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

Average Error: 27.5 → 0.5

Time: 10.9s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -3.2249422671409879 \cdot 10^{47} \lor \neg \left(x \le 3.6379943132031436 \cdot 10^{44}\right):\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}}\right) - 110.11392429848109\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999964, 78.6994924154000017\right), x, 137.51941641600001\right), x, y\right), x, z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000014, x, 263.50507472100003\right), x, 313.399215894\right), x, 47.066876606000001\right)} \cdot \left(x - 2\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r381067 = x;
        double r381068 = 2.0;
        double r381069 = r381067 - r381068;
        double r381070 = 4.16438922228;
        double r381071 = r381067 * r381070;
        double r381072 = 78.6994924154;
        double r381073 = r381071 + r381072;
        double r381074 = r381073 * r381067;
        double r381075 = 137.519416416;
        double r381076 = r381074 + r381075;
        double r381077 = r381076 * r381067;
        double r381078 = y;
        double r381079 = r381077 + r381078;
        double r381080 = r381079 * r381067;
        double r381081 = z;
        double r381082 = r381080 + r381081;
        double r381083 = r381069 * r381082;
        double r381084 = 43.3400022514;
        double r381085 = r381067 + r381084;
        double r381086 = r381085 * r381067;
        double r381087 = 263.505074721;
        double r381088 = r381086 + r381087;
        double r381089 = r381088 * r381067;
        double r381090 = 313.399215894;
        double r381091 = r381089 + r381090;
        double r381092 = r381091 * r381067;
        double r381093 = 47.066876606;
        double r381094 = r381092 + r381093;
        double r381095 = r381083 / r381094;
        return r381095;
}

↓

double f(double x, double y, double z) {
        double r381096 = x;
        double r381097 = -3.224942267140988e+47;
        bool r381098 = r381096 <= r381097;
        double r381099 = 3.6379943132031436e+44;
        bool r381100 = r381096 <= r381099;
        double r381101 = !r381100;
        bool r381102 = r381098 || r381101;
        double r381103 = 4.16438922228;
        double r381104 = y;
        double r381105 = 2.0;
        double r381106 = pow(r381096, r381105);
        double r381107 = r381104 / r381106;
        double r381108 = fma(r381096, r381103, r381107);
        double r381109 = 110.1139242984811;
        double r381110 = r381108 - r381109;
        double r381111 = 78.6994924154;
        double r381112 = fma(r381096, r381103, r381111);
        double r381113 = 137.519416416;
        double r381114 = fma(r381112, r381096, r381113);
        double r381115 = fma(r381114, r381096, r381104);
        double r381116 = z;
        double r381117 = fma(r381115, r381096, r381116);
        double r381118 = 43.3400022514;
        double r381119 = r381096 + r381118;
        double r381120 = 263.505074721;
        double r381121 = fma(r381119, r381096, r381120);
        double r381122 = 313.399215894;
        double r381123 = fma(r381121, r381096, r381122);
        double r381124 = 47.066876606;
        double r381125 = fma(r381123, r381096, r381124);
        double r381126 = r381117 / r381125;
        double r381127 = 2.0;
        double r381128 = r381096 - r381127;
        double r381129 = r381126 * r381128;
        double r381130 = r381102 ? r381110 : r381129;
        return r381130;
}

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 < -3.224942267140988e+47 or 3.6379943132031436e+44 < x

   1. Initial program 61.5

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

   2. Simplified 61.5

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

   3. Taylor expanded around inf 0.5

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

   4. Simplified 0.5

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

   if -3.224942267140988e+47 < x < 3.6379943132031436e+44

   1. Initial program 1.2

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

   2. Simplified 1.2

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

   4. Applied associate-/l* 0.4

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

   6. Applied associate-/r/ 0.4

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.2249422671409879 \cdot 10^{47} \lor \neg \left(x \le 3.6379943132031436 \cdot 10^{44}\right):\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}}\right) - 110.11392429848109\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999964, 78.6994924154000017\right), x, 137.51941641600001\right), x, y\right), x, z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000014, x, 263.50507472100003\right), x, 313.399215894\right), x, 47.066876606000001\right)} \cdot \left(x - 2\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(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)))