Average Error: 27.1 → 1.0
Time: 31.4s
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -14572255947977734100 \lor \neg \left(x \le 3.68584910506240575 \cdot 10^{31}\right):\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}}\right) - 110.11392429848109\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999964, 78.6994924154000017\right), x, 137.51941641600001\right), x, y\right), z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000014, x, 263.50507472100003\right), x, 313.399215894\right), x, 47.066876606000001\right)}\\ \end{array}\]
\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 -14572255947977734100 \lor \neg \left(x \le 3.68584910506240575 \cdot 10^{31}\right):\\
\;\;\;\;\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}}\right) - 110.11392429848109\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999964, 78.6994924154000017\right), x, 137.51941641600001\right), x, y\right), z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000014, x, 263.50507472100003\right), x, 313.399215894\right), x, 47.066876606000001\right)}\\

\end{array}
double f(double x, double y, double z) {
        double r244542 = x;
        double r244543 = 2.0;
        double r244544 = r244542 - r244543;
        double r244545 = 4.16438922228;
        double r244546 = r244542 * r244545;
        double r244547 = 78.6994924154;
        double r244548 = r244546 + r244547;
        double r244549 = r244548 * r244542;
        double r244550 = 137.519416416;
        double r244551 = r244549 + r244550;
        double r244552 = r244551 * r244542;
        double r244553 = y;
        double r244554 = r244552 + r244553;
        double r244555 = r244554 * r244542;
        double r244556 = z;
        double r244557 = r244555 + r244556;
        double r244558 = r244544 * r244557;
        double r244559 = 43.3400022514;
        double r244560 = r244542 + r244559;
        double r244561 = r244560 * r244542;
        double r244562 = 263.505074721;
        double r244563 = r244561 + r244562;
        double r244564 = r244563 * r244542;
        double r244565 = 313.399215894;
        double r244566 = r244564 + r244565;
        double r244567 = r244566 * r244542;
        double r244568 = 47.066876606;
        double r244569 = r244567 + r244568;
        double r244570 = r244558 / r244569;
        return r244570;
}
↓
double f(double x, double y, double z) {
        double r244571 = x;
        double r244572 = -1.4572255947977734e+19;
        bool r244573 = r244571 <= r244572;
        double r244574 = 3.6858491050624058e+31;
        bool r244575 = r244571 <= r244574;
        double r244576 = !r244575;
        bool r244577 = r244573 || r244576;
        double r244578 = 4.16438922228;
        double r244579 = y;
        double r244580 = 2.0;
        double r244581 = pow(r244571, r244580);
        double r244582 = r244579 / r244581;
        double r244583 = fma(r244571, r244578, r244582);
        double r244584 = 110.1139242984811;
        double r244585 = r244583 - r244584;
        double r244586 = 2.0;
        double r244587 = r244571 - r244586;
        double r244588 = 78.6994924154;
        double r244589 = fma(r244571, r244578, r244588);
        double r244590 = 137.519416416;
        double r244591 = fma(r244589, r244571, r244590);
        double r244592 = fma(r244591, r244571, r244579);
        double r244593 = z;
        double r244594 = fma(r244571, r244592, r244593);
        double r244595 = r244587 * r244594;
        double r244596 = 43.3400022514;
        double r244597 = r244571 + r244596;
        double r244598 = 263.505074721;
        double r244599 = fma(r244597, r244571, r244598);
        double r244600 = 313.399215894;
        double r244601 = fma(r244599, r244571, r244600);
        double r244602 = 47.066876606;
        double r244603 = fma(r244601, r244571, r244602);
        double r244604 = r244595 / r244603;
        double r244605 = r244577 ? r244585 : r244604;
        return r244605;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original27.1
Target0.6
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;x \lt -3.3261287258700048 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{elif}\;x \lt 9.4299917145546727 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.50507472100003 \cdot x + \left(43.3400022514000014 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.399215894\right) \cdot x + 47.066876606000001}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -1.4572255947977734e+19 or 3.6858491050624058e+31 < x

    1. Initial program 57.9

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

    2. Simplified53.5

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

    3. Taylor expanded around inf 1.8

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109}\]

    4. Simplified1.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}}\right) - 110.11392429848109}\]

    if -1.4572255947977734e+19 < x < 3.6858491050624058e+31

    1. Initial program 0.4

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

    2. Simplified0.4

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

    4. Applied associate-/r/0.6

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

    6. Applied associate-*l/0.4

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

    7. Simplified0.4

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -14572255947977734100 \lor \neg \left(x \le 3.68584910506240575 \cdot 10^{31}\right):\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}}\right) - 110.11392429848109\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999964, 78.6994924154000017\right), x, 137.51941641600001\right), x, y\right), z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000014, x, 263.50507472100003\right), x, 313.399215894\right), x, 47.066876606000001\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C"

  :herbie-target
  (if (< x -3.326128725870005e+62) (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811) (if (< x 9.429991714554673e+55) (* (/ (- x 2.0) 1.0) (/ (+ (* (+ (* (+ (* (+ (* 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 2.0) (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z)) (+ (* (+ (* (+ (* (+ x 43.3400022514) x) 263.505074721) x) 313.399215894) x) 47.066876606)))