Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B

Average Error: 19.6 → 0.1

Time: 5.4s

Precision: 64

Math

\[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1877563903899137105461248 \lor \neg \left(z \le 130680103.77611891925334930419921875\right):\\ \;\;\;\;x + \left(0.07512208616047560960637952121032867580652 \cdot \frac{y}{z} + 0.06929105992918889456166908757950295694172 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\sqrt[3]{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736} \cdot \sqrt[3]{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}}{\frac{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}{\sqrt[3]{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r388287 = x;
        double r388288 = y;
        double r388289 = z;
        double r388290 = 0.0692910599291889;
        double r388291 = r388289 * r388290;
        double r388292 = 0.4917317610505968;
        double r388293 = r388291 + r388292;
        double r388294 = r388293 * r388289;
        double r388295 = 0.279195317918525;
        double r388296 = r388294 + r388295;
        double r388297 = r388288 * r388296;
        double r388298 = 6.012459259764103;
        double r388299 = r388289 + r388298;
        double r388300 = r388299 * r388289;
        double r388301 = 3.350343815022304;
        double r388302 = r388300 + r388301;
        double r388303 = r388297 / r388302;
        double r388304 = r388287 + r388303;
        return r388304;
}

↓

double f(double x, double y, double z) {
        double r388305 = z;
        double r388306 = -1.877563903899137e+24;
        bool r388307 = r388305 <= r388306;
        double r388308 = 130680103.77611892;
        bool r388309 = r388305 <= r388308;
        double r388310 = !r388309;
        bool r388311 = r388307 || r388310;
        double r388312 = x;
        double r388313 = 0.07512208616047561;
        double r388314 = y;
        double r388315 = r388314 / r388305;
        double r388316 = r388313 * r388315;
        double r388317 = 0.0692910599291889;
        double r388318 = r388317 * r388314;
        double r388319 = r388316 + r388318;
        double r388320 = r388312 + r388319;
        double r388321 = r388305 * r388317;
        double r388322 = 0.4917317610505968;
        double r388323 = r388321 + r388322;
        double r388324 = r388323 * r388305;
        double r388325 = 0.279195317918525;
        double r388326 = r388324 + r388325;
        double r388327 = cbrt(r388326);
        double r388328 = r388327 * r388327;
        double r388329 = 6.012459259764103;
        double r388330 = r388305 + r388329;
        double r388331 = r388330 * r388305;
        double r388332 = 3.350343815022304;
        double r388333 = r388331 + r388332;
        double r388334 = r388333 / r388327;
        double r388335 = r388328 / r388334;
        double r388336 = r388314 * r388335;
        double r388337 = r388312 + r388336;
        double r388338 = r388311 ? r388320 : r388337;
        return r388338;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	19.6
Target	0.1
Herbie	0.1

\[\begin{array}{l} \mathbf{if}\;z \lt -8120153.6524566747248172760009765625:\\ \;\;\;\;\left(\frac{0.07512208616047560960637952121032867580652}{z} + 0.06929105992918889456166908757950295694172\right) \cdot y - \left(\frac{0.4046220386999212492717958866705885156989 \cdot y}{z \cdot z} - x\right)\\ \mathbf{elif}\;z \lt 657611897278737678336:\\ \;\;\;\;x + \left(y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)\right) \cdot \frac{1}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.07512208616047560960637952121032867580652}{z} + 0.06929105992918889456166908757950295694172\right) \cdot y - \left(\frac{0.4046220386999212492717958866705885156989 \cdot y}{z \cdot z} - x\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -1.877563903899137e+24 or 130680103.77611892 < z

   1. Initial program 41.1

      \[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]

   2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{x + \left(0.07512208616047560960637952121032867580652 \cdot \frac{y}{z} + 0.06929105992918889456166908757950295694172 \cdot y\right)}\]

   if -1.877563903899137e+24 < z < 130680103.77611892

   1. Initial program 0.3

      \[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 0.3

      \[\leadsto x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\color{blue}{1 \cdot \left(\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084\right)}}\]

   4. Applied times-frac 0.1

      \[\leadsto x + \color{blue}{\frac{y}{1} \cdot \frac{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}}\]

   5. Simplified 0.1

      \[\leadsto x + \color{blue}{y} \cdot \frac{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
   6. Using strategy rm

   7. Applied add-cube-cbrt 0.2

      \[\leadsto x + y \cdot \frac{\color{blue}{\left(\sqrt[3]{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736} \cdot \sqrt[3]{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}\right) \cdot \sqrt[3]{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}}}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]

   8. Applied associate-/l* 0.2

      \[\leadsto x + y \cdot \color{blue}{\frac{\sqrt[3]{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736} \cdot \sqrt[3]{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}}{\frac{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}{\sqrt[3]{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1877563903899137105461248 \lor \neg \left(z \le 130680103.77611891925334930419921875\right):\\ \;\;\;\;x + \left(0.07512208616047560960637952121032867580652 \cdot \frac{y}{z} + 0.06929105992918889456166908757950295694172 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\sqrt[3]{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736} \cdot \sqrt[3]{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}}{\frac{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}{\sqrt[3]{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< z -8120153.652456675) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) (- (/ (* 0.40462203869992125 y) (* z z)) x)) (if (< z 657611897278737680000) (+ x (* (* y (+ (* (+ (* z 0.0692910599291889) 0.4917317610505968) z) 0.279195317918525)) (/ 1 (+ (* (+ z 6.012459259764103) z) 3.350343815022304)))) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) (- (/ (* 0.40462203869992125 y) (* z z)) x))))

  (+ x (/ (* y (+ (* (+ (* z 0.0692910599291889) 0.4917317610505968) z) 0.279195317918525)) (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))