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

Average Error: 19.3 → 0.1

Time: 1.0m

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 r280131 = x;
        double r280132 = y;
        double r280133 = z;
        double r280134 = 0.0692910599291889;
        double r280135 = r280133 * r280134;
        double r280136 = 0.4917317610505968;
        double r280137 = r280135 + r280136;
        double r280138 = r280137 * r280133;
        double r280139 = 0.279195317918525;
        double r280140 = r280138 + r280139;
        double r280141 = r280132 * r280140;
        double r280142 = 6.012459259764103;
        double r280143 = r280133 + r280142;
        double r280144 = r280143 * r280133;
        double r280145 = 3.350343815022304;
        double r280146 = r280144 + r280145;
        double r280147 = r280141 / r280146;
        double r280148 = r280131 + r280147;
        return r280148;
}

↓

double f(double x, double y, double z) {
        double r280149 = z;
        double r280150 = -1.3610841356047813e+154;
        bool r280151 = r280149 <= r280150;
        double r280152 = 8120465736065972.0;
        bool r280153 = r280149 <= r280152;
        double r280154 = !r280153;
        bool r280155 = r280151 || r280154;
        double r280156 = y;
        double r280157 = 0.0692910599291889;
        double r280158 = r280156 * r280157;
        double r280159 = 0.07512208616047561;
        double r280160 = r280159 * r280156;
        double r280161 = r280160 / r280149;
        double r280162 = r280156 / r280149;
        double r280163 = r280162 / r280149;
        double r280164 = 0.40462203869992125;
        double r280165 = r280163 * r280164;
        double r280166 = r280161 - r280165;
        double r280167 = r280158 + r280166;
        double r280168 = x;
        double r280169 = r280167 + r280168;
        double r280170 = 0.279195317918525;
        double r280171 = 0.4917317610505968;
        double r280172 = r280149 * r280171;
        double r280173 = r280149 * r280149;
        double r280174 = r280157 * r280173;
        double r280175 = r280172 + r280174;
        double r280176 = r280170 + r280175;
        double r280177 = 3.350343815022304;
        double r280178 = 6.012459259764103;
        double r280179 = r280149 + r280178;
        double r280180 = r280179 * r280149;
        double r280181 = r280177 + r280180;
        double r280182 = r280176 / r280181;
        double r280183 = r280156 * r280182;
        double r280184 = r280183 + r280168;
        double r280185 = r280155 ? r280169 : r280184;
        return r280185;
}

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))))