Average Error: 27.1 → 0.8
Time: 10.5s
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -6.666887204842866 \cdot 10^{59} \lor \neg \left(x \le 116525157839104912\right):\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}} - 110.11392429848109\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x - 2 \cdot 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000014, x, 263.50507472100003\right), x, 313.399215894\right), x, 47.066876606000001\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999964, 78.6994924154000017\right), x, 137.51941641600001\right), x, y\right), x, z\right)} \cdot \left(x + 2\right)}\\ \end{array}\]
\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}
↓
\begin{array}{l}
\mathbf{if}\;x \le -6.666887204842866 \cdot 10^{59} \lor \neg \left(x \le 116525157839104912\right):\\
\;\;\;\;\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}} - 110.11392429848109\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{x \cdot x - 2 \cdot 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000014, x, 263.50507472100003\right), x, 313.399215894\right), x, 47.066876606000001\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999964, 78.6994924154000017\right), x, 137.51941641600001\right), x, y\right), x, z\right)} \cdot \left(x + 2\right)}\\

\end{array}
double f(double x, double y, double z) {
        double r403329 = x;
        double r403330 = 2.0;
        double r403331 = r403329 - r403330;
        double r403332 = 4.16438922228;
        double r403333 = r403329 * r403332;
        double r403334 = 78.6994924154;
        double r403335 = r403333 + r403334;
        double r403336 = r403335 * r403329;
        double r403337 = 137.519416416;
        double r403338 = r403336 + r403337;
        double r403339 = r403338 * r403329;
        double r403340 = y;
        double r403341 = r403339 + r403340;
        double r403342 = r403341 * r403329;
        double r403343 = z;
        double r403344 = r403342 + r403343;
        double r403345 = r403331 * r403344;
        double r403346 = 43.3400022514;
        double r403347 = r403329 + r403346;
        double r403348 = r403347 * r403329;
        double r403349 = 263.505074721;
        double r403350 = r403348 + r403349;
        double r403351 = r403350 * r403329;
        double r403352 = 313.399215894;
        double r403353 = r403351 + r403352;
        double r403354 = r403353 * r403329;
        double r403355 = 47.066876606;
        double r403356 = r403354 + r403355;
        double r403357 = r403345 / r403356;
        return r403357;
}
↓
double f(double x, double y, double z) {
        double r403358 = x;
        double r403359 = -6.666887204842866e+59;
        bool r403360 = r403358 <= r403359;
        double r403361 = 1.1652515783910491e+17;
        bool r403362 = r403358 <= r403361;
        double r403363 = !r403362;
        bool r403364 = r403360 || r403363;
        double r403365 = 4.16438922228;
        double r403366 = y;
        double r403367 = 2.0;
        double r403368 = pow(r403358, r403367);
        double r403369 = r403366 / r403368;
        double r403370 = 110.1139242984811;
        double r403371 = r403369 - r403370;
        double r403372 = fma(r403358, r403365, r403371);
        double r403373 = r403358 * r403358;
        double r403374 = 2.0;
        double r403375 = r403374 * r403374;
        double r403376 = r403373 - r403375;
        double r403377 = 43.3400022514;
        double r403378 = r403358 + r403377;
        double r403379 = 263.505074721;
        double r403380 = fma(r403378, r403358, r403379);
        double r403381 = 313.399215894;
        double r403382 = fma(r403380, r403358, r403381);
        double r403383 = 47.066876606;
        double r403384 = fma(r403382, r403358, r403383);
        double r403385 = 78.6994924154;
        double r403386 = fma(r403358, r403365, r403385);
        double r403387 = 137.519416416;
        double r403388 = fma(r403386, r403358, r403387);
        double r403389 = fma(r403388, r403358, r403366);
        double r403390 = z;
        double r403391 = fma(r403389, r403358, r403390);
        double r403392 = r403384 / r403391;
        double r403393 = r403358 + r403374;
        double r403394 = r403392 * r403393;
        double r403395 = r403376 / r403394;
        double r403396 = r403364 ? r403372 : r403395;
        return r403396;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original27.1
Target0.5
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;x \lt -3.3261287258700048 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{elif}\;x \lt 9.4299917145546727 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.50507472100003 \cdot x + \left(43.3400022514000014 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.399215894\right) \cdot x + 47.066876606000001}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -6.666887204842866e+59 or 1.1652515783910491e+17 < x

    1. Initial program 59.2

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]

    2. Simplified55.5

      \[\leadsto \color{blue}{\frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000014, x, 263.50507472100003\right), x, 313.399215894\right), x, 47.066876606000001\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999964, 78.6994924154000017\right), x, 137.51941641600001\right), x, y\right), x, z\right)}}}\]

    3. Taylor expanded around inf 1.0

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109}\]

    4. Simplified1.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}} - 110.11392429848109\right)}\]

    if -6.666887204842866e+59 < x < 1.1652515783910491e+17

    1. Initial program 1.0

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]

    2. Simplified0.6

      \[\leadsto \color{blue}{\frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000014, x, 263.50507472100003\right), x, 313.399215894\right), x, 47.066876606000001\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999964, 78.6994924154000017\right), x, 137.51941641600001\right), x, y\right), x, z\right)}}}\]
    3. Using strategy rm

    4. Applied flip--0.6

      \[\leadsto \frac{\color{blue}{\frac{x \cdot x - 2 \cdot 2}{x + 2}}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000014, x, 263.50507472100003\right), x, 313.399215894\right), x, 47.066876606000001\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999964, 78.6994924154000017\right), x, 137.51941641600001\right), x, y\right), x, z\right)}}\]

    5. Applied associate-/l/0.6

      \[\leadsto \color{blue}{\frac{x \cdot x - 2 \cdot 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000014, x, 263.50507472100003\right), x, 313.399215894\right), x, 47.066876606000001\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999964, 78.6994924154000017\right), x, 137.51941641600001\right), x, y\right), x, z\right)} \cdot \left(x + 2\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6.666887204842866 \cdot 10^{59} \lor \neg \left(x \le 116525157839104912\right):\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}} - 110.11392429848109\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x - 2 \cdot 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000014, x, 263.50507472100003\right), x, 313.399215894\right), x, 47.066876606000001\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999964, 78.6994924154000017\right), x, 137.51941641600001\right), x, y\right), x, z\right)} \cdot \left(x + 2\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C"
  :precision binary64

  :herbie-target
  (if (< x -3.326128725870005e+62) (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811) (if (< x 9.429991714554673e+55) (* (/ (- x 2) 1) (/ (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z) (+ (* (+ (+ (* 263.505074721 x) (+ (* 43.3400022514 (* x x)) (* x (* x x)))) 313.399215894) x) 47.066876606))) (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811)))

  (/ (* (- x 2) (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z)) (+ (* (+ (* (+ (* (+ x 43.3400022514) x) 263.505074721) x) 313.399215894) x) 47.066876606)))