Average Error: 29.1 → 1.3
Time: 19.9s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.48756405302082346 \cdot 10^{53}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{z}, \frac{t}{z}, 3.13060547622999996\right), x\right)\\ \mathbf{elif}\;z \le 5.2047260742974899 \cdot 10^{61}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), t\right), z, a\right), z, b\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{{z}^{2}} + 3.13060547622999996, x\right)\\ \end{array}\]
x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.48756405302082346 \cdot 10^{53}:\\
\;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{z}, \frac{t}{z}, 3.13060547622999996\right), x\right)\\

\mathbf{elif}\;z \le 5.2047260742974899 \cdot 10^{61}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), t\right), z, a\right), z, b\right), x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{t}{{z}^{2}} + 3.13060547622999996, x\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r318649 = x;
        double r318650 = y;
        double r318651 = z;
        double r318652 = 3.13060547623;
        double r318653 = r318651 * r318652;
        double r318654 = 11.1667541262;
        double r318655 = r318653 + r318654;
        double r318656 = r318655 * r318651;
        double r318657 = t;
        double r318658 = r318656 + r318657;
        double r318659 = r318658 * r318651;
        double r318660 = a;
        double r318661 = r318659 + r318660;
        double r318662 = r318661 * r318651;
        double r318663 = b;
        double r318664 = r318662 + r318663;
        double r318665 = r318650 * r318664;
        double r318666 = 15.234687407;
        double r318667 = r318651 + r318666;
        double r318668 = r318667 * r318651;
        double r318669 = 31.4690115749;
        double r318670 = r318668 + r318669;
        double r318671 = r318670 * r318651;
        double r318672 = 11.9400905721;
        double r318673 = r318671 + r318672;
        double r318674 = r318673 * r318651;
        double r318675 = 0.607771387771;
        double r318676 = r318674 + r318675;
        double r318677 = r318665 / r318676;
        double r318678 = r318649 + r318677;
        return r318678;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r318679 = z;
        double r318680 = -1.4875640530208235e+53;
        bool r318681 = r318679 <= r318680;
        double r318682 = y;
        double r318683 = 1.0;
        double r318684 = r318683 / r318679;
        double r318685 = t;
        double r318686 = r318685 / r318679;
        double r318687 = 3.13060547623;
        double r318688 = fma(r318684, r318686, r318687);
        double r318689 = x;
        double r318690 = fma(r318682, r318688, r318689);
        double r318691 = 5.20472607429749e+61;
        bool r318692 = r318679 <= r318691;
        double r318693 = 15.234687407;
        double r318694 = r318679 + r318693;
        double r318695 = 31.4690115749;
        double r318696 = fma(r318694, r318679, r318695);
        double r318697 = 11.9400905721;
        double r318698 = fma(r318696, r318679, r318697);
        double r318699 = 0.607771387771;
        double r318700 = fma(r318698, r318679, r318699);
        double r318701 = r318682 / r318700;
        double r318702 = 11.1667541262;
        double r318703 = fma(r318679, r318687, r318702);
        double r318704 = fma(r318679, r318703, r318685);
        double r318705 = a;
        double r318706 = fma(r318704, r318679, r318705);
        double r318707 = b;
        double r318708 = fma(r318706, r318679, r318707);
        double r318709 = fma(r318701, r318708, r318689);
        double r318710 = 2.0;
        double r318711 = pow(r318679, r318710);
        double r318712 = r318685 / r318711;
        double r318713 = r318712 + r318687;
        double r318714 = fma(r318682, r318713, r318689);
        double r318715 = r318692 ? r318709 : r318714;
        double r318716 = r318681 ? r318690 : r318715;
        return r318716;
}

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

Original29.1
Target1.1
Herbie1.3
\[\begin{array}{l} \mathbf{if}\;z \lt -6.4993449962526318 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.0669654369142868 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.4875640530208235e+53

    1. Initial program 61.6

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]

    2. Simplified60.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right), z, a\right), z, b\right), x\right)}\]

    3. Taylor expanded around inf 7.9

      \[\leadsto \color{blue}{x + \left(\frac{t \cdot y}{{z}^{2}} + 3.13060547622999996 \cdot y\right)}\]

    4. Simplified0.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{t}{{z}^{2}} + 3.13060547622999996, x\right)}\]
    5. Using strategy rm

    6. Applied sqr-pow0.9

      \[\leadsto \mathsf{fma}\left(y, \frac{t}{\color{blue}{{z}^{\left(\frac{2}{2}\right)} \cdot {z}^{\left(\frac{2}{2}\right)}}} + 3.13060547622999996, x\right)\]

    7. Applied *-un-lft-identity0.9

      \[\leadsto \mathsf{fma}\left(y, \frac{\color{blue}{1 \cdot t}}{{z}^{\left(\frac{2}{2}\right)} \cdot {z}^{\left(\frac{2}{2}\right)}} + 3.13060547622999996, x\right)\]

    8. Applied times-frac0.9

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{{z}^{\left(\frac{2}{2}\right)}} \cdot \frac{t}{{z}^{\left(\frac{2}{2}\right)}}} + 3.13060547622999996, x\right)\]

    9. Applied fma-def0.9

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\frac{1}{{z}^{\left(\frac{2}{2}\right)}}, \frac{t}{{z}^{\left(\frac{2}{2}\right)}}, 3.13060547622999996\right)}, x\right)\]

    if -1.4875640530208235e+53 < z < 5.20472607429749e+61

    1. Initial program 2.9

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]

    2. Simplified1.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right), z, a\right), z, b\right), x\right)}\]

    3. Taylor expanded around 0 1.7

      \[\leadsto \mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{t + \left(11.166754126200001 \cdot z + 3.13060547622999996 \cdot {z}^{2}\right)}, z, a\right), z, b\right), x\right)\]

    4. Simplified1.7

      \[\leadsto \mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), t\right)}, z, a\right), z, b\right), x\right)\]

    if 5.20472607429749e+61 < z

    1. Initial program 62.6

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]

    2. Simplified61.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right), z, a\right), z, b\right), x\right)}\]

    3. Taylor expanded around inf 8.3

      \[\leadsto \color{blue}{x + \left(\frac{t \cdot y}{{z}^{2}} + 3.13060547622999996 \cdot y\right)}\]

    4. Simplified0.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{t}{{z}^{2}} + 3.13060547622999996, x\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.48756405302082346 \cdot 10^{53}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{z}, \frac{t}{z}, 3.13060547622999996\right), x\right)\\ \mathbf{elif}\;z \le 5.2047260742974899 \cdot 10^{61}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), t\right), z, a\right), z, b\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{{z}^{2}} + 3.13060547622999996, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 +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))))