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

Average Error: 26.5 → 0.6

Time: 22.5s

Precision: 64

Math

\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -5.657540384046955638503103437127492884033 \cdot 10^{70} \lor \neg \left(x \le 2.326727261056874248511482631265264838987 \cdot 10^{74}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{else}:\\ \;\;\;\;\left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r287012 = x;
        double r287013 = 2.0;
        double r287014 = r287012 - r287013;
        double r287015 = 4.16438922228;
        double r287016 = r287012 * r287015;
        double r287017 = 78.6994924154;
        double r287018 = r287016 + r287017;
        double r287019 = r287018 * r287012;
        double r287020 = 137.519416416;
        double r287021 = r287019 + r287020;
        double r287022 = r287021 * r287012;
        double r287023 = y;
        double r287024 = r287022 + r287023;
        double r287025 = r287024 * r287012;
        double r287026 = z;
        double r287027 = r287025 + r287026;
        double r287028 = r287014 * r287027;
        double r287029 = 43.3400022514;
        double r287030 = r287012 + r287029;
        double r287031 = r287030 * r287012;
        double r287032 = 263.505074721;
        double r287033 = r287031 + r287032;
        double r287034 = r287033 * r287012;
        double r287035 = 313.399215894;
        double r287036 = r287034 + r287035;
        double r287037 = r287036 * r287012;
        double r287038 = 47.066876606;
        double r287039 = r287037 + r287038;
        double r287040 = r287028 / r287039;
        return r287040;
}

↓

double f(double x, double y, double z) {
        double r287041 = x;
        double r287042 = -5.657540384046956e+70;
        bool r287043 = r287041 <= r287042;
        double r287044 = 2.326727261056874e+74;
        bool r287045 = r287041 <= r287044;
        double r287046 = !r287045;
        bool r287047 = r287043 || r287046;
        double r287048 = y;
        double r287049 = 2.0;
        double r287050 = pow(r287041, r287049);
        double r287051 = r287048 / r287050;
        double r287052 = 4.16438922228;
        double r287053 = r287052 * r287041;
        double r287054 = r287051 + r287053;
        double r287055 = 110.1139242984811;
        double r287056 = r287054 - r287055;
        double r287057 = 2.0;
        double r287058 = r287041 - r287057;
        double r287059 = r287041 * r287052;
        double r287060 = 78.6994924154;
        double r287061 = r287059 + r287060;
        double r287062 = r287061 * r287041;
        double r287063 = 137.519416416;
        double r287064 = r287062 + r287063;
        double r287065 = r287064 * r287041;
        double r287066 = r287065 + r287048;
        double r287067 = r287066 * r287041;
        double r287068 = z;
        double r287069 = r287067 + r287068;
        double r287070 = 43.3400022514;
        double r287071 = r287041 + r287070;
        double r287072 = r287071 * r287041;
        double r287073 = 263.505074721;
        double r287074 = r287072 + r287073;
        double r287075 = r287074 * r287041;
        double r287076 = 313.399215894;
        double r287077 = r287075 + r287076;
        double r287078 = r287077 * r287041;
        double r287079 = 47.066876606;
        double r287080 = r287078 + r287079;
        double r287081 = r287069 / r287080;
        double r287082 = r287058 * r287081;
        double r287083 = r287047 ? r287056 : r287082;
        return r287083;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	26.5
Target	0.5
Herbie	0.6

\[\begin{array}{l} \mathbf{if}\;x \lt -3.326128725870004842699683658678411714981 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{elif}\;x \lt 9.429991714554672672712552870340896976735 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.5050747210000281484099105000495910645 \cdot x + \left(43.3400022514000013984514225739985704422 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if x < -5.657540384046956e+70 or 2.326727261056874e+74 < x

   1. Initial program 64.0

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]

   2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229}\]

   if -5.657540384046956e+70 < x < 2.326727261056874e+74

   1. Initial program 3.5

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]
   2. Using strategy rm

   3. Applied associate-/l* 1.1

      \[\leadsto \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}}}\]
   4. Using strategy rm

   5. Applied associate-/r/ 1.2

      \[\leadsto \color{blue}{\frac{x - 2}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825} \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}\]
   6. Using strategy rm

   7. Applied div-inv 1.2

      \[\leadsto \color{blue}{\left(\left(x - 2\right) \cdot \frac{1}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\right)} \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)\]

   8. Applied associate-*l* 1.2

      \[\leadsto \color{blue}{\left(x - 2\right) \cdot \left(\frac{1}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825} \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)\right)}\]

   9. Simplified 0.9

      \[\leadsto \left(x - 2\right) \cdot \color{blue}{\frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.657540384046955638503103437127492884033 \cdot 10^{70} \lor \neg \left(x \le 2.326727261056874248511482631265264838987 \cdot 10^{74}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{else}:\\ \;\;\;\;\left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C"
  :precision binary64

  :herbie-target
  (if (< x -3.3261287258700048e62) (- (+ (/ y (* x x)) (* 4.16438922227999964 x)) 110.11392429848109) (if (< x 9.4299917145546727e55) (* (/ (- x 2) 1) (/ (+ (* (+ (* (+ (* (+ (* x 4.16438922227999964) 78.6994924154000017) x) 137.51941641600001) x) y) x) z) (+ (* (+ (+ (* 263.50507472100003 x) (+ (* 43.3400022514000014 (* x x)) (* x (* x x)))) 313.399215894) x) 47.066876606000001))) (- (+ (/ y (* x x)) (* 4.16438922227999964 x)) 110.11392429848109)))

  (/ (* (- x 2) (+ (* (+ (* (+ (* (+ (* x 4.16438922227999964) 78.6994924154000017) x) 137.51941641600001) x) y) x) z)) (+ (* (+ (* (+ (* (+ x 43.3400022514000014) x) 263.50507472100003) x) 313.399215894) x) 47.066876606000001)))