Average Error: 29.8 → 1.5
Time: 1.0m
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -8.121332546661932 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{t}{z}}{z} + 3.13060547623, x\right)\\ \mathbf{elif}\;z \le 1.43696004741907 \cdot 10^{+17}:\\ \;\;\;\;x + \frac{y \cdot \left(b + z \cdot \left(a + z \cdot \left(\left(3.13060547623 \cdot z + 11.1667541262\right) \cdot z + t\right)\right)\right)}{\left(\left(z \cdot \left(15.234687407 + z\right) + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1}{z} \cdot \frac{t}{z} + 3.13060547623, x\right)\\ \end{array}\]
x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}
↓
\begin{array}{l}
\mathbf{if}\;z \le -8.121332546661932 \cdot 10^{+24}:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{\frac{t}{z}}{z} + 3.13060547623, x\right)\\

\mathbf{elif}\;z \le 1.43696004741907 \cdot 10^{+17}:\\
\;\;\;\;x + \frac{y \cdot \left(b + z \cdot \left(a + z \cdot \left(\left(3.13060547623 \cdot z + 11.1667541262\right) \cdot z + t\right)\right)\right)}{\left(\left(z \cdot \left(15.234687407 + z\right) + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{1}{z} \cdot \frac{t}{z} + 3.13060547623, x\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r14387453 = x;
        double r14387454 = y;
        double r14387455 = z;
        double r14387456 = 3.13060547623;
        double r14387457 = r14387455 * r14387456;
        double r14387458 = 11.1667541262;
        double r14387459 = r14387457 + r14387458;
        double r14387460 = r14387459 * r14387455;
        double r14387461 = t;
        double r14387462 = r14387460 + r14387461;
        double r14387463 = r14387462 * r14387455;
        double r14387464 = a;
        double r14387465 = r14387463 + r14387464;
        double r14387466 = r14387465 * r14387455;
        double r14387467 = b;
        double r14387468 = r14387466 + r14387467;
        double r14387469 = r14387454 * r14387468;
        double r14387470 = 15.234687407;
        double r14387471 = r14387455 + r14387470;
        double r14387472 = r14387471 * r14387455;
        double r14387473 = 31.4690115749;
        double r14387474 = r14387472 + r14387473;
        double r14387475 = r14387474 * r14387455;
        double r14387476 = 11.9400905721;
        double r14387477 = r14387475 + r14387476;
        double r14387478 = r14387477 * r14387455;
        double r14387479 = 0.607771387771;
        double r14387480 = r14387478 + r14387479;
        double r14387481 = r14387469 / r14387480;
        double r14387482 = r14387453 + r14387481;
        return r14387482;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r14387483 = z;
        double r14387484 = -8.121332546661932e+24;
        bool r14387485 = r14387483 <= r14387484;
        double r14387486 = y;
        double r14387487 = t;
        double r14387488 = r14387487 / r14387483;
        double r14387489 = r14387488 / r14387483;
        double r14387490 = 3.13060547623;
        double r14387491 = r14387489 + r14387490;
        double r14387492 = x;
        double r14387493 = fma(r14387486, r14387491, r14387492);
        double r14387494 = 1.43696004741907e+17;
        bool r14387495 = r14387483 <= r14387494;
        double r14387496 = b;
        double r14387497 = a;
        double r14387498 = r14387490 * r14387483;
        double r14387499 = 11.1667541262;
        double r14387500 = r14387498 + r14387499;
        double r14387501 = r14387500 * r14387483;
        double r14387502 = r14387501 + r14387487;
        double r14387503 = r14387483 * r14387502;
        double r14387504 = r14387497 + r14387503;
        double r14387505 = r14387483 * r14387504;
        double r14387506 = r14387496 + r14387505;
        double r14387507 = r14387486 * r14387506;
        double r14387508 = 15.234687407;
        double r14387509 = r14387508 + r14387483;
        double r14387510 = r14387483 * r14387509;
        double r14387511 = 31.4690115749;
        double r14387512 = r14387510 + r14387511;
        double r14387513 = r14387512 * r14387483;
        double r14387514 = 11.9400905721;
        double r14387515 = r14387513 + r14387514;
        double r14387516 = r14387515 * r14387483;
        double r14387517 = 0.607771387771;
        double r14387518 = r14387516 + r14387517;
        double r14387519 = r14387507 / r14387518;
        double r14387520 = r14387492 + r14387519;
        double r14387521 = 1.0;
        double r14387522 = r14387521 / r14387483;
        double r14387523 = r14387522 * r14387488;
        double r14387524 = r14387523 + r14387490;
        double r14387525 = fma(r14387486, r14387524, r14387492);
        double r14387526 = r14387495 ? r14387520 : r14387525;
        double r14387527 = r14387485 ? r14387493 : r14387526;
        return r14387527;
}

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

Original29.8
Target1.0
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;z \lt -6.499344996252632 \cdot 10^{+53}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1.0}\\ \mathbf{elif}\;z \lt 7.066965436914287 \cdot 10^{+59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1.0}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -8.121332546661932e+24

    1. Initial program 59.1

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\]

    2. Simplified55.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), a\right), z, b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)}\]

    3. Taylor expanded around inf 9.0

      \[\leadsto \color{blue}{x + \left(\frac{t \cdot y}{{z}^{2}} + 3.13060547623 \cdot y\right)}\]

    4. Simplified2.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623 + \frac{\frac{t}{z}}{z}, x\right)}\]

    if -8.121332546661932e+24 < z < 1.43696004741907e+17

    1. Initial program 0.6

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\]

    if 1.43696004741907e+17 < z

    1. Initial program 57.1

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\]

    2. Simplified54.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), a\right), z, b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)}\]

    3. Taylor expanded around inf 10.3

      \[\leadsto \color{blue}{x + \left(\frac{t \cdot y}{{z}^{2}} + 3.13060547623 \cdot y\right)}\]

    4. Simplified2.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623 + \frac{\frac{t}{z}}{z}, x\right)}\]
    5. Using strategy rm

    6. Applied add-sqr-sqrt2.8

      \[\leadsto \mathsf{fma}\left(y, 3.13060547623 + \frac{\frac{t}{z}}{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}, x\right)\]

    7. Applied add-sqr-sqrt2.8

      \[\leadsto \mathsf{fma}\left(y, 3.13060547623 + \frac{\frac{t}{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}}{\sqrt{z} \cdot \sqrt{z}}, x\right)\]

    8. Applied *-un-lft-identity2.8

      \[\leadsto \mathsf{fma}\left(y, 3.13060547623 + \frac{\frac{\color{blue}{1 \cdot t}}{\sqrt{z} \cdot \sqrt{z}}}{\sqrt{z} \cdot \sqrt{z}}, x\right)\]

    9. Applied times-frac2.8

      \[\leadsto \mathsf{fma}\left(y, 3.13060547623 + \frac{\color{blue}{\frac{1}{\sqrt{z}} \cdot \frac{t}{\sqrt{z}}}}{\sqrt{z} \cdot \sqrt{z}}, x\right)\]

    10. Applied times-frac2.8

      \[\leadsto \mathsf{fma}\left(y, 3.13060547623 + \color{blue}{\frac{\frac{1}{\sqrt{z}}}{\sqrt{z}} \cdot \frac{\frac{t}{\sqrt{z}}}{\sqrt{z}}}, x\right)\]

    11. Simplified2.8

      \[\leadsto \mathsf{fma}\left(y, 3.13060547623 + \color{blue}{\frac{1}{z}} \cdot \frac{\frac{t}{\sqrt{z}}}{\sqrt{z}}, x\right)\]

    12. Simplified2.8

      \[\leadsto \mathsf{fma}\left(y, 3.13060547623 + \frac{1}{z} \cdot \color{blue}{\frac{t}{z}}, x\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -8.121332546661932 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{t}{z}}{z} + 3.13060547623, x\right)\\ \mathbf{elif}\;z \le 1.43696004741907 \cdot 10^{+17}:\\ \;\;\;\;x + \frac{y \cdot \left(b + z \cdot \left(a + z \cdot \left(\left(3.13060547623 \cdot z + 11.1667541262\right) \cdot z + t\right)\right)\right)}{\left(\left(z \cdot \left(15.234687407 + z\right) + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1}{z} \cdot \frac{t}{z} + 3.13060547623, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 +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))))