Average Error: 29.6 → 4.5
Time: 6.8s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.82773053729933007 \cdot 10^{53} \lor \neg \left(z \le 1.6003447527747322 \cdot 10^{61}\right):\\ \;\;\;\;x + \left(\left(3.13060547622999996 \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\\ \end{array}\]
x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.82773053729933007 \cdot 10^{53} \lor \neg \left(z \le 1.6003447527747322 \cdot 10^{61}\right):\\
\;\;\;\;x + \left(\left(3.13060547622999996 \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)\\

\mathbf{else}:\\
\;\;\;\;x + y \cdot \frac{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r354270 = x;
        double r354271 = y;
        double r354272 = z;
        double r354273 = 3.13060547623;
        double r354274 = r354272 * r354273;
        double r354275 = 11.1667541262;
        double r354276 = r354274 + r354275;
        double r354277 = r354276 * r354272;
        double r354278 = t;
        double r354279 = r354277 + r354278;
        double r354280 = r354279 * r354272;
        double r354281 = a;
        double r354282 = r354280 + r354281;
        double r354283 = r354282 * r354272;
        double r354284 = b;
        double r354285 = r354283 + r354284;
        double r354286 = r354271 * r354285;
        double r354287 = 15.234687407;
        double r354288 = r354272 + r354287;
        double r354289 = r354288 * r354272;
        double r354290 = 31.4690115749;
        double r354291 = r354289 + r354290;
        double r354292 = r354291 * r354272;
        double r354293 = 11.9400905721;
        double r354294 = r354292 + r354293;
        double r354295 = r354294 * r354272;
        double r354296 = 0.607771387771;
        double r354297 = r354295 + r354296;
        double r354298 = r354286 / r354297;
        double r354299 = r354270 + r354298;
        return r354299;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r354300 = z;
        double r354301 = -1.82773053729933e+53;
        bool r354302 = r354300 <= r354301;
        double r354303 = 1.6003447527747322e+61;
        bool r354304 = r354300 <= r354303;
        double r354305 = !r354304;
        bool r354306 = r354302 || r354305;
        double r354307 = x;
        double r354308 = 3.13060547623;
        double r354309 = y;
        double r354310 = r354308 * r354309;
        double r354311 = t;
        double r354312 = r354311 * r354309;
        double r354313 = 2.0;
        double r354314 = pow(r354300, r354313);
        double r354315 = r354312 / r354314;
        double r354316 = r354310 + r354315;
        double r354317 = 36.527041698806414;
        double r354318 = r354309 / r354300;
        double r354319 = r354317 * r354318;
        double r354320 = r354316 - r354319;
        double r354321 = r354307 + r354320;
        double r354322 = r354300 * r354308;
        double r354323 = 11.1667541262;
        double r354324 = r354322 + r354323;
        double r354325 = r354324 * r354300;
        double r354326 = r354325 + r354311;
        double r354327 = r354326 * r354300;
        double r354328 = a;
        double r354329 = r354327 + r354328;
        double r354330 = r354329 * r354300;
        double r354331 = b;
        double r354332 = r354330 + r354331;
        double r354333 = 15.234687407;
        double r354334 = r354300 + r354333;
        double r354335 = r354334 * r354300;
        double r354336 = 31.4690115749;
        double r354337 = r354335 + r354336;
        double r354338 = r354337 * r354300;
        double r354339 = 11.9400905721;
        double r354340 = r354338 + r354339;
        double r354341 = r354340 * r354300;
        double r354342 = 0.607771387771;
        double r354343 = r354341 + r354342;
        double r354344 = r354332 / r354343;
        double r354345 = r354309 * r354344;
        double r354346 = r354307 + r354345;
        double r354347 = r354306 ? r354321 : r354346;
        return r354347;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.6
Target1.0
Herbie4.5
\[\begin{array}{l} \mathbf{if}\;z \lt -6.4993449962526318 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.0669654369142868 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.82773053729933e+53 or 1.6003447527747322e+61 < z

    1. Initial program 62.1

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]

    2. Taylor expanded around inf 8.4

      \[\leadsto x + \color{blue}{\left(\left(3.13060547622999996 \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)}\]

    if -1.82773053729933e+53 < z < 1.6003447527747322e+61

    1. Initial program 3.0

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity3.0

      \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\color{blue}{1 \cdot \left(\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004\right)}}\]

    4. Applied times-frac1.4

      \[\leadsto x + \color{blue}{\frac{y}{1} \cdot \frac{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}}\]

    5. Simplified1.4

      \[\leadsto x + \color{blue}{y} \cdot \frac{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification4.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.82773053729933007 \cdot 10^{53} \lor \neg \left(z \le 1.6003447527747322 \cdot 10^{61}\right):\\ \;\;\;\;x + \left(\left(3.13060547622999996 \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :herbie-target
  (if (< z -6.499344996252632e+53) (+ x (* (+ (- 3.13060547623 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1))) (if (< z 7.066965436914287e+59) (+ x (/ y (/ (+ (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z) 0.607771387771) (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b)))) (+ x (* (+ (- 3.13060547623 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1)))))

  (+ x (/ (* y (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b)) (+ (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z) 0.607771387771))))