Average Error: 26.5 → 0.8
Time: 11.4s
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.5558868413567101 \cdot 10^{22} \lor \neg \left(x \le 1.1453649274389147 \cdot 10^{36}\right):\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}}\right) - 110.113924298481081\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\frac{\left({\left(\mathsf{fma}\left(x, 4.16438922227999964, 78.6994924154000017\right) \cdot x\right)}^{3} + {137.51941641600001}^{3}\right) \cdot x}{\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) + \left(137.51941641600001 \cdot 137.51941641600001 - \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot 137.51941641600001\right)} + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\\ \end{array}\]
\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 -3.5558868413567101 \cdot 10^{22} \lor \neg \left(x \le 1.1453649274389147 \cdot 10^{36}\right):\\
\;\;\;\;\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}}\right) - 110.113924298481081\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\frac{\left({\left(\mathsf{fma}\left(x, 4.16438922227999964, 78.6994924154000017\right) \cdot x\right)}^{3} + {137.51941641600001}^{3}\right) \cdot x}{\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) + \left(137.51941641600001 \cdot 137.51941641600001 - \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot 137.51941641600001\right)} + 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}\\

\end{array}
double f(double x, double y, double z) {
        double r458452 = x;
        double r458453 = 2.0;
        double r458454 = r458452 - r458453;
        double r458455 = 4.16438922228;
        double r458456 = r458452 * r458455;
        double r458457 = 78.6994924154;
        double r458458 = r458456 + r458457;
        double r458459 = r458458 * r458452;
        double r458460 = 137.519416416;
        double r458461 = r458459 + r458460;
        double r458462 = r458461 * r458452;
        double r458463 = y;
        double r458464 = r458462 + r458463;
        double r458465 = r458464 * r458452;
        double r458466 = z;
        double r458467 = r458465 + r458466;
        double r458468 = r458454 * r458467;
        double r458469 = 43.3400022514;
        double r458470 = r458452 + r458469;
        double r458471 = r458470 * r458452;
        double r458472 = 263.505074721;
        double r458473 = r458471 + r458472;
        double r458474 = r458473 * r458452;
        double r458475 = 313.399215894;
        double r458476 = r458474 + r458475;
        double r458477 = r458476 * r458452;
        double r458478 = 47.066876606;
        double r458479 = r458477 + r458478;
        double r458480 = r458468 / r458479;
        return r458480;
}
↓
double f(double x, double y, double z) {
        double r458481 = x;
        double r458482 = -3.55588684135671e+22;
        bool r458483 = r458481 <= r458482;
        double r458484 = 1.1453649274389147e+36;
        bool r458485 = r458481 <= r458484;
        double r458486 = !r458485;
        bool r458487 = r458483 || r458486;
        double r458488 = 4.16438922228;
        double r458489 = y;
        double r458490 = 2.0;
        double r458491 = pow(r458481, r458490);
        double r458492 = r458489 / r458491;
        double r458493 = fma(r458481, r458488, r458492);
        double r458494 = 110.11392429848108;
        double r458495 = r458493 - r458494;
        double r458496 = 2.0;
        double r458497 = r458481 - r458496;
        double r458498 = 78.6994924154;
        double r458499 = fma(r458481, r458488, r458498);
        double r458500 = r458499 * r458481;
        double r458501 = 3.0;
        double r458502 = pow(r458500, r458501);
        double r458503 = 137.519416416;
        double r458504 = pow(r458503, r458501);
        double r458505 = r458502 + r458504;
        double r458506 = r458505 * r458481;
        double r458507 = r458481 * r458488;
        double r458508 = r458507 + r458498;
        double r458509 = r458508 * r458481;
        double r458510 = r458509 * r458509;
        double r458511 = r458503 * r458503;
        double r458512 = r458509 * r458503;
        double r458513 = r458511 - r458512;
        double r458514 = r458510 + r458513;
        double r458515 = r458506 / r458514;
        double r458516 = r458515 + r458489;
        double r458517 = r458516 * r458481;
        double r458518 = z;
        double r458519 = r458517 + r458518;
        double r458520 = r458497 * r458519;
        double r458521 = 43.3400022514;
        double r458522 = r458481 + r458521;
        double r458523 = r458522 * r458481;
        double r458524 = 263.505074721;
        double r458525 = r458523 + r458524;
        double r458526 = r458525 * r458481;
        double r458527 = 313.399215894;
        double r458528 = r458526 + r458527;
        double r458529 = r458528 * r458481;
        double r458530 = 47.066876606;
        double r458531 = r458529 + r458530;
        double r458532 = r458520 / r458531;
        double r458533 = r458487 ? r458495 : r458532;
        return r458533;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original26.5
Target0.4
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;x \lt -3.3261287258700048 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{elif}\;x \lt 9.4299917145546727 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.50507472100003 \cdot x + \left(43.3400022514000014 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.399215894\right) \cdot x + 47.066876606000001}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -3.55588684135671e+22 or 1.1453649274389147e+36 < x

    1. Initial program 58.3

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

    3. Applied flip--58.3

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

    4. Applied associate-*l/60.6

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

    5. Applied associate-/l/60.6

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

    6. Simplified60.6

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

    7. Taylor expanded around inf 1.1

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.113924298481081}\]

    8. Simplified1.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}}\right) - 110.113924298481081}\]

    if -3.55588684135671e+22 < x < 1.1453649274389147e+36

    1. Initial program 0.6

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

    3. Applied flip3-+0.6

      \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\color{blue}{\frac{{\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right)}^{3} + {137.51941641600001}^{3}}{\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) + \left(137.51941641600001 \cdot 137.51941641600001 - \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot 137.51941641600001\right)}} \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]

    4. Applied associate-*l/0.6

      \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\color{blue}{\frac{\left({\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right)}^{3} + {137.51941641600001}^{3}\right) \cdot x}{\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) + \left(137.51941641600001 \cdot 137.51941641600001 - \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot 137.51941641600001\right)}} + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]

    5. Simplified0.6

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.5558868413567101 \cdot 10^{22} \lor \neg \left(x \le 1.1453649274389147 \cdot 10^{36}\right):\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}}\right) - 110.113924298481081\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\frac{\left({\left(\mathsf{fma}\left(x, 4.16438922227999964, 78.6994924154000017\right) \cdot x\right)}^{3} + {137.51941641600001}^{3}\right) \cdot x}{\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) + \left(137.51941641600001 \cdot 137.51941641600001 - \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot 137.51941641600001\right)} + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +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)))