Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, D

Average Error: 29.2 → 0.9

Time: 26.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -4.232625433896362687402834018909691734934 \cdot 10^{64} \lor \neg \left(z \le 731737881160239040954368\right):\\ \;\;\;\;\mathsf{fma}\left(3.130605476229999961645944495103321969509, y, \mathsf{fma}\left(\frac{y}{z}, \frac{t}{z}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.130605476229999961645944495103321969509, z, 11.16675412620000074070958362426608800888\right), z, t\right), a\right) \cdot \frac{z}{\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z, 15.2346874069999991263557603815570473671 + z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)} + \frac{b}{\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z, 15.2346874069999991263557603815570473671 + z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)}, x\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r326646 = x;
        double r326647 = y;
        double r326648 = z;
        double r326649 = 3.13060547623;
        double r326650 = r326648 * r326649;
        double r326651 = 11.1667541262;
        double r326652 = r326650 + r326651;
        double r326653 = r326652 * r326648;
        double r326654 = t;
        double r326655 = r326653 + r326654;
        double r326656 = r326655 * r326648;
        double r326657 = a;
        double r326658 = r326656 + r326657;
        double r326659 = r326658 * r326648;
        double r326660 = b;
        double r326661 = r326659 + r326660;
        double r326662 = r326647 * r326661;
        double r326663 = 15.234687407;
        double r326664 = r326648 + r326663;
        double r326665 = r326664 * r326648;
        double r326666 = 31.4690115749;
        double r326667 = r326665 + r326666;
        double r326668 = r326667 * r326648;
        double r326669 = 11.9400905721;
        double r326670 = r326668 + r326669;
        double r326671 = r326670 * r326648;
        double r326672 = 0.607771387771;
        double r326673 = r326671 + r326672;
        double r326674 = r326662 / r326673;
        double r326675 = r326646 + r326674;
        return r326675;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r326676 = z;
        double r326677 = -4.2326254338963627e+64;
        bool r326678 = r326676 <= r326677;
        double r326679 = 7.31737881160239e+23;
        bool r326680 = r326676 <= r326679;
        double r326681 = !r326680;
        bool r326682 = r326678 || r326681;
        double r326683 = 3.13060547623;
        double r326684 = y;
        double r326685 = r326684 / r326676;
        double r326686 = t;
        double r326687 = r326686 / r326676;
        double r326688 = x;
        double r326689 = fma(r326685, r326687, r326688);
        double r326690 = fma(r326683, r326684, r326689);
        double r326691 = 11.1667541262;
        double r326692 = fma(r326683, r326676, r326691);
        double r326693 = fma(r326692, r326676, r326686);
        double r326694 = a;
        double r326695 = fma(r326676, r326693, r326694);
        double r326696 = 15.234687407;
        double r326697 = r326696 + r326676;
        double r326698 = 31.4690115749;
        double r326699 = fma(r326676, r326697, r326698);
        double r326700 = 11.9400905721;
        double r326701 = fma(r326699, r326676, r326700);
        double r326702 = 0.607771387771;
        double r326703 = fma(r326676, r326701, r326702);
        double r326704 = r326676 / r326703;
        double r326705 = r326695 * r326704;
        double r326706 = b;
        double r326707 = r326706 / r326703;
        double r326708 = r326705 + r326707;
        double r326709 = fma(r326684, r326708, r326688);
        double r326710 = r326682 ? r326690 : r326709;
        return r326710;
}

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

Original	29.2
Target	0.9
Herbie	0.9

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

Derivation

1. Split input into 2 regimes

2. if z < -4.2326254338963627e+64 or 7.31737881160239e+23 < z

   1. Initial program 60.5

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

   2. Simplified 58.2

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

   4. Applied clear-num 58.2

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

   5. Simplified 58.2

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

   6. Taylor expanded around inf 8.0

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

   7. Simplified 1.3

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

   if -4.2326254338963627e+64 < z < 7.31737881160239e+23

   1. Initial program 2.2

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

   2. Simplified 0.9

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

   4. Applied clear-num 1.0

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

   5. Simplified 1.0

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

   7. Applied associate-/r/ 1.0

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

   8. Simplified 1.0

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

   10. Applied fma-udef 1.0

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

   11. Applied distribute-lft-in 1.0

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

   12. Simplified 0.5

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

   13. Simplified 0.5

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.232625433896362687402834018909691734934 \cdot 10^{64} \lor \neg \left(z \le 731737881160239040954368\right):\\ \;\;\;\;\mathsf{fma}\left(3.130605476229999961645944495103321969509, y, \mathsf{fma}\left(\frac{y}{z}, \frac{t}{z}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.130605476229999961645944495103321969509, z, 11.16675412620000074070958362426608800888\right), z, t\right), a\right) \cdot \frac{z}{\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z, 15.2346874069999991263557603815570473671 + z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)} + \frac{b}{\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z, 15.2346874069999991263557603815570473671 + z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +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))))