Average Error: 26.3 → 0.8
Time: 18.6s
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4892908567716605206340181583462400 \lor \neg \left(x \le 14213415011424090370543837758445552842830000\right):\\ \;\;\;\;\mathsf{fma}\left(4.16438922227999963610045597306452691555, x, \frac{y}{{x}^{2}} - 110.1139242984810948655649553984403610229\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000013984514225739985704422, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), x, 137.5194164160000127594685181975364685059\right), x, y\right), x, z\right)}}{x - 2}}\\ \end{array}\]
\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 -4892908567716605206340181583462400 \lor \neg \left(x \le 14213415011424090370543837758445552842830000\right):\\
\;\;\;\;\mathsf{fma}\left(4.16438922227999963610045597306452691555, x, \frac{y}{{x}^{2}} - 110.1139242984810948655649553984403610229\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000013984514225739985704422, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), x, 137.5194164160000127594685181975364685059\right), x, y\right), x, z\right)}}{x - 2}}\\

\end{array}
double f(double x, double y, double z) {
        double r441516 = x;
        double r441517 = 2.0;
        double r441518 = r441516 - r441517;
        double r441519 = 4.16438922228;
        double r441520 = r441516 * r441519;
        double r441521 = 78.6994924154;
        double r441522 = r441520 + r441521;
        double r441523 = r441522 * r441516;
        double r441524 = 137.519416416;
        double r441525 = r441523 + r441524;
        double r441526 = r441525 * r441516;
        double r441527 = y;
        double r441528 = r441526 + r441527;
        double r441529 = r441528 * r441516;
        double r441530 = z;
        double r441531 = r441529 + r441530;
        double r441532 = r441518 * r441531;
        double r441533 = 43.3400022514;
        double r441534 = r441516 + r441533;
        double r441535 = r441534 * r441516;
        double r441536 = 263.505074721;
        double r441537 = r441535 + r441536;
        double r441538 = r441537 * r441516;
        double r441539 = 313.399215894;
        double r441540 = r441538 + r441539;
        double r441541 = r441540 * r441516;
        double r441542 = 47.066876606;
        double r441543 = r441541 + r441542;
        double r441544 = r441532 / r441543;
        return r441544;
}
↓
double f(double x, double y, double z) {
        double r441545 = x;
        double r441546 = -4.892908567716605e+33;
        bool r441547 = r441545 <= r441546;
        double r441548 = 1.421341501142409e+43;
        bool r441549 = r441545 <= r441548;
        double r441550 = !r441549;
        bool r441551 = r441547 || r441550;
        double r441552 = 4.16438922228;
        double r441553 = y;
        double r441554 = 2.0;
        double r441555 = pow(r441545, r441554);
        double r441556 = r441553 / r441555;
        double r441557 = 110.1139242984811;
        double r441558 = r441556 - r441557;
        double r441559 = fma(r441552, r441545, r441558);
        double r441560 = 1.0;
        double r441561 = 43.3400022514;
        double r441562 = r441545 + r441561;
        double r441563 = 263.505074721;
        double r441564 = fma(r441562, r441545, r441563);
        double r441565 = 313.399215894;
        double r441566 = fma(r441564, r441545, r441565);
        double r441567 = 47.066876606;
        double r441568 = fma(r441566, r441545, r441567);
        double r441569 = 78.6994924154;
        double r441570 = fma(r441545, r441552, r441569);
        double r441571 = 137.519416416;
        double r441572 = fma(r441570, r441545, r441571);
        double r441573 = fma(r441572, r441545, r441553);
        double r441574 = z;
        double r441575 = fma(r441573, r441545, r441574);
        double r441576 = r441568 / r441575;
        double r441577 = 2.0;
        double r441578 = r441545 - r441577;
        double r441579 = r441576 / r441578;
        double r441580 = r441560 / r441579;
        double r441581 = r441551 ? r441559 : r441580;
        return r441581;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original26.3
Target0.4
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;x \lt -3.326128725870004842699683658678411714981 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{elif}\;x \lt 9.429991714554672672712552870340896976735 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.5050747210000281484099105000495910645 \cdot x + \left(43.3400022514000013984514225739985704422 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -4.892908567716605e+33 or 1.421341501142409e+43 < x

    1. Initial program 60.0

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

    2. Taylor expanded around inf 1.1

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

    3. Simplified1.1

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

    if -4.892908567716605e+33 < x < 1.421341501142409e+43

    1. Initial program 0.8

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

    2. Simplified0.6

      \[\leadsto \color{blue}{\frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000013984514225739985704422, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), x, 137.5194164160000127594685181975364685059\right), x, y\right), x, z\right)}}}\]
    3. Using strategy rm

    4. Applied clear-num0.6

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000013984514225739985704422, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), x, 137.5194164160000127594685181975364685059\right), x, y\right), x, z\right)}}{x - 2}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4892908567716605206340181583462400 \lor \neg \left(x \le 14213415011424090370543837758445552842830000\right):\\ \;\;\;\;\mathsf{fma}\left(4.16438922227999963610045597306452691555, x, \frac{y}{{x}^{2}} - 110.1139242984810948655649553984403610229\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000013984514225739985704422, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), x, 137.5194164160000127594685181975364685059\right), x, y\right), x, z\right)}}{x - 2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 +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)))