Average Error: 26.8 → 1.0
Time: 11.4s
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5554983803297941420000 \lor \neg \left(x \le 1.39857869664759388 \cdot 10^{51}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 2}{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}}\\ \end{array}\]
\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 -5554983803297941420000 \lor \neg \left(x \le 1.39857869664759388 \cdot 10^{51}\right):\\
\;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\

\mathbf{else}:\\
\;\;\;\;\frac{x - 2}{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\sqrt{\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 r319541 = x;
        double r319542 = 2.0;
        double r319543 = r319541 - r319542;
        double r319544 = 4.16438922228;
        double r319545 = r319541 * r319544;
        double r319546 = 78.6994924154;
        double r319547 = r319545 + r319546;
        double r319548 = r319547 * r319541;
        double r319549 = 137.519416416;
        double r319550 = r319548 + r319549;
        double r319551 = r319550 * r319541;
        double r319552 = y;
        double r319553 = r319551 + r319552;
        double r319554 = r319553 * r319541;
        double r319555 = z;
        double r319556 = r319554 + r319555;
        double r319557 = r319543 * r319556;
        double r319558 = 43.3400022514;
        double r319559 = r319541 + r319558;
        double r319560 = r319559 * r319541;
        double r319561 = 263.505074721;
        double r319562 = r319560 + r319561;
        double r319563 = r319562 * r319541;
        double r319564 = 313.399215894;
        double r319565 = r319563 + r319564;
        double r319566 = r319565 * r319541;
        double r319567 = 47.066876606;
        double r319568 = r319566 + r319567;
        double r319569 = r319557 / r319568;
        return r319569;
}
↓
double f(double x, double y, double z) {
        double r319570 = x;
        double r319571 = -5.554983803297941e+21;
        bool r319572 = r319570 <= r319571;
        double r319573 = 1.3985786966475939e+51;
        bool r319574 = r319570 <= r319573;
        double r319575 = !r319574;
        bool r319576 = r319572 || r319575;
        double r319577 = y;
        double r319578 = 2.0;
        double r319579 = pow(r319570, r319578);
        double r319580 = r319577 / r319579;
        double r319581 = 4.16438922228;
        double r319582 = r319581 * r319570;
        double r319583 = r319580 + r319582;
        double r319584 = 110.1139242984811;
        double r319585 = r319583 - r319584;
        double r319586 = 2.0;
        double r319587 = r319570 - r319586;
        double r319588 = 43.3400022514;
        double r319589 = r319570 + r319588;
        double r319590 = r319589 * r319570;
        double r319591 = 263.505074721;
        double r319592 = r319590 + r319591;
        double r319593 = r319592 * r319570;
        double r319594 = 313.399215894;
        double r319595 = r319593 + r319594;
        double r319596 = r319595 * r319570;
        double r319597 = 47.066876606;
        double r319598 = r319596 + r319597;
        double r319599 = sqrt(r319598);
        double r319600 = r319587 / r319599;
        double r319601 = r319570 * r319581;
        double r319602 = 78.6994924154;
        double r319603 = r319601 + r319602;
        double r319604 = r319603 * r319570;
        double r319605 = 137.519416416;
        double r319606 = r319604 + r319605;
        double r319607 = r319606 * r319570;
        double r319608 = r319607 + r319577;
        double r319609 = r319608 * r319570;
        double r319610 = z;
        double r319611 = r319609 + r319610;
        double r319612 = r319611 / r319599;
        double r319613 = r319600 * r319612;
        double r319614 = r319576 ? r319585 : r319613;
        return r319614;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original26.8
Target0.5
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 < -5.554983803297941e+21 or 1.3985786966475939e+51 < x

    1. Initial program 59.5

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

    2. Taylor expanded around inf 1.3

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

    if -5.554983803297941e+21 < x < 1.3985786966475939e+51

    1. Initial program 0.8

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

    3. Applied add-sqr-sqrt1.0

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

    4. Applied times-frac0.8

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5554983803297941420000 \lor \neg \left(x \le 1.39857869664759388 \cdot 10^{51}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 2}{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(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)))