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

Average Error: 20.4 → 0.1

Time: 16.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -405129914590555930114134145845166080 \lor \neg \left(z \le 188100201.1050882637500762939453125\right):\\ \;\;\;\;\mathsf{fma}\left(0.06929105992918889456166908757950295694172, y, \mathsf{fma}\left(0.07512208616047560960637952121032867580652, \frac{y}{z}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), 0.2791953179185249767080279070796677842736\right)}{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)} + x\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r234389 = x;
        double r234390 = y;
        double r234391 = z;
        double r234392 = 0.0692910599291889;
        double r234393 = r234391 * r234392;
        double r234394 = 0.4917317610505968;
        double r234395 = r234393 + r234394;
        double r234396 = r234395 * r234391;
        double r234397 = 0.279195317918525;
        double r234398 = r234396 + r234397;
        double r234399 = r234390 * r234398;
        double r234400 = 6.012459259764103;
        double r234401 = r234391 + r234400;
        double r234402 = r234401 * r234391;
        double r234403 = 3.350343815022304;
        double r234404 = r234402 + r234403;
        double r234405 = r234399 / r234404;
        double r234406 = r234389 + r234405;
        return r234406;
}

↓

double f(double x, double y, double z) {
        double r234407 = z;
        double r234408 = -4.051299145905559e+35;
        bool r234409 = r234407 <= r234408;
        double r234410 = 188100201.10508826;
        bool r234411 = r234407 <= r234410;
        double r234412 = !r234411;
        bool r234413 = r234409 || r234412;
        double r234414 = 0.0692910599291889;
        double r234415 = y;
        double r234416 = 0.07512208616047561;
        double r234417 = r234415 / r234407;
        double r234418 = x;
        double r234419 = fma(r234416, r234417, r234418);
        double r234420 = fma(r234414, r234415, r234419);
        double r234421 = 0.4917317610505968;
        double r234422 = fma(r234407, r234414, r234421);
        double r234423 = 0.279195317918525;
        double r234424 = fma(r234407, r234422, r234423);
        double r234425 = 6.012459259764103;
        double r234426 = r234407 + r234425;
        double r234427 = 3.350343815022304;
        double r234428 = fma(r234426, r234407, r234427);
        double r234429 = r234424 / r234428;
        double r234430 = r234415 * r234429;
        double r234431 = r234430 + r234418;
        double r234432 = r234413 ? r234420 : r234431;
        return r234432;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	20.4
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 < -4.051299145905559e+35 or 188100201.10508826 < z

   1. Initial program 43.2

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

   2. Simplified 35.9

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

   4. Applied clear-num 36.1

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

   6. Applied fma-udef 36.1

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

   7. Simplified 35.9

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

   8. Taylor expanded around inf 0.0

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

   9. Simplified 0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.06929105992918889456166908757950295694172, y, \mathsf{fma}\left(0.07512208616047560960637952121032867580652, \frac{y}{z}, x\right)\right)}\]

   if -4.051299145905559e+35 < z < 188100201.10508826

   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. Simplified 0.1

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

   4. Applied clear-num 0.2

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

   6. Applied fma-udef 0.2

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

   7. Simplified 0.1

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

   9. Applied div-inv 0.2

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

   10. Applied associate-*l* 0.4

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

   11. Simplified 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -405129914590555930114134145845166080 \lor \neg \left(z \le 188100201.1050882637500762939453125\right):\\ \;\;\;\;\mathsf{fma}\left(0.06929105992918889456166908757950295694172, y, \mathsf{fma}\left(0.07512208616047560960637952121032867580652, \frac{y}{z}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), 0.2791953179185249767080279070796677842736\right)}{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +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))))