Average Error: 26.7 → 0.6
Time: 1.4m
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -18446644311559690167397500658474620551170:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, \frac{y}{x \cdot x} - 110.1139242984810948655649553984403610229\right)\\ \mathbf{elif}\;x \le 5.400940663197947339625629709488573945958 \cdot 10^{71}:\\ \;\;\;\;\left(x - 2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922227999963610045597306452691555, x, 78.69949241540000173245061887428164482117\right), x, 137.5194164160000127594685181975364685059\right), x, y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 43.3400022514000013984514225739985704422 + x, 263.5050747210000281484099105000495910645\right), 313.3992158940000081202015280723571777344\right), 47.06687660600000100430406746454536914825\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, \frac{y}{x \cdot x} - 110.1139242984810948655649553984403610229\right)\\ \end{array}\]
\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}
↓
\begin{array}{l}
\mathbf{if}\;x \le -18446644311559690167397500658474620551170:\\
\;\;\;\;\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, \frac{y}{x \cdot x} - 110.1139242984810948655649553984403610229\right)\\

\mathbf{elif}\;x \le 5.400940663197947339625629709488573945958 \cdot 10^{71}:\\
\;\;\;\;\left(x - 2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922227999963610045597306452691555, x, 78.69949241540000173245061887428164482117\right), x, 137.5194164160000127594685181975364685059\right), x, y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 43.3400022514000013984514225739985704422 + x, 263.5050747210000281484099105000495910645\right), 313.3992158940000081202015280723571777344\right), 47.06687660600000100430406746454536914825\right)}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, \frac{y}{x \cdot x} - 110.1139242984810948655649553984403610229\right)\\

\end{array}
double f(double x, double y, double z) {
        double r19381359 = x;
        double r19381360 = 2.0;
        double r19381361 = r19381359 - r19381360;
        double r19381362 = 4.16438922228;
        double r19381363 = r19381359 * r19381362;
        double r19381364 = 78.6994924154;
        double r19381365 = r19381363 + r19381364;
        double r19381366 = r19381365 * r19381359;
        double r19381367 = 137.519416416;
        double r19381368 = r19381366 + r19381367;
        double r19381369 = r19381368 * r19381359;
        double r19381370 = y;
        double r19381371 = r19381369 + r19381370;
        double r19381372 = r19381371 * r19381359;
        double r19381373 = z;
        double r19381374 = r19381372 + r19381373;
        double r19381375 = r19381361 * r19381374;
        double r19381376 = 43.3400022514;
        double r19381377 = r19381359 + r19381376;
        double r19381378 = r19381377 * r19381359;
        double r19381379 = 263.505074721;
        double r19381380 = r19381378 + r19381379;
        double r19381381 = r19381380 * r19381359;
        double r19381382 = 313.399215894;
        double r19381383 = r19381381 + r19381382;
        double r19381384 = r19381383 * r19381359;
        double r19381385 = 47.066876606;
        double r19381386 = r19381384 + r19381385;
        double r19381387 = r19381375 / r19381386;
        return r19381387;
}
↓
double f(double x, double y, double z) {
        double r19381388 = x;
        double r19381389 = -1.844664431155969e+40;
        bool r19381390 = r19381388 <= r19381389;
        double r19381391 = 4.16438922228;
        double r19381392 = y;
        double r19381393 = r19381388 * r19381388;
        double r19381394 = r19381392 / r19381393;
        double r19381395 = 110.1139242984811;
        double r19381396 = r19381394 - r19381395;
        double r19381397 = fma(r19381388, r19381391, r19381396);
        double r19381398 = 5.400940663197947e+71;
        bool r19381399 = r19381388 <= r19381398;
        double r19381400 = 2.0;
        double r19381401 = r19381388 - r19381400;
        double r19381402 = 78.6994924154;
        double r19381403 = fma(r19381391, r19381388, r19381402);
        double r19381404 = 137.519416416;
        double r19381405 = fma(r19381403, r19381388, r19381404);
        double r19381406 = fma(r19381405, r19381388, r19381392);
        double r19381407 = z;
        double r19381408 = fma(r19381388, r19381406, r19381407);
        double r19381409 = 43.3400022514;
        double r19381410 = r19381409 + r19381388;
        double r19381411 = 263.505074721;
        double r19381412 = fma(r19381388, r19381410, r19381411);
        double r19381413 = 313.399215894;
        double r19381414 = fma(r19381388, r19381412, r19381413);
        double r19381415 = 47.066876606;
        double r19381416 = fma(r19381388, r19381414, r19381415);
        double r19381417 = r19381408 / r19381416;
        double r19381418 = r19381401 * r19381417;
        double r19381419 = r19381399 ? r19381418 : r19381397;
        double r19381420 = r19381390 ? r19381397 : r19381419;
        return r19381420;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original26.7
Target0.6
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;x \lt -3.326128725870004842699683658678411714981 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{elif}\;x \lt 9.429991714554672672712552870340896976735 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.5050747210000281484099105000495910645 \cdot x + \left(43.3400022514000013984514225739985704422 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -1.844664431155969e+40 or 5.400940663197947e+71 < x

    1. Initial program 62.4

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]

    2. Simplified59.7

      \[\leadsto \color{blue}{\frac{x - 2}{\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(43.3400022514000013984514225739985704422 + x, x, 263.5050747210000281484099105000495910645\right), 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), x, 137.5194164160000127594685181975364685059\right), y\right), z\right)}\]

    3. Taylor expanded around inf 0.5

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229}\]

    4. Simplified0.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, \frac{y}{x \cdot x} - 110.1139242984810948655649553984403610229\right)}\]

    if -1.844664431155969e+40 < x < 5.400940663197947e+71

    1. Initial program 2.3

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]

    2. Simplified1.0

      \[\leadsto \color{blue}{\frac{x - 2}{\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(43.3400022514000013984514225739985704422 + x, x, 263.5050747210000281484099105000495910645\right), 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), x, 137.5194164160000127594685181975364685059\right), y\right), z\right)}\]
    3. Using strategy rm

    4. Applied div-inv1.0

      \[\leadsto \color{blue}{\left(\left(x - 2\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(43.3400022514000013984514225739985704422 + x, x, 263.5050747210000281484099105000495910645\right), 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}\right)} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), x, 137.5194164160000127594685181975364685059\right), y\right), z\right)\]

    5. Applied associate-*l*1.0

      \[\leadsto \color{blue}{\left(x - 2\right) \cdot \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(43.3400022514000013984514225739985704422 + x, x, 263.5050747210000281484099105000495910645\right), 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), x, 137.5194164160000127594685181975364685059\right), y\right), z\right)\right)}\]

    6. Simplified0.7

      \[\leadsto \left(x - 2\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922227999963610045597306452691555, x, 78.69949241540000173245061887428164482117\right), x, 137.5194164160000127594685181975364685059\right), x, y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 43.3400022514000013984514225739985704422 + x, 263.5050747210000281484099105000495910645\right), 313.3992158940000081202015280723571777344\right), 47.06687660600000100430406746454536914825\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -18446644311559690167397500658474620551170:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, \frac{y}{x \cdot x} - 110.1139242984810948655649553984403610229\right)\\ \mathbf{elif}\;x \le 5.400940663197947339625629709488573945958 \cdot 10^{71}:\\ \;\;\;\;\left(x - 2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922227999963610045597306452691555, x, 78.69949241540000173245061887428164482117\right), x, 137.5194164160000127594685181975364685059\right), x, y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 43.3400022514000013984514225739985704422 + x, 263.5050747210000281484099105000495910645\right), 313.3992158940000081202015280723571777344\right), 47.06687660600000100430406746454536914825\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, \frac{y}{x \cdot x} - 110.1139242984810948655649553984403610229\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C"

  :herbie-target
  (if (< x -3.326128725870005e+62) (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811) (if (< x 9.429991714554673e+55) (* (/ (- x 2.0) 1.0) (/ (+ (* (+ (* (+ (* (+ (* 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 2.0) (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z)) (+ (* (+ (* (+ (* (+ x 43.3400022514) x) 263.505074721) x) 313.399215894) x) 47.066876606)))