Average Error: 26.4 → 0.9
Time: 13.7s
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -7.689434859723142729524296101916669810814 \cdot 10^{48} \lor \neg \left(x \le 219853368066851898595505554325504\right):\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, \frac{y}{{x}^{2}}\right) - 110.1139242984810806547102401964366436005\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \end{array}\]
\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 -7.689434859723142729524296101916669810814 \cdot 10^{48} \lor \neg \left(x \le 219853368066851898595505554325504\right):\\
\;\;\;\;\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, \frac{y}{{x}^{2}}\right) - 110.1139242984810806547102401964366436005\\

\mathbf{else}:\\
\;\;\;\;\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}\\

\end{array}
double f(double x, double y, double z) {
        double r604166 = x;
        double r604167 = 2.0;
        double r604168 = r604166 - r604167;
        double r604169 = 4.16438922228;
        double r604170 = r604166 * r604169;
        double r604171 = 78.6994924154;
        double r604172 = r604170 + r604171;
        double r604173 = r604172 * r604166;
        double r604174 = 137.519416416;
        double r604175 = r604173 + r604174;
        double r604176 = r604175 * r604166;
        double r604177 = y;
        double r604178 = r604176 + r604177;
        double r604179 = r604178 * r604166;
        double r604180 = z;
        double r604181 = r604179 + r604180;
        double r604182 = r604168 * r604181;
        double r604183 = 43.3400022514;
        double r604184 = r604166 + r604183;
        double r604185 = r604184 * r604166;
        double r604186 = 263.505074721;
        double r604187 = r604185 + r604186;
        double r604188 = r604187 * r604166;
        double r604189 = 313.399215894;
        double r604190 = r604188 + r604189;
        double r604191 = r604190 * r604166;
        double r604192 = 47.066876606;
        double r604193 = r604191 + r604192;
        double r604194 = r604182 / r604193;
        return r604194;
}
↓
double f(double x, double y, double z) {
        double r604195 = x;
        double r604196 = -7.689434859723143e+48;
        bool r604197 = r604195 <= r604196;
        double r604198 = 2.198533680668519e+32;
        bool r604199 = r604195 <= r604198;
        double r604200 = !r604199;
        bool r604201 = r604197 || r604200;
        double r604202 = 4.16438922228;
        double r604203 = y;
        double r604204 = 2.0;
        double r604205 = pow(r604195, r604204);
        double r604206 = r604203 / r604205;
        double r604207 = fma(r604195, r604202, r604206);
        double r604208 = 110.11392429848108;
        double r604209 = r604207 - r604208;
        double r604210 = 2.0;
        double r604211 = r604195 - r604210;
        double r604212 = r604195 * r604202;
        double r604213 = 78.6994924154;
        double r604214 = r604212 + r604213;
        double r604215 = r604214 * r604195;
        double r604216 = 137.519416416;
        double r604217 = r604215 + r604216;
        double r604218 = r604217 * r604195;
        double r604219 = r604218 + r604203;
        double r604220 = r604219 * r604195;
        double r604221 = z;
        double r604222 = r604220 + r604221;
        double r604223 = r604211 * r604222;
        double r604224 = 43.3400022514;
        double r604225 = r604195 + r604224;
        double r604226 = r604225 * r604195;
        double r604227 = 263.505074721;
        double r604228 = r604226 + r604227;
        double r604229 = r604228 * r604195;
        double r604230 = 313.399215894;
        double r604231 = r604229 + r604230;
        double r604232 = r604231 * r604195;
        double r604233 = 47.066876606;
        double r604234 = r604232 + r604233;
        double r604235 = r604223 / r604234;
        double r604236 = r604201 ? r604209 : r604235;
        return r604236;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original26.4
Target0.5
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;x \lt -3.326128725870004842699683658678411714981 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{elif}\;x \lt 9.429991714554672672712552870340896976735 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.5050747210000281484099105000495910645 \cdot x + \left(43.3400022514000013984514225739985704422 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -7.689434859723143e+48 or 2.198533680668519e+32 < x

    1. Initial program 60.1

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

    2. Simplified56.4

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

    4. Applied flip--56.4

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

    5. Applied associate-/l/56.4

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

    6. Taylor expanded around inf 0.9

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810806547102401964366436005}\]

    7. Simplified0.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, \frac{y}{{x}^{2}}\right) - 110.1139242984810806547102401964366436005}\]

    if -7.689434859723143e+48 < x < 2.198533680668519e+32

    1. Initial program 0.8

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.689434859723142729524296101916669810814 \cdot 10^{48} \lor \neg \left(x \le 219853368066851898595505554325504\right):\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, \frac{y}{{x}^{2}}\right) - 110.1139242984810806547102401964366436005\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 +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)))