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

Average Error: 20.1 → 0.1

Time: 19.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -95371158.39650727808475494384765625:\\ \;\;\;\;\mathsf{fma}\left(0.06929105992918889456166908757950295694172, y, \mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, x\right)\right)\\ \mathbf{elif}\;z \le 114313904.497305810451507568359375:\\ \;\;\;\;x + \frac{y \cdot \left(z \cdot \left(0.06929105992918889456166908757950295694172 \cdot z + 0.4917317610505967939715787906607147306204\right) + 0.2791953179185249767080279070796677842736\right)}{3.350343815022303939343828460550867021084 + z \cdot \left(z + 6.012459259764103336465268512256443500519\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.06929105992918889456166908757950295694172, y, \mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, x\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r17418892 = x;
        double r17418893 = y;
        double r17418894 = z;
        double r17418895 = 0.0692910599291889;
        double r17418896 = r17418894 * r17418895;
        double r17418897 = 0.4917317610505968;
        double r17418898 = r17418896 + r17418897;
        double r17418899 = r17418898 * r17418894;
        double r17418900 = 0.279195317918525;
        double r17418901 = r17418899 + r17418900;
        double r17418902 = r17418893 * r17418901;
        double r17418903 = 6.012459259764103;
        double r17418904 = r17418894 + r17418903;
        double r17418905 = r17418904 * r17418894;
        double r17418906 = 3.350343815022304;
        double r17418907 = r17418905 + r17418906;
        double r17418908 = r17418902 / r17418907;
        double r17418909 = r17418892 + r17418908;
        return r17418909;
}

↓

double f(double x, double y, double z) {
        double r17418910 = z;
        double r17418911 = -95371158.39650728;
        bool r17418912 = r17418910 <= r17418911;
        double r17418913 = 0.0692910599291889;
        double r17418914 = y;
        double r17418915 = r17418914 / r17418910;
        double r17418916 = 0.07512208616047561;
        double r17418917 = x;
        double r17418918 = fma(r17418915, r17418916, r17418917);
        double r17418919 = fma(r17418913, r17418914, r17418918);
        double r17418920 = 114313904.49730581;
        bool r17418921 = r17418910 <= r17418920;
        double r17418922 = r17418913 * r17418910;
        double r17418923 = 0.4917317610505968;
        double r17418924 = r17418922 + r17418923;
        double r17418925 = r17418910 * r17418924;
        double r17418926 = 0.279195317918525;
        double r17418927 = r17418925 + r17418926;
        double r17418928 = r17418914 * r17418927;
        double r17418929 = 3.350343815022304;
        double r17418930 = 6.012459259764103;
        double r17418931 = r17418910 + r17418930;
        double r17418932 = r17418910 * r17418931;
        double r17418933 = r17418929 + r17418932;
        double r17418934 = r17418928 / r17418933;
        double r17418935 = r17418917 + r17418934;
        double r17418936 = r17418921 ? r17418935 : r17418919;
        double r17418937 = r17418912 ? r17418919 : r17418936;
        return r17418937;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	20.1
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 < -95371158.39650728 or 114313904.49730581 < z

   1. Initial program 40.6

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

   2. Simplified 33.7

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

   3. Taylor expanded around 0 33.8

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

   4. Simplified 33.7

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

   6. Applied clear-num 33.8

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

   8. Applied div-inv 33.8

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

   9. Applied associate-/r* 33.7

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

   10. Taylor expanded around inf 0.0

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

   11. Simplified 0.0

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

   if -95371158.39650728 < z < 114313904.49730581

   1. Initial program 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -95371158.39650727808475494384765625:\\ \;\;\;\;\mathsf{fma}\left(0.06929105992918889456166908757950295694172, y, \mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, x\right)\right)\\ \mathbf{elif}\;z \le 114313904.497305810451507568359375:\\ \;\;\;\;x + \frac{y \cdot \left(z \cdot \left(0.06929105992918889456166908757950295694172 \cdot z + 0.4917317610505967939715787906607147306204\right) + 0.2791953179185249767080279070796677842736\right)}{3.350343815022303939343828460550867021084 + z \cdot \left(z + 6.012459259764103336465268512256443500519\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.06929105992918889456166908757950295694172, y, \mathsf{fma}\left(\frac{y}{z}, 0.07512208616047560960637952121032867580652, x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(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))))