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

Average Error: 26.7 → 0.9

Time: 20.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -32986745579037009546076749824 \lor \neg \left(x \le 2501382024595968964723726090240\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\frac{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x\right) \cdot \left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x\right) - 263.5050747210000281484099105000495910645 \cdot 263.5050747210000281484099105000495910645\right) \cdot x}{\left(x + 43.3400022514000013984514225739985704422\right) \cdot x - 263.5050747210000281484099105000495910645} + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r284536 = x;
        double r284537 = 2.0;
        double r284538 = r284536 - r284537;
        double r284539 = 4.16438922228;
        double r284540 = r284536 * r284539;
        double r284541 = 78.6994924154;
        double r284542 = r284540 + r284541;
        double r284543 = r284542 * r284536;
        double r284544 = 137.519416416;
        double r284545 = r284543 + r284544;
        double r284546 = r284545 * r284536;
        double r284547 = y;
        double r284548 = r284546 + r284547;
        double r284549 = r284548 * r284536;
        double r284550 = z;
        double r284551 = r284549 + r284550;
        double r284552 = r284538 * r284551;
        double r284553 = 43.3400022514;
        double r284554 = r284536 + r284553;
        double r284555 = r284554 * r284536;
        double r284556 = 263.505074721;
        double r284557 = r284555 + r284556;
        double r284558 = r284557 * r284536;
        double r284559 = 313.399215894;
        double r284560 = r284558 + r284559;
        double r284561 = r284560 * r284536;
        double r284562 = 47.066876606;
        double r284563 = r284561 + r284562;
        double r284564 = r284552 / r284563;
        return r284564;
}

↓

double f(double x, double y, double z) {
        double r284565 = x;
        double r284566 = -3.298674557903701e+28;
        bool r284567 = r284565 <= r284566;
        double r284568 = 2.501382024595969e+30;
        bool r284569 = r284565 <= r284568;
        double r284570 = !r284569;
        bool r284571 = r284567 || r284570;
        double r284572 = y;
        double r284573 = 2.0;
        double r284574 = pow(r284565, r284573);
        double r284575 = r284572 / r284574;
        double r284576 = 4.16438922228;
        double r284577 = r284576 * r284565;
        double r284578 = r284575 + r284577;
        double r284579 = 110.1139242984811;
        double r284580 = r284578 - r284579;
        double r284581 = 2.0;
        double r284582 = r284565 - r284581;
        double r284583 = r284565 * r284576;
        double r284584 = 78.6994924154;
        double r284585 = r284583 + r284584;
        double r284586 = r284585 * r284565;
        double r284587 = 137.519416416;
        double r284588 = r284586 + r284587;
        double r284589 = r284588 * r284565;
        double r284590 = r284589 + r284572;
        double r284591 = r284590 * r284565;
        double r284592 = z;
        double r284593 = r284591 + r284592;
        double r284594 = r284582 * r284593;
        double r284595 = 43.3400022514;
        double r284596 = r284565 + r284595;
        double r284597 = r284596 * r284565;
        double r284598 = r284597 * r284597;
        double r284599 = 263.505074721;
        double r284600 = r284599 * r284599;
        double r284601 = r284598 - r284600;
        double r284602 = r284601 * r284565;
        double r284603 = r284597 - r284599;
        double r284604 = r284602 / r284603;
        double r284605 = 313.399215894;
        double r284606 = r284604 + r284605;
        double r284607 = r284606 * r284565;
        double r284608 = 47.066876606;
        double r284609 = r284607 + r284608;
        double r284610 = r284594 / r284609;
        double r284611 = r284571 ? r284580 : r284610;
        return r284611;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	26.7
Target	0.5
Herbie	0.9

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

Derivation

1. Split input into 2 regimes

2. if x < -3.298674557903701e+28 or 2.501382024595969e+30 < x

   1. Initial program 58.5

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

   2. Taylor expanded around inf 1.2

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

   if -3.298674557903701e+28 < x < 2.501382024595969e+30

   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

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

   4. Applied associate-*l/ 0.6

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -32986745579037009546076749824 \lor \neg \left(x \le 2501382024595968964723726090240\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\frac{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x\right) \cdot \left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x\right) - 263.5050747210000281484099105000495910645 \cdot 263.5050747210000281484099105000495910645\right) \cdot x}{\left(x + 43.3400022514000013984514225739985704422\right) \cdot x - 263.5050747210000281484099105000495910645} + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 
(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)))