Average Error: 29.0 → 1.1
Time: 1.3m
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.337724750251567597194107095239406966919 \cdot 10^{47}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, \frac{t}{z}, \mathsf{fma}\left(3.130605476229999961645944495103321969509, y, x\right)\right)\\ \mathbf{elif}\;z \le 2727078133961668334404484726784:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), a\right), b\right), y \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 15.2346874069999991263557603815570473671 + z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, t \cdot \frac{1}{z \cdot z} + 3.130605476229999961645944495103321969509, x\right)\\ \end{array}\]
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 -2.337724750251567597194107095239406966919 \cdot 10^{47}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{z}, \frac{t}{z}, \mathsf{fma}\left(3.130605476229999961645944495103321969509, y, x\right)\right)\\

\mathbf{elif}\;z \le 2727078133961668334404484726784:\\
\;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), a\right), b\right), y \cdot \frac{1}{\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\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y, t \cdot \frac{1}{z \cdot z} + 3.130605476229999961645944495103321969509, x\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r10984497 = x;
        double r10984498 = y;
        double r10984499 = z;
        double r10984500 = 3.13060547623;
        double r10984501 = r10984499 * r10984500;
        double r10984502 = 11.1667541262;
        double r10984503 = r10984501 + r10984502;
        double r10984504 = r10984503 * r10984499;
        double r10984505 = t;
        double r10984506 = r10984504 + r10984505;
        double r10984507 = r10984506 * r10984499;
        double r10984508 = a;
        double r10984509 = r10984507 + r10984508;
        double r10984510 = r10984509 * r10984499;
        double r10984511 = b;
        double r10984512 = r10984510 + r10984511;
        double r10984513 = r10984498 * r10984512;
        double r10984514 = 15.234687407;
        double r10984515 = r10984499 + r10984514;
        double r10984516 = r10984515 * r10984499;
        double r10984517 = 31.4690115749;
        double r10984518 = r10984516 + r10984517;
        double r10984519 = r10984518 * r10984499;
        double r10984520 = 11.9400905721;
        double r10984521 = r10984519 + r10984520;
        double r10984522 = r10984521 * r10984499;
        double r10984523 = 0.607771387771;
        double r10984524 = r10984522 + r10984523;
        double r10984525 = r10984513 / r10984524;
        double r10984526 = r10984497 + r10984525;
        return r10984526;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r10984527 = z;
        double r10984528 = -2.3377247502515676e+47;
        bool r10984529 = r10984527 <= r10984528;
        double r10984530 = y;
        double r10984531 = r10984530 / r10984527;
        double r10984532 = t;
        double r10984533 = r10984532 / r10984527;
        double r10984534 = 3.13060547623;
        double r10984535 = x;
        double r10984536 = fma(r10984534, r10984530, r10984535);
        double r10984537 = fma(r10984531, r10984533, r10984536);
        double r10984538 = 2.7270781339616683e+30;
        bool r10984539 = r10984527 <= r10984538;
        double r10984540 = 11.1667541262;
        double r10984541 = fma(r10984527, r10984534, r10984540);
        double r10984542 = fma(r10984541, r10984527, r10984532);
        double r10984543 = a;
        double r10984544 = fma(r10984527, r10984542, r10984543);
        double r10984545 = b;
        double r10984546 = fma(r10984527, r10984544, r10984545);
        double r10984547 = 1.0;
        double r10984548 = 15.234687407;
        double r10984549 = r10984548 + r10984527;
        double r10984550 = 31.4690115749;
        double r10984551 = fma(r10984527, r10984549, r10984550);
        double r10984552 = 11.9400905721;
        double r10984553 = fma(r10984551, r10984527, r10984552);
        double r10984554 = 0.607771387771;
        double r10984555 = fma(r10984553, r10984527, r10984554);
        double r10984556 = r10984547 / r10984555;
        double r10984557 = r10984530 * r10984556;
        double r10984558 = fma(r10984546, r10984557, r10984535);
        double r10984559 = r10984527 * r10984527;
        double r10984560 = r10984547 / r10984559;
        double r10984561 = r10984532 * r10984560;
        double r10984562 = r10984561 + r10984534;
        double r10984563 = fma(r10984530, r10984562, r10984535);
        double r10984564 = r10984539 ? r10984558 : r10984563;
        double r10984565 = r10984529 ? r10984537 : r10984564;
        return r10984565;
}

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

Original29.0
Target1.0
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;z \lt -6.499344996252631754123144978817242590467 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.066965436914286795694558389038333165002 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -2.3377247502515676e+47

    1. Initial program 60.7

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

    2. Simplified58.9

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

    3. Taylor expanded around inf 8.5

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

    4. Simplified1.2

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

    if -2.3377247502515676e+47 < z < 2.7270781339616683e+30

    1. Initial program 1.6

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

    2. Simplified0.9

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

    4. Applied div-inv0.9

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

    if 2.7270781339616683e+30 < z

    1. Initial program 59.3

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

    2. Simplified57.7

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

    3. Taylor expanded around inf 8.5

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

    4. Simplified1.4

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

    5. Taylor expanded around 0 8.5

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

    6. Simplified1.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \frac{t}{z \cdot z}, x\right)}\]
    7. Using strategy rm

    8. Applied div-inv1.4

      \[\leadsto \mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \color{blue}{t \cdot \frac{1}{z \cdot z}}, x\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.337724750251567597194107095239406966919 \cdot 10^{47}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, \frac{t}{z}, \mathsf{fma}\left(3.130605476229999961645944495103321969509, y, x\right)\right)\\ \mathbf{elif}\;z \le 2727078133961668334404484726784:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), a\right), b\right), y \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 15.2346874069999991263557603815570473671 + z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, t \cdot \frac{1}{z \cdot z} + 3.130605476229999961645944495103321969509, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(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))))