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

Average Error: 26.7 → 1.0

Time: 20.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -34042328811361574912 \lor \neg \left(x \le 1.079429159403752641178373627064126450738 \cdot 10^{44}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\frac{\left(\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x\right) \cdot \left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x\right) - 313.3992158940000081202015280723571777344 \cdot 313.3992158940000081202015280723571777344\right) \cdot x}{\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x - 313.3992158940000081202015280723571777344} + 47.06687660600000100430406746454536914825}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r272552 = x;
        double r272553 = 2.0;
        double r272554 = r272552 - r272553;
        double r272555 = 4.16438922228;
        double r272556 = r272552 * r272555;
        double r272557 = 78.6994924154;
        double r272558 = r272556 + r272557;
        double r272559 = r272558 * r272552;
        double r272560 = 137.519416416;
        double r272561 = r272559 + r272560;
        double r272562 = r272561 * r272552;
        double r272563 = y;
        double r272564 = r272562 + r272563;
        double r272565 = r272564 * r272552;
        double r272566 = z;
        double r272567 = r272565 + r272566;
        double r272568 = r272554 * r272567;
        double r272569 = 43.3400022514;
        double r272570 = r272552 + r272569;
        double r272571 = r272570 * r272552;
        double r272572 = 263.505074721;
        double r272573 = r272571 + r272572;
        double r272574 = r272573 * r272552;
        double r272575 = 313.399215894;
        double r272576 = r272574 + r272575;
        double r272577 = r272576 * r272552;
        double r272578 = 47.066876606;
        double r272579 = r272577 + r272578;
        double r272580 = r272568 / r272579;
        return r272580;
}

↓

double f(double x, double y, double z) {
        double r272581 = x;
        double r272582 = -3.4042328811361575e+19;
        bool r272583 = r272581 <= r272582;
        double r272584 = 1.0794291594037526e+44;
        bool r272585 = r272581 <= r272584;
        double r272586 = !r272585;
        bool r272587 = r272583 || r272586;
        double r272588 = y;
        double r272589 = 2.0;
        double r272590 = pow(r272581, r272589);
        double r272591 = r272588 / r272590;
        double r272592 = 4.16438922228;
        double r272593 = r272592 * r272581;
        double r272594 = r272591 + r272593;
        double r272595 = 110.1139242984811;
        double r272596 = r272594 - r272595;
        double r272597 = 2.0;
        double r272598 = r272581 - r272597;
        double r272599 = r272581 * r272592;
        double r272600 = 78.6994924154;
        double r272601 = r272599 + r272600;
        double r272602 = r272601 * r272581;
        double r272603 = 137.519416416;
        double r272604 = r272602 + r272603;
        double r272605 = r272604 * r272581;
        double r272606 = r272605 + r272588;
        double r272607 = r272606 * r272581;
        double r272608 = z;
        double r272609 = r272607 + r272608;
        double r272610 = r272598 * r272609;
        double r272611 = 43.3400022514;
        double r272612 = r272581 + r272611;
        double r272613 = r272612 * r272581;
        double r272614 = 263.505074721;
        double r272615 = r272613 + r272614;
        double r272616 = r272615 * r272581;
        double r272617 = r272616 * r272616;
        double r272618 = 313.399215894;
        double r272619 = r272618 * r272618;
        double r272620 = r272617 - r272619;
        double r272621 = r272620 * r272581;
        double r272622 = r272616 - r272618;
        double r272623 = r272621 / r272622;
        double r272624 = 47.066876606;
        double r272625 = r272623 + r272624;
        double r272626 = r272610 / r272625;
        double r272627 = r272587 ? r272596 : r272626;
        return r272627;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	26.7
Target	0.5
Herbie	1.0

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

Derivation

1. Split input into 2 regimes

2. if x < -3.4042328811361575e+19 or 1.0794291594037526e+44 < x

   1. Initial program 58.6

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

   2. Taylor expanded around inf 1.6

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

   if -3.4042328811361575e+19 < x < 1.0794291594037526e+44

   1. Initial program 0.6

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

   3. Applied flip-+ 0.6

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

   4. Applied associate-*l/ 0.6

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

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -34042328811361574912 \lor \neg \left(x \le 1.079429159403752641178373627064126450738 \cdot 10^{44}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\frac{\left(\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x\right) \cdot \left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x\right) - 313.3992158940000081202015280723571777344 \cdot 313.3992158940000081202015280723571777344\right) \cdot x}{\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x - 313.3992158940000081202015280723571777344} + 47.06687660600000100430406746454536914825}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(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)))