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

Average Error: 20.2 → 0.1

Time: 18.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -5.110187933824355434057475957993495901711 \cdot 10^{45} \lor \neg \left(z \le 267536142.104578495025634765625\right):\\ \;\;\;\;\mathsf{fma}\left(0.07512208616047566511753075246815569698811, \frac{y}{z}, \mathsf{fma}\left(0.06929105992918889456166908757950295694172, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right)}{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r217340 = x;
        double r217341 = y;
        double r217342 = z;
        double r217343 = 0.0692910599291889;
        double r217344 = r217342 * r217343;
        double r217345 = 0.4917317610505968;
        double r217346 = r217344 + r217345;
        double r217347 = r217346 * r217342;
        double r217348 = 0.279195317918525;
        double r217349 = r217347 + r217348;
        double r217350 = r217341 * r217349;
        double r217351 = 6.012459259764103;
        double r217352 = r217342 + r217351;
        double r217353 = r217352 * r217342;
        double r217354 = 3.350343815022304;
        double r217355 = r217353 + r217354;
        double r217356 = r217350 / r217355;
        double r217357 = r217340 + r217356;
        return r217357;
}

↓

double f(double x, double y, double z) {
        double r217358 = z;
        double r217359 = -5.1101879338243554e+45;
        bool r217360 = r217358 <= r217359;
        double r217361 = 267536142.1045785;
        bool r217362 = r217358 <= r217361;
        double r217363 = !r217362;
        bool r217364 = r217360 || r217363;
        double r217365 = 0.07512208616047567;
        double r217366 = y;
        double r217367 = r217366 / r217358;
        double r217368 = 0.0692910599291889;
        double r217369 = x;
        double r217370 = fma(r217368, r217366, r217369);
        double r217371 = fma(r217365, r217367, r217370);
        double r217372 = 0.4917317610505968;
        double r217373 = fma(r217358, r217368, r217372);
        double r217374 = 0.279195317918525;
        double r217375 = fma(r217373, r217358, r217374);
        double r217376 = 6.012459259764103;
        double r217377 = r217358 + r217376;
        double r217378 = 3.350343815022304;
        double r217379 = fma(r217377, r217358, r217378);
        double r217380 = r217375 / r217379;
        double r217381 = r217366 * r217380;
        double r217382 = r217369 + r217381;
        double r217383 = r217364 ? r217371 : r217382;
        return r217383;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	20.2
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 < -5.1101879338243554e+45 or 267536142.1045785 < z

   1. Initial program 43.7

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

   2. Simplified 36.6

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

   4. Applied add-cube-cbrt 36.8

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

   5. Applied *-un-lft-identity 36.8

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{1 \cdot y}}{\left(\sqrt[3]{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)}}, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right), x\right)\]

   6. Applied times-frac 36.8

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\sqrt[3]{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)}} \cdot \frac{y}{\sqrt[3]{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)}}}, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right), x\right)\]
   7. Using strategy rm

   8. Applied expm1-log1p-u 37.4

      \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)}\right)\right)}} \cdot \frac{y}{\sqrt[3]{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)}}, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right), x\right)\]

   9. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{x + \left(0.07512208616047566511753075246815569698811 \cdot \frac{y}{z} + 0.06929105992918889456166908757950295694172 \cdot y\right)}\]

   10. Simplified 0.0

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.07512208616047566511753075246815569698811, \frac{y}{z}, \mathsf{fma}\left(0.06929105992918889456166908757950295694172, y, x\right)\right)}\]

   if -5.1101879338243554e+45 < z < 267536142.1045785

   1. Initial program 0.4

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

   3. Applied *-un-lft-identity 0.4

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

   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. Simplified 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.110187933824355434057475957993495901711 \cdot 10^{45} \lor \neg \left(z \le 267536142.104578495025634765625\right):\\ \;\;\;\;\mathsf{fma}\left(0.07512208616047566511753075246815569698811, \frac{y}{z}, \mathsf{fma}\left(0.06929105992918889456166908757950295694172, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right)}{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(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))))