25501254);
        double r25501258 = r25501256 * r25501257;
        double r25501259 = r25501258 - r25501254;
        double r25501260 = 0.91893853320467;
        double r25501261 = r25501259 + r25501260;
        double r25501262 = y;
        double r25501263 = 0.0007936500793651;
        double r25501264 = r25501262 + r25501263;
        double r25501265 = z;
        double r25501266 = r25501264 * r25501265;
        double r25501267 = 0.0027777777777778;
        double r25501268 = r25501266 - r25501267;
        double r25501269 = r25501268 * r25501265;
        double r25501270 = 0.083333333333333;
        double r25501271 = r25501269 + r25501270;
        double r25501272 = r25501271 / r25501254;
        double r25501273 = r25501261 + r25501272;
        return r25501273;
}

↓

double f(double x, double y, double z) {
        double r25501274 = x;
        double r25501275 = 2849501.9306197646;
        bool r25501276 = r25501274 <= r25501275;
        double r25501277 = z;
        double r25501278 = y;
        double r25501279 = 0.0007936500793651;
        double r25501280 = r25501278 + r25501279;
        double r25501281 = r25501277 * r25501280;
        double r25501282 = 0.0027777777777778;
        double r25501283 = r25501281 - r25501282;
        double r25501284 = r25501277 * r25501283;
        double r25501285 = 0.083333333333333;
        double r25501286 = r25501284 + r25501285;
        double r25501287 = r25501286 / r25501274;
        double r25501288 = 0.91893853320467;
        double r25501289 = log(r25501274);
        double r25501290 = cbrt(r25501289);
        double r25501291 = r25501290 * r25501290;
        double r25501292 = 0.5;
        double r25501293 = r25501274 - r25501292;
        double r25501294 = r25501291 * r25501293;
        double r25501295 = r25501290 * r25501294;
        double r25501296 = r25501295 - r25501274;
        double r25501297 = r25501288 + r25501296;
        double r25501298 = r25501287 + r25501297;
        double r25501299 = r25501274 / r25501277;
        double r25501300 = r25501277 / r25501299;
        double r25501301 = r25501300 * r25501280;
        double r25501302 = r25501277 * r25501282;
        double r25501303 = r25501302 / r25501274;
        double r25501304 = r25501301 - r25501303;
        double r25501305 = 0.6666666666666666;
        double r25501306 = cbrt(r25501305);
        double r25501307 = r25501306 * r25501306;
        double r25501308 = pow(r25501274, r25501307);
        double r25501309 = pow(r25501308, r25501306);
        double r25501310 = log(r25501309);
        double r25501311 = r25501310 * r25501293;
        double r25501312 = cbrt(r25501274);
        double r25501313 = log(r25501312);
        double r25501314 = r25501313 * r25501293;
        double r25501315 = r25501314 - r25501274;
        double r25501316 = r25501315 + r25501288;
        double r25501317 = r25501311 + r25501316;
        double r25501318 = r25501304 + r25501317;
        double r25501319 = r25501276 ? r25501298 : r25501318;
        return r25501319;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	5.7
Target	1.1
Herbie	0.6

\[\left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467 - x\right)\right) + \frac{0.083333333333333}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)\]

Derivation

1. Split input into 2 regimes

2. if x < 2849501.9306197646

   1. Initial program 0.1

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 0.1

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}\right) \cdot \sqrt[3]{\log x}\right)} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

   4. Applied associate-*r* 0.1

      \[\leadsto \left(\left(\color{blue}{\left(\left(x - 0.5\right) \cdot \left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}\right)\right) \cdot \sqrt[3]{\log x}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

   if 2849501.9306197646 < x

   1. Initial program 9.7

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 9.7

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

   4. Applied log-prod 9.7

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \log \left(\sqrt[3]{x}\right)\right)} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

   5. Applied distribute-rgt-in 9.7

      \[\leadsto \left(\left(\color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right)\right)} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

   6. Applied associate--l+ 9.7

      \[\leadsto \left(\color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right)\right)} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

   7. Applied associate-+l+ 9.7

      \[\leadsto \color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467\right)\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]
   8. Using strategy rm

   9. Applied pow1/3 9.7

      \[\leadsto \left(\log \left(\sqrt[3]{x} \cdot \color{blue}{{x}^{\frac{1}{3}}}\right) \cdot \left(x - 0.5\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467\right)\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

   10. Applied pow1/3 9.7

       \[\leadsto \left(\log \left(\color{blue}{{x}^{\frac{1}{3}}} \cdot {x}^{\frac{1}{3}}\right) \cdot \left(x - 0.5\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467\right)\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

   11. Applied pow-prod-up 9.7

       \[\leadsto \left(\log \color{blue}{\left({x}^{\left(\frac{1}{3} + \frac{1}{3}\right)}\right)} \cdot \left(x - 0.5\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467\right)\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

   12. Simplified 9.7

       \[\leadsto \left(\log \left({x}^{\color{blue}{\frac{2}{3}}}\right) \cdot \left(x - 0.5\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467\right)\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]
   13. Using strategy rm

   14. Applied add-cube-cbrt 10.2

       \[\leadsto \left(\log \left({x}^{\color{blue}{\left(\left(\sqrt[3]{\frac{2}{3}} \cdot \sqrt[3]{\frac{2}{3}}\right) \cdot \sqrt[3]{\frac{2}{3}}\right)}}\right) \cdot \left(x - 0.5\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467\right)\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

   15. Applied pow-unpow 10.0

       \[\leadsto \left(\log \color{blue}{\left({\left({x}^{\left(\sqrt[3]{\frac{2}{3}} \cdot \sqrt[3]{\frac{2}{3}}\right)}\right)}^{\left(\sqrt[3]{\frac{2}{3}}\right)}\right)} \cdot \left(x - 0.5\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467\right)\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

   16. Taylor expanded around inf 10.1

       \[\leadsto \left(\log \left({\left({x}^{\left(\sqrt[3]{\frac{2}{3}} \cdot \sqrt[3]{\frac{2}{3}}\right)}\right)}^{\left(\sqrt[3]{\frac{2}{3}}\right)}\right) \cdot \left(x - 0.5\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467\right)\right) + \color{blue}{\left(\left(0.0007936500793651 \cdot \frac{{z}^{2}}{x} + \frac{{z}^{2} \cdot y}{x}\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)}\]

   17. Simplified 0.9

       \[\leadsto \left(\log \left({\left({x}^{\left(\sqrt[3]{\frac{2}{3}} \cdot \sqrt[3]{\frac{2}{3}}\right)}\right)}^{\left(\sqrt[3]{\frac{2}{3}}\right)}\right) \cdot \left(x - 0.5\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467\right)\right) + \color{blue}{\left(\frac{z}{\frac{x}{z}} \cdot \left(0.0007936500793651 + y\right) - \frac{z \cdot 0.0027777777777778}{x}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 2849501.9306197646:\\ \;\;\;\;\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 + \left(\sqrt[3]{\log x} \cdot \left(\left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}\right) \cdot \left(x - 0.5\right)\right) - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{\frac{x}{z}} \cdot \left(y + 0.0007936500793651\right) - \frac{z \cdot 0.0027777777777778}{x}\right) + \left(\log \left({\left({x}^{\left(\sqrt[3]{\frac{2}{3}} \cdot \sqrt[3]{\frac{2}{3}}\right)}\right)}^{\left(\sqrt[3]{\frac{2}{3}}\right)}\right) \cdot \left(x - 0.5\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"

  :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)))