Average Error: 29.0 → 1.5
Time: 12.4s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.74079175780789884 \cdot 10^{36} \lor \neg \left(z \le 10094.0500466464382\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{y}{z}, \mathsf{fma}\left(3.13060547622999996, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right), z, a\right), z, b\right), x\right)\\ \end{array}\]
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 -1.74079175780789884 \cdot 10^{36} \lor \neg \left(z \le 10094.0500466464382\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{y}{z}, \mathsf{fma}\left(3.13060547622999996, y, x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y \cdot \frac{1}{\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)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r467361 = x;
        double r467362 = y;
        double r467363 = z;
        double r467364 = 3.13060547623;
        double r467365 = r467363 * r467364;
        double r467366 = 11.1667541262;
        double r467367 = r467365 + r467366;
        double r467368 = r467367 * r467363;
        double r467369 = t;
        double r467370 = r467368 + r467369;
        double r467371 = r467370 * r467363;
        double r467372 = a;
        double r467373 = r467371 + r467372;
        double r467374 = r467373 * r467363;
        double r467375 = b;
        double r467376 = r467374 + r467375;
        double r467377 = r467362 * r467376;
        double r467378 = 15.234687407;
        double r467379 = r467363 + r467378;
        double r467380 = r467379 * r467363;
        double r467381 = 31.4690115749;
        double r467382 = r467380 + r467381;
        double r467383 = r467382 * r467363;
        double r467384 = 11.9400905721;
        double r467385 = r467383 + r467384;
        double r467386 = r467385 * r467363;
        double r467387 = 0.607771387771;
        double r467388 = r467386 + r467387;
        double r467389 = r467377 / r467388;
        double r467390 = r467361 + r467389;
        return r467390;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r467391 = z;
        double r467392 = -1.7407917578078988e+36;
        bool r467393 = r467391 <= r467392;
        double r467394 = 10094.050046646438;
        bool r467395 = r467391 <= r467394;
        double r467396 = !r467395;
        bool r467397 = r467393 || r467396;
        double r467398 = t;
        double r467399 = r467398 / r467391;
        double r467400 = y;
        double r467401 = r467400 / r467391;
        double r467402 = 3.13060547623;
        double r467403 = x;
        double r467404 = fma(r467402, r467400, r467403);
        double r467405 = fma(r467399, r467401, r467404);
        double r467406 = 1.0;
        double r467407 = 15.234687407;
        double r467408 = r467391 + r467407;
        double r467409 = 31.4690115749;
        double r467410 = fma(r467408, r467391, r467409);
        double r467411 = 11.9400905721;
        double r467412 = fma(r467410, r467391, r467411);
        double r467413 = 0.607771387771;
        double r467414 = fma(r467412, r467391, r467413);
        double r467415 = r467406 / r467414;
        double r467416 = r467400 * r467415;
        double r467417 = 11.1667541262;
        double r467418 = fma(r467391, r467402, r467417);
        double r467419 = fma(r467418, r467391, r467398);
        double r467420 = a;
        double r467421 = fma(r467419, r467391, r467420);
        double r467422 = b;
        double r467423 = fma(r467421, r467391, r467422);
        double r467424 = fma(r467416, r467423, r467403);
        double r467425 = r467397 ? r467405 : r467424;
        return r467425;
}

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
Target1.1
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;z \lt -6.4993449962526318 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.0669654369142868 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.7407917578078988e+36 or 10094.050046646438 < z

    1. Initial program 57.3

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

    2. Simplified55.4

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

    3. Taylor expanded around inf 9.3

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

    4. Simplified2.4

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

    if -1.7407917578078988e+36 < z < 10094.050046646438

    1. Initial program 0.7

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

    2. Simplified0.5

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

    4. Applied div-inv0.6

      \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right), z, a\right), z, b\right), x\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.74079175780789884 \cdot 10^{36} \lor \neg \left(z \le 10094.0500466464382\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{y}{z}, \mathsf{fma}\left(3.13060547622999996, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right), z, a\right), z, b\right), x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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))))