Average Error: 29.2 → 1.3
Time: 23.7s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.168216123229296831008511442933072558801 \cdot 10^{55} \lor \neg \left(z \le 9.979529776664346341474863534760464067821 \cdot 10^{57}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{{z}^{2}} + 3.130605476229999961645944495103321969509, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), z, a\right), z, b\right) \cdot \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.2346874069999991263557603815570473671, z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)} + x\\ \end{array}\]
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 -3.168216123229296831008511442933072558801 \cdot 10^{55} \lor \neg \left(z \le 9.979529776664346341474863534760464067821 \cdot 10^{57}\right):\\
\;\;\;\;\mathsf{fma}\left(y, \frac{t}{{z}^{2}} + 3.130605476229999961645944495103321969509, x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), z, a\right), z, b\right) \cdot \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.2346874069999991263557603815570473671, z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)} + x\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r201455 = x;
        double r201456 = y;
        double r201457 = z;
        double r201458 = 3.13060547623;
        double r201459 = r201457 * r201458;
        double r201460 = 11.1667541262;
        double r201461 = r201459 + r201460;
        double r201462 = r201461 * r201457;
        double r201463 = t;
        double r201464 = r201462 + r201463;
        double r201465 = r201464 * r201457;
        double r201466 = a;
        double r201467 = r201465 + r201466;
        double r201468 = r201467 * r201457;
        double r201469 = b;
        double r201470 = r201468 + r201469;
        double r201471 = r201456 * r201470;
        double r201472 = 15.234687407;
        double r201473 = r201457 + r201472;
        double r201474 = r201473 * r201457;
        double r201475 = 31.4690115749;
        double r201476 = r201474 + r201475;
        double r201477 = r201476 * r201457;
        double r201478 = 11.9400905721;
        double r201479 = r201477 + r201478;
        double r201480 = r201479 * r201457;
        double r201481 = 0.607771387771;
        double r201482 = r201480 + r201481;
        double r201483 = r201471 / r201482;
        double r201484 = r201455 + r201483;
        return r201484;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r201485 = z;
        double r201486 = -3.168216123229297e+55;
        bool r201487 = r201485 <= r201486;
        double r201488 = 9.979529776664346e+57;
        bool r201489 = r201485 <= r201488;
        double r201490 = !r201489;
        bool r201491 = r201487 || r201490;
        double r201492 = y;
        double r201493 = t;
        double r201494 = 2.0;
        double r201495 = pow(r201485, r201494);
        double r201496 = r201493 / r201495;
        double r201497 = 3.13060547623;
        double r201498 = r201496 + r201497;
        double r201499 = x;
        double r201500 = fma(r201492, r201498, r201499);
        double r201501 = 11.1667541262;
        double r201502 = fma(r201485, r201497, r201501);
        double r201503 = fma(r201502, r201485, r201493);
        double r201504 = a;
        double r201505 = fma(r201503, r201485, r201504);
        double r201506 = b;
        double r201507 = fma(r201505, r201485, r201506);
        double r201508 = 15.234687407;
        double r201509 = r201485 + r201508;
        double r201510 = 31.4690115749;
        double r201511 = fma(r201509, r201485, r201510);
        double r201512 = 11.9400905721;
        double r201513 = fma(r201511, r201485, r201512);
        double r201514 = 0.607771387771;
        double r201515 = fma(r201513, r201485, r201514);
        double r201516 = r201492 / r201515;
        double r201517 = r201507 * r201516;
        double r201518 = r201517 + r201499;
        double r201519 = r201491 ? r201500 : r201518;
        return r201519;
}

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

Target

Original29.2
Target1.1
Herbie1.3
\[\begin{array}{l} \mathbf{if}\;z \lt -6.499344996252631754123144978817242590467 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.066965436914286795694558389038333165002 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -3.168216123229297e+55 or 9.979529776664346e+57 < z

    1. Initial program 61.9

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

    2. Simplified60.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.2346874069999991263557603815570473671, z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), z, a\right), z, b\right), x\right)}\]

    3. Taylor expanded around inf 8.1

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

    4. Simplified0.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{t}{{z}^{2}} + 3.130605476229999961645944495103321969509, x\right)}\]

    if -3.168216123229297e+55 < z < 9.979529776664346e+57

    1. Initial program 3.4

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

    2. Simplified1.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.2346874069999991263557603815570473671, z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), z, a\right), z, b\right), x\right)}\]
    3. Using strategy rm

    4. Applied clear-num1.9

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.2346874069999991263557603815570473671, z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)}{y}}}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), z, a\right), z, b\right), x\right)\]
    5. Using strategy rm

    6. Applied fma-udef1.9

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.2346874069999991263557603815570473671, z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)}{y}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), z, a\right), z, b\right) + x}\]

    7. Simplified1.8

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), z, a\right), z, b\right)}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.2346874069999991263557603815570473671, z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)}{y}}} + x\]
    8. Using strategy rm

    9. Applied div-inv1.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), z, a\right), z, b\right) \cdot \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.2346874069999991263557603815570473671, z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)}{y}}} + x\]

    10. Simplified1.8

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.168216123229296831008511442933072558801 \cdot 10^{55} \lor \neg \left(z \le 9.979529776664346341474863534760464067821 \cdot 10^{57}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{{z}^{2}} + 3.130605476229999961645944495103321969509, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), z, a\right), z, b\right) \cdot \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.2346874069999991263557603815570473671, z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :herbie-target
  (if (< z -6.499344996252632e+53) (+ x (* (+ (- 3.13060547623 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1))) (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)))))

  (+ 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))))