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

Average Error: 20.5 → 0.5

Time: 4.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1766059654986.306884765625:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.07512208616047560960637952121032867580652}{z}, y, \mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, x\right)\right)\\ \mathbf{elif}\;z \le 9.349284559663863576071788271886616134097 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, 6.012459259764103336465268512256443500519, \mathsf{fma}\left(z, z, 3.350343815022303939343828460550867021084\right)\right)}, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.07512208616047560960637952121032867580652, \frac{y}{z}, 0.06929105992918889456166908757950295694172 \cdot y - 0.4046220386999212492717958866705885156989 \cdot \frac{y}{{z}^{2}}\right) + x\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r398380 = x;
        double r398381 = y;
        double r398382 = z;
        double r398383 = 0.0692910599291889;
        double r398384 = r398382 * r398383;
        double r398385 = 0.4917317610505968;
        double r398386 = r398384 + r398385;
        double r398387 = r398386 * r398382;
        double r398388 = 0.279195317918525;
        double r398389 = r398387 + r398388;
        double r398390 = r398381 * r398389;
        double r398391 = 6.012459259764103;
        double r398392 = r398382 + r398391;
        double r398393 = r398392 * r398382;
        double r398394 = 3.350343815022304;
        double r398395 = r398393 + r398394;
        double r398396 = r398390 / r398395;
        double r398397 = r398380 + r398396;
        return r398397;
}

↓

double f(double x, double y, double z) {
        double r398398 = z;
        double r398399 = -1766059654986.307;
        bool r398400 = r398398 <= r398399;
        double r398401 = 0.07512208616047561;
        double r398402 = r398401 / r398398;
        double r398403 = y;
        double r398404 = 0.0692910599291889;
        double r398405 = x;
        double r398406 = fma(r398403, r398404, r398405);
        double r398407 = fma(r398402, r398403, r398406);
        double r398408 = 9.349284559663864e-13;
        bool r398409 = r398398 <= r398408;
        double r398410 = 6.012459259764103;
        double r398411 = 3.350343815022304;
        double r398412 = fma(r398398, r398398, r398411);
        double r398413 = fma(r398398, r398410, r398412);
        double r398414 = r398403 / r398413;
        double r398415 = 0.4917317610505968;
        double r398416 = fma(r398398, r398404, r398415);
        double r398417 = 0.279195317918525;
        double r398418 = fma(r398416, r398398, r398417);
        double r398419 = fma(r398414, r398418, r398405);
        double r398420 = r398403 / r398398;
        double r398421 = r398404 * r398403;
        double r398422 = 0.40462203869992125;
        double r398423 = 2.0;
        double r398424 = pow(r398398, r398423);
        double r398425 = r398403 / r398424;
        double r398426 = r398422 * r398425;
        double r398427 = r398421 - r398426;
        double r398428 = fma(r398401, r398420, r398427);
        double r398429 = r398428 + r398405;
        double r398430 = r398409 ? r398419 : r398429;
        double r398431 = r398400 ? r398407 : r398430;
        return r398431;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	20.5
Target	0.2
Herbie	0.5

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

Derivation

1. Split input into 3 regimes

2. if z < -1766059654986.307

   1. Initial program 41.8

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

   2. Simplified 35.0

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

   3. Taylor expanded around inf 0.0

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

   4. Simplified 0.0

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

   if -1766059654986.307 < z < 9.349284559663864e-13

   1. Initial program 0.2

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

   2. Simplified 0.1

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

   3. Taylor expanded around 0 0.1

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

   4. Simplified 0.1

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

   if 9.349284559663864e-13 < z

   1. Initial program 38.9

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

   2. Simplified 32.9

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

   3. Taylor expanded around 0 32.9

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

   4. Simplified 32.9

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

   6. Applied clear-num 32.9

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

   8. Applied fma-udef 32.9

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

   9. Simplified 32.9

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

   10. Taylor expanded around inf 1.5

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

   11. Simplified 1.5

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.07512208616047560960637952121032867580652, \frac{y}{z}, 0.06929105992918889456166908757950295694172 \cdot y - 0.4046220386999212492717958866705885156989 \cdot \frac{y}{{z}^{2}}\right)} + x\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1766059654986.306884765625:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.07512208616047560960637952121032867580652}{z}, y, \mathsf{fma}\left(y, 0.06929105992918889456166908757950295694172, x\right)\right)\\ \mathbf{elif}\;z \le 9.349284559663863576071788271886616134097 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, 6.012459259764103336465268512256443500519, \mathsf{fma}\left(z, z, 3.350343815022303939343828460550867021084\right)\right)}, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.07512208616047560960637952121032867580652, \frac{y}{z}, 0.06929105992918889456166908757950295694172 \cdot y - 0.4046220386999212492717958866705885156989 \cdot \frac{y}{{z}^{2}}\right) + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 +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))))