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

Average Error: 19.3 → 0.1

Time: 59.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.361084135604781330698244032176934602371 \cdot 10^{154} \lor \neg \left(z \le 8120465736065972\right):\\ \;\;\;\;\left(y \cdot 0.06929105992918889456166908757950295694172 + \left(\frac{0.07512208616047560960637952121032867580652 \cdot y}{z} - \frac{\frac{y}{z}}{z} \cdot 0.4046220386999212492717958866705885156989\right)\right) + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{0.2791953179185249767080279070796677842736 + \left(z \cdot 0.4917317610505967939715787906607147306204 + 0.06929105992918889456166908757950295694172 \cdot \left(z \cdot z\right)\right)}{3.350343815022303939343828460550867021084 + \left(z + 6.012459259764103336465268512256443500519\right) \cdot z} + x\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r325417 = x;
        double r325418 = y;
        double r325419 = z;
        double r325420 = 0.0692910599291889;
        double r325421 = r325419 * r325420;
        double r325422 = 0.4917317610505968;
        double r325423 = r325421 + r325422;
        double r325424 = r325423 * r325419;
        double r325425 = 0.279195317918525;
        double r325426 = r325424 + r325425;
        double r325427 = r325418 * r325426;
        double r325428 = 6.012459259764103;
        double r325429 = r325419 + r325428;
        double r325430 = r325429 * r325419;
        double r325431 = 3.350343815022304;
        double r325432 = r325430 + r325431;
        double r325433 = r325427 / r325432;
        double r325434 = r325417 + r325433;
        return r325434;
}

↓

double f(double x, double y, double z) {
        double r325435 = z;
        double r325436 = -1.3610841356047813e+154;
        bool r325437 = r325435 <= r325436;
        double r325438 = 8120465736065972.0;
        bool r325439 = r325435 <= r325438;
        double r325440 = !r325439;
        bool r325441 = r325437 || r325440;
        double r325442 = y;
        double r325443 = 0.0692910599291889;
        double r325444 = r325442 * r325443;
        double r325445 = 0.07512208616047561;
        double r325446 = r325445 * r325442;
        double r325447 = r325446 / r325435;
        double r325448 = r325442 / r325435;
        double r325449 = r325448 / r325435;
        double r325450 = 0.40462203869992125;
        double r325451 = r325449 * r325450;
        double r325452 = r325447 - r325451;
        double r325453 = r325444 + r325452;
        double r325454 = x;
        double r325455 = r325453 + r325454;
        double r325456 = 0.279195317918525;
        double r325457 = 0.4917317610505968;
        double r325458 = r325435 * r325457;
        double r325459 = r325435 * r325435;
        double r325460 = r325443 * r325459;
        double r325461 = r325458 + r325460;
        double r325462 = r325456 + r325461;
        double r325463 = 3.350343815022304;
        double r325464 = 6.012459259764103;
        double r325465 = r325435 + r325464;
        double r325466 = r325465 * r325435;
        double r325467 = r325463 + r325466;
        double r325468 = r325462 / r325467;
        double r325469 = r325442 * r325468;
        double r325470 = r325469 + r325454;
        double r325471 = r325441 ? r325455 : r325470;
        return r325471;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	19.3
Target	0.2
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 < -1.3610841356047813e+154 or 8120465736065972.0 < z

   1. Initial program 48.4

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

   2. Simplified 43.1

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

   3. Taylor expanded around 0 43.1

      \[\leadsto \frac{\color{blue}{\left(0.4917317610505967939715787906607147306204 \cdot z + 0.06929105992918889456166908757950295694172 \cdot {z}^{2}\right)} + 0.2791953179185249767080279070796677842736}{z \cdot \left(z + 6.012459259764103336465268512256443500519\right) + 3.350343815022303939343828460550867021084} \cdot y + x\]

   4. Simplified 43.1

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

   6. Applied div-inv 43.2

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

   7. Applied associate-*l* 44.1

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

   8. Simplified 44.1

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

   9. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(\left(0.06929105992918889456166908757950295694172 \cdot y + 0.07512208616047560960637952121032867580652 \cdot \frac{y}{z}\right) - 0.4046220386999212492717958866705885156989 \cdot \frac{y}{{z}^{2}}\right)} + x\]

   10. Simplified 0.0

       \[\leadsto \color{blue}{\left(\left(\frac{0.07512208616047560960637952121032867580652 \cdot y}{z} - 0.4046220386999212492717958866705885156989 \cdot \frac{\frac{y}{z}}{z}\right) + 0.06929105992918889456166908757950295694172 \cdot y\right)} + x\]

   if -1.3610841356047813e+154 < z < 8120465736065972.0

   1. Initial program 3.2

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

   2. Simplified 0.1

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

   3. Taylor expanded around 0 0.1

      \[\leadsto \frac{\color{blue}{\left(0.4917317610505967939715787906607147306204 \cdot z + 0.06929105992918889456166908757950295694172 \cdot {z}^{2}\right)} + 0.2791953179185249767080279070796677842736}{z \cdot \left(z + 6.012459259764103336465268512256443500519\right) + 3.350343815022303939343828460550867021084} \cdot y + x\]

   4. Simplified 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.361084135604781330698244032176934602371 \cdot 10^{154} \lor \neg \left(z \le 8120465736065972\right):\\ \;\;\;\;\left(y \cdot 0.06929105992918889456166908757950295694172 + \left(\frac{0.07512208616047560960637952121032867580652 \cdot y}{z} - \frac{\frac{y}{z}}{z} \cdot 0.4046220386999212492717958866705885156989\right)\right) + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{0.2791953179185249767080279070796677842736 + \left(z \cdot 0.4917317610505967939715787906607147306204 + 0.06929105992918889456166908757950295694172 \cdot \left(z \cdot z\right)\right)}{3.350343815022303939343828460550867021084 + \left(z + 6.012459259764103336465268512256443500519\right) \cdot z} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B"

  :herbie-target
  (if (< z -8120153.652456675) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) (- (/ (* 0.40462203869992125 y) (* z z)) x)) (if (< z 6.576118972787377e+20) (+ x (* (* y (+ (* (+ (* z 0.0692910599291889) 0.4917317610505968) z) 0.279195317918525)) (/ 1.0 (+ (* (+ 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))))