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

Average Error: 27.5 → 0.5

Time: 10.8s

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 r337398 = x;
        double r337399 = 2.0;
        double r337400 = r337398 - r337399;
        double r337401 = 4.16438922228;
        double r337402 = r337398 * r337401;
        double r337403 = 78.6994924154;
        double r337404 = r337402 + r337403;
        double r337405 = r337404 * r337398;
        double r337406 = 137.519416416;
        double r337407 = r337405 + r337406;
        double r337408 = r337407 * r337398;
        double r337409 = y;
        double r337410 = r337408 + r337409;
        double r337411 = r337410 * r337398;
        double r337412 = z;
        double r337413 = r337411 + r337412;
        double r337414 = r337400 * r337413;
        double r337415 = 43.3400022514;
        double r337416 = r337398 + r337415;
        double r337417 = r337416 * r337398;
        double r337418 = 263.505074721;
        double r337419 = r337417 + r337418;
        double r337420 = r337419 * r337398;
        double r337421 = 313.399215894;
        double r337422 = r337420 + r337421;
        double r337423 = r337422 * r337398;
        double r337424 = 47.066876606;
        double r337425 = r337423 + r337424;
        double r337426 = r337414 / r337425;
        return r337426;
}

↓

double f(double x, double y, double z) {
        double r337427 = x;
        double r337428 = -3.224942267140988e+47;
        bool r337429 = r337427 <= r337428;
        double r337430 = 3.6379943132031436e+44;
        bool r337431 = r337427 <= r337430;
        double r337432 = !r337431;
        bool r337433 = r337429 || r337432;
        double r337434 = 4.16438922228;
        double r337435 = y;
        double r337436 = 2.0;
        double r337437 = pow(r337427, r337436);
        double r337438 = r337435 / r337437;
        double r337439 = fma(r337427, r337434, r337438);
        double r337440 = 110.1139242984811;
        double r337441 = r337439 - r337440;
        double r337442 = 78.6994924154;
        double r337443 = fma(r337427, r337434, r337442);
        double r337444 = 137.519416416;
        double r337445 = fma(r337443, r337427, r337444);
        double r337446 = fma(r337445, r337427, r337435);
        double r337447 = z;
        double r337448 = fma(r337446, r337427, r337447);
        double r337449 = 43.3400022514;
        double r337450 = r337427 + r337449;
        double r337451 = 263.505074721;
        double r337452 = fma(r337450, r337427, r337451);
        double r337453 = 313.399215894;
        double r337454 = fma(r337452, r337427, r337453);
        double r337455 = 47.066876606;
        double r337456 = fma(r337454, r337427, r337455);
        double r337457 = r337448 / r337456;
        double r337458 = 2.0;
        double r337459 = r337427 - r337458;
        double r337460 = r337457 * r337459;
        double r337461 = r337433 ? r337441 : r337460;
        return r337461;
}

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)))