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

Average Error: 27.5 → 0.5

Time: 12.2s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -3.67035503993276071 \cdot 10^{43} \lor \neg \left(x \le 3.7248869126440915 \cdot 10^{43}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{else}:\\ \;\;\;\;\left(x - 2\right) \cdot \frac{\left(\frac{\left({\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right)}^{3} + {137.51941641600001}^{3}\right) \cdot x}{\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) + \left(137.51941641600001 \cdot 137.51941641600001 - \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot 137.51941641600001\right)} + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r1093012 = x;
        double r1093013 = 2.0;
        double r1093014 = r1093012 - r1093013;
        double r1093015 = 4.16438922228;
        double r1093016 = r1093012 * r1093015;
        double r1093017 = 78.6994924154;
        double r1093018 = r1093016 + r1093017;
        double r1093019 = r1093018 * r1093012;
        double r1093020 = 137.519416416;
        double r1093021 = r1093019 + r1093020;
        double r1093022 = r1093021 * r1093012;
        double r1093023 = y;
        double r1093024 = r1093022 + r1093023;
        double r1093025 = r1093024 * r1093012;
        double r1093026 = z;
        double r1093027 = r1093025 + r1093026;
        double r1093028 = r1093014 * r1093027;
        double r1093029 = 43.3400022514;
        double r1093030 = r1093012 + r1093029;
        double r1093031 = r1093030 * r1093012;
        double r1093032 = 263.505074721;
        double r1093033 = r1093031 + r1093032;
        double r1093034 = r1093033 * r1093012;
        double r1093035 = 313.399215894;
        double r1093036 = r1093034 + r1093035;
        double r1093037 = r1093036 * r1093012;
        double r1093038 = 47.066876606;
        double r1093039 = r1093037 + r1093038;
        double r1093040 = r1093028 / r1093039;
        return r1093040;
}

↓

double f(double x, double y, double z) {
        double r1093041 = x;
        double r1093042 = -3.6703550399327607e+43;
        bool r1093043 = r1093041 <= r1093042;
        double r1093044 = 3.7248869126440915e+43;
        bool r1093045 = r1093041 <= r1093044;
        double r1093046 = !r1093045;
        bool r1093047 = r1093043 || r1093046;
        double r1093048 = y;
        double r1093049 = 2.0;
        double r1093050 = pow(r1093041, r1093049);
        double r1093051 = r1093048 / r1093050;
        double r1093052 = 4.16438922228;
        double r1093053 = r1093052 * r1093041;
        double r1093054 = r1093051 + r1093053;
        double r1093055 = 110.1139242984811;
        double r1093056 = r1093054 - r1093055;
        double r1093057 = 2.0;
        double r1093058 = r1093041 - r1093057;
        double r1093059 = r1093041 * r1093052;
        double r1093060 = 78.6994924154;
        double r1093061 = r1093059 + r1093060;
        double r1093062 = r1093061 * r1093041;
        double r1093063 = 3.0;
        double r1093064 = pow(r1093062, r1093063);
        double r1093065 = 137.519416416;
        double r1093066 = pow(r1093065, r1093063);
        double r1093067 = r1093064 + r1093066;
        double r1093068 = r1093067 * r1093041;
        double r1093069 = r1093062 * r1093062;
        double r1093070 = r1093065 * r1093065;
        double r1093071 = r1093062 * r1093065;
        double r1093072 = r1093070 - r1093071;
        double r1093073 = r1093069 + r1093072;
        double r1093074 = r1093068 / r1093073;
        double r1093075 = r1093074 + r1093048;
        double r1093076 = r1093075 * r1093041;
        double r1093077 = z;
        double r1093078 = r1093076 + r1093077;
        double r1093079 = 43.3400022514;
        double r1093080 = r1093041 + r1093079;
        double r1093081 = r1093080 * r1093041;
        double r1093082 = 263.505074721;
        double r1093083 = r1093081 + r1093082;
        double r1093084 = r1093083 * r1093041;
        double r1093085 = 313.399215894;
        double r1093086 = r1093084 + r1093085;
        double r1093087 = r1093086 * r1093041;
        double r1093088 = 47.066876606;
        double r1093089 = r1093087 + r1093088;
        double r1093090 = r1093078 / r1093089;
        double r1093091 = r1093058 * r1093090;
        double r1093092 = r1093047 ? r1093056 : r1093091;
        return r1093092;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	27.5
Target	0.5
Herbie	0.5

\[\begin{array}{l} \mathbf{if}\;x \lt -3.3261287258700048 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{elif}\;x \lt 9.4299917145546727 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.50507472100003 \cdot x + \left(43.3400022514000014 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.399215894\right) \cdot x + 47.066876606000001}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if x < -3.6703550399327607e+43 or 3.7248869126440915e+43 < x

   1. Initial program 61.0

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]

   2. Taylor expanded around inf 0.6

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109}\]

   if -3.6703550399327607e+43 < x < 3.7248869126440915e+43

   1. Initial program 1.1

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 1.1

      \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\color{blue}{1 \cdot \left(\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001\right)}}\]

   4. Applied times-frac 0.4

      \[\leadsto \color{blue}{\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}}\]

   5. Simplified 0.4

      \[\leadsto \color{blue}{\left(x - 2\right)} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
   6. Using strategy rm

   7. Applied flip3-+ 0.5

      \[\leadsto \left(x - 2\right) \cdot \frac{\left(\color{blue}{\frac{{\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right)}^{3} + {137.51941641600001}^{3}}{\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) + \left(137.51941641600001 \cdot 137.51941641600001 - \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot 137.51941641600001\right)}} \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]

   8. Applied associate-*l/ 0.5

      \[\leadsto \left(x - 2\right) \cdot \frac{\left(\color{blue}{\frac{\left({\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right)}^{3} + {137.51941641600001}^{3}\right) \cdot x}{\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) + \left(137.51941641600001 \cdot 137.51941641600001 - \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot 137.51941641600001\right)}} + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.67035503993276071 \cdot 10^{43} \lor \neg \left(x \le 3.7248869126440915 \cdot 10^{43}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{else}:\\ \;\;\;\;\left(x - 2\right) \cdot \frac{\left(\frac{\left({\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right)}^{3} + {137.51941641600001}^{3}\right) \cdot x}{\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) + \left(137.51941641600001 \cdot 137.51941641600001 - \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot 137.51941641600001\right)} + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C"
  :precision binary64

  :herbie-target
  (if (< x -3.326128725870005e+62) (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811) (if (< x 9.429991714554673e+55) (* (/ (- x 2) 1) (/ (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z) (+ (* (+ (+ (* 263.505074721 x) (+ (* 43.3400022514 (* x x)) (* x (* x x)))) 313.399215894) x) 47.066876606))) (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811)))

  (/ (* (- x 2) (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z)) (+ (* (+ (* (+ (* (+ x 43.3400022514) x) 263.505074721) x) 313.399215894) x) 47.066876606)))