Average Error: 26.7 → 0.7
Time: 24.9s
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -41468195884167556148326040140678955008 \lor \neg \left(x \le 4.185814178876573776200097393718801150489 \cdot 10^{51}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 2}{\frac{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}}\\ \end{array}\]
\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}
↓
\begin{array}{l}
\mathbf{if}\;x \le -41468195884167556148326040140678955008 \lor \neg \left(x \le 4.185814178876573776200097393718801150489 \cdot 10^{51}\right):\\
\;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\

\mathbf{else}:\\
\;\;\;\;\frac{x - 2}{\frac{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}}\\

\end{array}
double f(double x, double y, double z) {
        double r281538 = x;
        double r281539 = 2.0;
        double r281540 = r281538 - r281539;
        double r281541 = 4.16438922228;
        double r281542 = r281538 * r281541;
        double r281543 = 78.6994924154;
        double r281544 = r281542 + r281543;
        double r281545 = r281544 * r281538;
        double r281546 = 137.519416416;
        double r281547 = r281545 + r281546;
        double r281548 = r281547 * r281538;
        double r281549 = y;
        double r281550 = r281548 + r281549;
        double r281551 = r281550 * r281538;
        double r281552 = z;
        double r281553 = r281551 + r281552;
        double r281554 = r281540 * r281553;
        double r281555 = 43.3400022514;
        double r281556 = r281538 + r281555;
        double r281557 = r281556 * r281538;
        double r281558 = 263.505074721;
        double r281559 = r281557 + r281558;
        double r281560 = r281559 * r281538;
        double r281561 = 313.399215894;
        double r281562 = r281560 + r281561;
        double r281563 = r281562 * r281538;
        double r281564 = 47.066876606;
        double r281565 = r281563 + r281564;
        double r281566 = r281554 / r281565;
        return r281566;
}
↓
double f(double x, double y, double z) {
        double r281567 = x;
        double r281568 = -4.1468195884167556e+37;
        bool r281569 = r281567 <= r281568;
        double r281570 = 4.185814178876574e+51;
        bool r281571 = r281567 <= r281570;
        double r281572 = !r281571;
        bool r281573 = r281569 || r281572;
        double r281574 = y;
        double r281575 = 2.0;
        double r281576 = pow(r281567, r281575);
        double r281577 = r281574 / r281576;
        double r281578 = 4.16438922228;
        double r281579 = r281578 * r281567;
        double r281580 = r281577 + r281579;
        double r281581 = 110.1139242984811;
        double r281582 = r281580 - r281581;
        double r281583 = 2.0;
        double r281584 = r281567 - r281583;
        double r281585 = 43.3400022514;
        double r281586 = r281567 + r281585;
        double r281587 = r281586 * r281567;
        double r281588 = 263.505074721;
        double r281589 = r281587 + r281588;
        double r281590 = r281589 * r281567;
        double r281591 = 313.399215894;
        double r281592 = r281590 + r281591;
        double r281593 = r281592 * r281567;
        double r281594 = 47.066876606;
        double r281595 = r281593 + r281594;
        double r281596 = r281567 * r281578;
        double r281597 = 78.6994924154;
        double r281598 = r281596 + r281597;
        double r281599 = r281598 * r281567;
        double r281600 = 137.519416416;
        double r281601 = r281599 + r281600;
        double r281602 = r281601 * r281567;
        double r281603 = r281602 + r281574;
        double r281604 = r281603 * r281567;
        double r281605 = z;
        double r281606 = r281604 + r281605;
        double r281607 = r281595 / r281606;
        double r281608 = r281584 / r281607;
        double r281609 = r281573 ? r281582 : r281608;
        return r281609;
}

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

Original26.7
Target0.4
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;x \lt -3.326128725870004842699683658678411714981 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{elif}\;x \lt 9.429991714554672672712552870340896976735 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.5050747210000281484099105000495910645 \cdot x + \left(43.3400022514000013984514225739985704422 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -4.1468195884167556e+37 or 4.185814178876574e+51 < x

    1. Initial program 60.8

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]

    2. Taylor expanded around inf 0.7

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229}\]

    if -4.1468195884167556e+37 < x < 4.185814178876574e+51

    1. Initial program 0.9

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]
    2. Using strategy rm

    3. Applied associate-/l*0.7

      \[\leadsto \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -41468195884167556148326040140678955008 \lor \neg \left(x \le 4.185814178876573776200097393718801150489 \cdot 10^{51}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 2}{\frac{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C"
  :precision binary64

  :herbie-target
  (if (< x -3.3261287258700048e62) (- (+ (/ y (* x x)) (* 4.16438922227999964 x)) 110.11392429848109) (if (< x 9.4299917145546727e55) (* (/ (- x 2) 1) (/ (+ (* (+ (* (+ (* (+ (* x 4.16438922227999964) 78.6994924154000017) x) 137.51941641600001) x) y) x) z) (+ (* (+ (+ (* 263.50507472100003 x) (+ (* 43.3400022514000014 (* x x)) (* x (* x x)))) 313.399215894) x) 47.066876606000001))) (- (+ (/ y (* x x)) (* 4.16438922227999964 x)) 110.11392429848109)))

  (/ (* (- x 2) (+ (* (+ (* (+ (* (+ (* x 4.16438922227999964) 78.6994924154000017) x) 137.51941641600001) x) y) x) z)) (+ (* (+ (* (+ (* (+ x 43.3400022514000014) x) 263.50507472100003) x) 313.399215894) x) 47.066876606000001)))