Average Error: 27.1 → 0.5
Time: 8.6s
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.26409396866929665 \cdot 10^{68} \lor \neg \left(x \le 1.32155597286449126 \cdot 10^{55}\right):\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}} - 110.113924298481081\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x - 2 \cdot 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000014, x, 263.50507472100003\right), x, 313.399215894\right), x, 47.066876606000001\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999964, 78.6994924154000017\right), x, 137.51941641600001\right), x, y\right), x, z\right)} \cdot \left(x + 2\right)}\\ \end{array}\]
\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 -4.26409396866929665 \cdot 10^{68} \lor \neg \left(x \le 1.32155597286449126 \cdot 10^{55}\right):\\
\;\;\;\;\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}} - 110.113924298481081\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{x \cdot x - 2 \cdot 2}{\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)}{\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)}\\

\end{array}
double f(double x, double y, double z) {
        double r426476 = x;
        double r426477 = 2.0;
        double r426478 = r426476 - r426477;
        double r426479 = 4.16438922228;
        double r426480 = r426476 * r426479;
        double r426481 = 78.6994924154;
        double r426482 = r426480 + r426481;
        double r426483 = r426482 * r426476;
        double r426484 = 137.519416416;
        double r426485 = r426483 + r426484;
        double r426486 = r426485 * r426476;
        double r426487 = y;
        double r426488 = r426486 + r426487;
        double r426489 = r426488 * r426476;
        double r426490 = z;
        double r426491 = r426489 + r426490;
        double r426492 = r426478 * r426491;
        double r426493 = 43.3400022514;
        double r426494 = r426476 + r426493;
        double r426495 = r426494 * r426476;
        double r426496 = 263.505074721;
        double r426497 = r426495 + r426496;
        double r426498 = r426497 * r426476;
        double r426499 = 313.399215894;
        double r426500 = r426498 + r426499;
        double r426501 = r426500 * r426476;
        double r426502 = 47.066876606;
        double r426503 = r426501 + r426502;
        double r426504 = r426492 / r426503;
        return r426504;
}
↓
double f(double x, double y, double z) {
        double r426505 = x;
        double r426506 = -4.2640939686692966e+68;
        bool r426507 = r426505 <= r426506;
        double r426508 = 1.3215559728644913e+55;
        bool r426509 = r426505 <= r426508;
        double r426510 = !r426509;
        bool r426511 = r426507 || r426510;
        double r426512 = 4.16438922228;
        double r426513 = y;
        double r426514 = 2.0;
        double r426515 = pow(r426505, r426514);
        double r426516 = r426513 / r426515;
        double r426517 = 110.11392429848108;
        double r426518 = r426516 - r426517;
        double r426519 = fma(r426505, r426512, r426518);
        double r426520 = r426505 * r426505;
        double r426521 = 2.0;
        double r426522 = r426521 * r426521;
        double r426523 = r426520 - r426522;
        double r426524 = 43.3400022514;
        double r426525 = r426505 + r426524;
        double r426526 = 263.505074721;
        double r426527 = fma(r426525, r426505, r426526);
        double r426528 = 313.399215894;
        double r426529 = fma(r426527, r426505, r426528);
        double r426530 = 47.066876606;
        double r426531 = fma(r426529, r426505, r426530);
        double r426532 = 78.6994924154;
        double r426533 = fma(r426505, r426512, r426532);
        double r426534 = 137.519416416;
        double r426535 = fma(r426533, r426505, r426534);
        double r426536 = fma(r426535, r426505, r426513);
        double r426537 = z;
        double r426538 = fma(r426536, r426505, r426537);
        double r426539 = r426531 / r426538;
        double r426540 = r426505 + r426521;
        double r426541 = r426539 * r426540;
        double r426542 = r426523 / r426541;
        double r426543 = r426511 ? r426519 : r426542;
        return r426543;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original27.1
Target0.4
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;x \lt -3.3261287258700048 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{elif}\;x \lt 9.4299917145546727 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.50507472100003 \cdot x + \left(43.3400022514000014 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.399215894\right) \cdot x + 47.066876606000001}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -4.2640939686692966e+68 or 1.3215559728644913e+55 < x

    1. Initial program 63.6

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

    2. Simplified60.4

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

    4. Applied flip--60.4

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

    5. Applied associate-/l/60.4

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

    6. Taylor expanded around inf 0.3

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

    7. Simplified0.3

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

    if -4.2640939686692966e+68 < x < 1.3215559728644913e+55

    1. Initial program 2.1

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

    2. Simplified0.7

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

    4. Applied flip--0.7

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

    5. Applied associate-/l/0.7

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.26409396866929665 \cdot 10^{68} \lor \neg \left(x \le 1.32155597286449126 \cdot 10^{55}\right):\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}} - 110.113924298481081\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x - 2 \cdot 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000014, x, 263.50507472100003\right), x, 313.399215894\right), x, 47.066876606000001\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999964, 78.6994924154000017\right), x, 137.51941641600001\right), x, y\right), x, z\right)} \cdot \left(x + 2\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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)))