Average Error: 26.4 → 1.0
Time: 8.1s
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.31518288772234492 \cdot 10^{47} \lor \neg \left(x \le 3.1500200017107755 \cdot 10^{35}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 2}{\frac{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}}{\frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}}}}\\ \end{array}\]
\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 -3.31518288772234492 \cdot 10^{47} \lor \neg \left(x \le 3.1500200017107755 \cdot 10^{35}\right):\\
\;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\

\mathbf{else}:\\
\;\;\;\;\frac{x - 2}{\frac{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}}{\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 r451649 = x;
        double r451650 = 2.0;
        double r451651 = r451649 - r451650;
        double r451652 = 4.16438922228;
        double r451653 = r451649 * r451652;
        double r451654 = 78.6994924154;
        double r451655 = r451653 + r451654;
        double r451656 = r451655 * r451649;
        double r451657 = 137.519416416;
        double r451658 = r451656 + r451657;
        double r451659 = r451658 * r451649;
        double r451660 = y;
        double r451661 = r451659 + r451660;
        double r451662 = r451661 * r451649;
        double r451663 = z;
        double r451664 = r451662 + r451663;
        double r451665 = r451651 * r451664;
        double r451666 = 43.3400022514;
        double r451667 = r451649 + r451666;
        double r451668 = r451667 * r451649;
        double r451669 = 263.505074721;
        double r451670 = r451668 + r451669;
        double r451671 = r451670 * r451649;
        double r451672 = 313.399215894;
        double r451673 = r451671 + r451672;
        double r451674 = r451673 * r451649;
        double r451675 = 47.066876606;
        double r451676 = r451674 + r451675;
        double r451677 = r451665 / r451676;
        return r451677;
}
↓
double f(double x, double y, double z) {
        double r451678 = x;
        double r451679 = -3.315182887722345e+47;
        bool r451680 = r451678 <= r451679;
        double r451681 = 3.1500200017107755e+35;
        bool r451682 = r451678 <= r451681;
        double r451683 = !r451682;
        bool r451684 = r451680 || r451683;
        double r451685 = y;
        double r451686 = 2.0;
        double r451687 = pow(r451678, r451686);
        double r451688 = r451685 / r451687;
        double r451689 = 4.16438922228;
        double r451690 = r451689 * r451678;
        double r451691 = r451688 + r451690;
        double r451692 = 110.1139242984811;
        double r451693 = r451691 - r451692;
        double r451694 = 2.0;
        double r451695 = r451678 - r451694;
        double r451696 = 43.3400022514;
        double r451697 = r451678 + r451696;
        double r451698 = r451697 * r451678;
        double r451699 = 263.505074721;
        double r451700 = r451698 + r451699;
        double r451701 = r451700 * r451678;
        double r451702 = 313.399215894;
        double r451703 = r451701 + r451702;
        double r451704 = r451703 * r451678;
        double r451705 = 47.066876606;
        double r451706 = r451704 + r451705;
        double r451707 = sqrt(r451706);
        double r451708 = r451678 * r451689;
        double r451709 = 78.6994924154;
        double r451710 = r451708 + r451709;
        double r451711 = r451710 * r451678;
        double r451712 = 137.519416416;
        double r451713 = r451711 + r451712;
        double r451714 = r451713 * r451678;
        double r451715 = r451714 + r451685;
        double r451716 = r451715 * r451678;
        double r451717 = z;
        double r451718 = r451716 + r451717;
        double r451719 = r451718 / r451707;
        double r451720 = r451707 / r451719;
        double r451721 = r451695 / r451720;
        double r451722 = r451684 ? r451693 : r451721;
        return r451722;
}

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

Original26.4
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 < -3.315182887722345e+47 or 3.1500200017107755e+35 < x

    1. Initial program 60.1

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

    2. Taylor expanded around inf 1.1

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

    if -3.315182887722345e+47 < x < 3.1500200017107755e+35

    1. Initial program 1.0

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

    3. Applied associate-/l*0.6

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

    5. Applied add-sqr-sqrt0.8

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

    6. Applied associate-/l*0.9

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.31518288772234492 \cdot 10^{47} \lor \neg \left(x \le 3.1500200017107755 \cdot 10^{35}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 2}{\frac{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}}{\frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 
(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)))