Average Error: 29.4 → 1.2
Time: 30.7s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -8.012125234246043972472451502679502100871 \cdot 10^{48}:\\ \;\;\;\;\left(\left(\frac{t}{z} \cdot \frac{y}{z} + 3.130605476229999961645944495103321969509 \cdot y\right) - \frac{36.52704169880641416057187598198652267456}{\frac{z}{y}}\right) + x\\ \mathbf{elif}\;z \le 6604017825197738179408704188920627200:\\ \;\;\;\;x + \frac{y}{\frac{z \cdot \left(11.94009057210000079862766142468899488449 + \left(31.46901157490000144889563671313226222992 + z \cdot \left(z + 15.2346874069999991263557603815570473671\right)\right) \cdot z\right) + 0.6077713877710000378584709324059076607227}{b + \left(z \cdot \left(t + \left(3.130605476229999961645944495103321969509 \cdot z + 11.16675412620000074070958362426608800888\right) \cdot z\right) + a\right) \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{t}{z} \cdot \frac{y}{z} + 3.130605476229999961645944495103321969509 \cdot y\right) - \frac{36.52704169880641416057187598198652267456}{\frac{z}{y}}\right) + x\\ \end{array}\]
x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}
↓
\begin{array}{l}
\mathbf{if}\;z \le -8.012125234246043972472451502679502100871 \cdot 10^{48}:\\
\;\;\;\;\left(\left(\frac{t}{z} \cdot \frac{y}{z} + 3.130605476229999961645944495103321969509 \cdot y\right) - \frac{36.52704169880641416057187598198652267456}{\frac{z}{y}}\right) + x\\

\mathbf{elif}\;z \le 6604017825197738179408704188920627200:\\
\;\;\;\;x + \frac{y}{\frac{z \cdot \left(11.94009057210000079862766142468899488449 + \left(31.46901157490000144889563671313226222992 + z \cdot \left(z + 15.2346874069999991263557603815570473671\right)\right) \cdot z\right) + 0.6077713877710000378584709324059076607227}{b + \left(z \cdot \left(t + \left(3.130605476229999961645944495103321969509 \cdot z + 11.16675412620000074070958362426608800888\right) \cdot z\right) + a\right) \cdot z}}\\

\mathbf{else}:\\
\;\;\;\;\left(\left(\frac{t}{z} \cdot \frac{y}{z} + 3.130605476229999961645944495103321969509 \cdot y\right) - \frac{36.52704169880641416057187598198652267456}{\frac{z}{y}}\right) + x\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r18463525 = x;
        double r18463526 = y;
        double r18463527 = z;
        double r18463528 = 3.13060547623;
        double r18463529 = r18463527 * r18463528;
        double r18463530 = 11.1667541262;
        double r18463531 = r18463529 + r18463530;
        double r18463532 = r18463531 * r18463527;
        double r18463533 = t;
        double r18463534 = r18463532 + r18463533;
        double r18463535 = r18463534 * r18463527;
        double r18463536 = a;
        double r18463537 = r18463535 + r18463536;
        double r18463538 = r18463537 * r18463527;
        double r18463539 = b;
        double r18463540 = r18463538 + r18463539;
        double r18463541 = r18463526 * r18463540;
        double r18463542 = 15.234687407;
        double r18463543 = r18463527 + r18463542;
        double r18463544 = r18463543 * r18463527;
        double r18463545 = 31.4690115749;
        double r18463546 = r18463544 + r18463545;
        double r18463547 = r18463546 * r18463527;
        double r18463548 = 11.9400905721;
        double r18463549 = r18463547 + r18463548;
        double r18463550 = r18463549 * r18463527;
        double r18463551 = 0.607771387771;
        double r18463552 = r18463550 + r18463551;
        double r18463553 = r18463541 / r18463552;
        double r18463554 = r18463525 + r18463553;
        return r18463554;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r18463555 = z;
        double r18463556 = -8.012125234246044e+48;
        bool r18463557 = r18463555 <= r18463556;
        double r18463558 = t;
        double r18463559 = r18463558 / r18463555;
        double r18463560 = y;
        double r18463561 = r18463560 / r18463555;
        double r18463562 = r18463559 * r18463561;
        double r18463563 = 3.13060547623;
        double r18463564 = r18463563 * r18463560;
        double r18463565 = r18463562 + r18463564;
        double r18463566 = 36.527041698806414;
        double r18463567 = r18463555 / r18463560;
        double r18463568 = r18463566 / r18463567;
        double r18463569 = r18463565 - r18463568;
        double r18463570 = x;
        double r18463571 = r18463569 + r18463570;
        double r18463572 = 6.604017825197738e+36;
        bool r18463573 = r18463555 <= r18463572;
        double r18463574 = 11.9400905721;
        double r18463575 = 31.4690115749;
        double r18463576 = 15.234687407;
        double r18463577 = r18463555 + r18463576;
        double r18463578 = r18463555 * r18463577;
        double r18463579 = r18463575 + r18463578;
        double r18463580 = r18463579 * r18463555;
        double r18463581 = r18463574 + r18463580;
        double r18463582 = r18463555 * r18463581;
        double r18463583 = 0.607771387771;
        double r18463584 = r18463582 + r18463583;
        double r18463585 = b;
        double r18463586 = r18463563 * r18463555;
        double r18463587 = 11.1667541262;
        double r18463588 = r18463586 + r18463587;
        double r18463589 = r18463588 * r18463555;
        double r18463590 = r18463558 + r18463589;
        double r18463591 = r18463555 * r18463590;
        double r18463592 = a;
        double r18463593 = r18463591 + r18463592;
        double r18463594 = r18463593 * r18463555;
        double r18463595 = r18463585 + r18463594;
        double r18463596 = r18463584 / r18463595;
        double r18463597 = r18463560 / r18463596;
        double r18463598 = r18463570 + r18463597;
        double r18463599 = r18463573 ? r18463598 : r18463571;
        double r18463600 = r18463557 ? r18463571 : r18463599;
        return r18463600;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.4
Target1.1
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;z \lt -6.499344996252631754123144978817242590467 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.066965436914286795694558389038333165002 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -8.012125234246044e+48 or 6.604017825197738e+36 < z

    1. Initial program 60.4

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]

    2. Taylor expanded around inf 8.5

      \[\leadsto x + \color{blue}{\left(\left(\frac{t \cdot y}{{z}^{2}} + 3.130605476229999961645944495103321969509 \cdot y\right) - 36.52704169880641416057187598198652267456 \cdot \frac{y}{z}\right)}\]

    3. Simplified1.5

      \[\leadsto x + \color{blue}{\left(\left(3.130605476229999961645944495103321969509 \cdot y + \frac{y}{z} \cdot \frac{t}{z}\right) - \frac{36.52704169880641416057187598198652267456}{\frac{z}{y}}\right)}\]

    if -8.012125234246044e+48 < z < 6.604017825197738e+36

    1. Initial program 1.8

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]
    2. Using strategy rm

    3. Applied associate-/l*0.9

      \[\leadsto x + \color{blue}{\frac{y}{\frac{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -8.012125234246043972472451502679502100871 \cdot 10^{48}:\\ \;\;\;\;\left(\left(\frac{t}{z} \cdot \frac{y}{z} + 3.130605476229999961645944495103321969509 \cdot y\right) - \frac{36.52704169880641416057187598198652267456}{\frac{z}{y}}\right) + x\\ \mathbf{elif}\;z \le 6604017825197738179408704188920627200:\\ \;\;\;\;x + \frac{y}{\frac{z \cdot \left(11.94009057210000079862766142468899488449 + \left(31.46901157490000144889563671313226222992 + z \cdot \left(z + 15.2346874069999991263557603815570473671\right)\right) \cdot z\right) + 0.6077713877710000378584709324059076607227}{b + \left(z \cdot \left(t + \left(3.130605476229999961645944495103321969509 \cdot z + 11.16675412620000074070958362426608800888\right) \cdot z\right) + a\right) \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{t}{z} \cdot \frac{y}{z} + 3.130605476229999961645944495103321969509 \cdot y\right) - \frac{36.52704169880641416057187598198652267456}{\frac{z}{y}}\right) + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, D"

  :herbie-target
  (if (< z -6.499344996252632e+53) (+ x (* (+ (- 3.13060547623 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1.0))) (if (< z 7.066965436914287e+59) (+ x (/ y (/ (+ (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z) 0.607771387771) (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b)))) (+ x (* (+ (- 3.13060547623 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1.0)))))

  (+ x (/ (* y (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b)) (+ (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z) 0.607771387771))))